Pointwise convergence of the Lloyd algorithm in higher dimension

TL;DR

This paper proves the pointwise convergence of the Lloyd algorithm for vector quantization in higher dimensions, even with unbounded distributions, and introduces a variant ensuring bounded grids.

Contribution

It establishes convergence of the Lloyd algorithm under minimal assumptions and proposes a variant that guarantees boundedness of the iterated grids.

Findings

Convergence of Lloyd algorithm is proven without continuity assumptions.

A splitting method ensures convergence even for unbounded distributions.

A variant guarantees the boundedness of the resulting grids.

Abstract

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.