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Greedy vector quantization

We investigate the greedy version of the $L^p$-optimal vector quantization problem for an $\mathbb{R}^d$-valued random vector $X\!\in L^p$. We show the existence of a sequence $(a_N)_{N\ge 1}$ such that $a_N$ minimizes $a\mapsto\big \|\min_{1\le i\le N-1}|X-a_i|\wedge |X-a|\big\|_{L^p}$ ($L^p$-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 $a^{(N)}=(a_1,\ldots,a_{_N})$ ($i.e.$ the $L^p$-mean quantization error at level $N$ induced by $a^{(N)}$ goes 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, $p\le q <p+d$. 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.