Pointwise convergence of the Lloyd algorithm in higher dimension

Gilles Pagès (1) , Jun Yu (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We establish the pointwise convergence of the iterative Lloyd algorithm, also known as $k$-means algorithm, when the quadratic quantization error of the starting grid (with size $N\ge 2$) is lower than the minimal quantization error with respect to the input distribution is lower at level $N-1$. Such a protocol is known as the splitting method and allows for convergence even when the input distribution has an unbounded support. We also show under very light assumption that the resulting limiting grid still has full size $N$. These results are obtained without continuity assumption on the input distribution. A variant of the procedure taking advantage of the asymptotic of the optimal quantizer radius is proposed which always guarantees the boundedness of the iterated grids.

Mots clés: Lloyd algorithm; $k$-means algorithm; centroidal Voronoi Tessellation; optimal vector quantization; stationary quantizers; splitting method; radius of a quantizer

Domaines: Probabilités [math.PR]

Gilles Pagès, Jun Yu. Pointwise convergence of the Lloyd algorithm in higher dimension. SIAM Journal on Control and Optimization, 2016, 54 (5). ⟨hal-00922957⟩