Greedy vector quantization.

Summary We investigate the greedy version of the L^p-optimal vector quantization problem for an R^d-valued random vector X\in L^p. We show the existence of a sequence (a_N) such that a_N minimizes a\mapsto\big \|\min_{1\le i\le N-1}|X-a_i|\wedge |X-a|\big\|_{p}: the L^p-mean quantization error at level N induced by (a_1,\ldots,a_{N-1},a). We show that this sequence produces L^p-rate optimal N-tuples a^{(N)}=(a_1,\ldots,a_{_N}): their L^p-mean quantization errors at level $N$ go to 0 at rate N^{-\frac 1d}. 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 L^q-norms, if p\le q.