We investigate the greedy version of the Lp-optimal vector quantization
problem for an Rd-valued random vector X∈Lp. We show the
existence of a sequence (aN)N≥1 such that aN minimizes
a↦min1≤i≤N−1∣X−ai∣∧∣X−a∣Lp
(Lp-mean quantization error at level N induced by
(a1,…,aN−1,a)). We show that this sequence produces Lp-rate
optimal N-tuples a(N)=(a1,…,aN) (i.e. the Lp-mean
quantization error at level N induced by a(N) goes to 0 at rate
N−d1). Greedy optimal sequences also satisfy, under natural
additional assumptions, the distortion mismatch property: the N-tuples
a(N) remain rate optimal with respect to the Lq-norms, p≤q<p+d.
Finally, we propose optimization methods to compute greedy sequences, adapted
from usual Lloyd's I and Competitive Learning Vector Quantization procedures,
either in their deterministic (implementable when d=1) or stochastic
versions.Comment: 31 pages, 4 figures, few typos corrected (now an extended version of
an eponym paper to appear in Journal of Approximation