Greedy vector quantization

We investigate the greedy version of the L-optimal vector quantization problem for an Rvalued random vector X ∈ L. We show the existence of a sequence (aN )N≥1 such that aN minimizes a 7→ ∥min1≤i≤N−1 |X−ai| ∧ |X−a| ∥∥ Lp (L-mean quantization error at level N induced by (a1, . . . , aN−1, a)). We show that this sequence produces L -rate optimal N -tuples a = (a1, . . . , aN ) (i.e. the L -mean quantization error at level N induced by a goes to 0 at rate N 1 d ). Greedy optimal sequences also satisfy, under natural additional assumptions, the distortion mismatch property: the N -tuples a remain rate optimal with respect to the L-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.