A space quantization method for numerical integration

We propose a new method (SQM) for numerical integration of L-n functions (alpha is an element of (0, 2]) defined on a convex subset C of R-d with respect to a continuous distribution mu. It relies on a space quantization of C by a n-tuple x: (x(1),..., x(n))is an element of C-n. integral f d mu is approximated by a weighted sum of the f(x(i))'s. The integration error bound depends on the distortion E-n(2,mu)(x) of the Voronoi tessellation of x. This notion comes from Information Theoretists. Its main properties (existence of a minimizing n-tuple in C-n, asymptotics of minc(Cn) E-n(alpha,mu) as n --> +infinity) are presented for a wide class of measures mu. A simple stochastic optimization procedure is proposed to compute, in any dimension d, x* and the characteristics of its Voronoi tessellation. Some new results on the Competitive Learning Vector Quantization algorithm (when alpha=2)are obtained as a by-product. Some tests, simulations and provisional remarks are proposed as a conclusion. (C) 1997 Elsevier Science B.V. All rights reserved.