A space quantization method for numerical integration

We propose a new method (SQM) for numerical integration of Cα functions (α ∈ (0,2]) defined on a convex subset C of Rd with respect to a continuous distribution μ. It relies on a space quantization of C by a n-tuplex:=(x1,…,xn) ∈ Cn. ∫fdμ is approximated by a weighted sum of the f(xi)ʹs. The integration error bound depends on the distortion Enα,μ(x) of the Voronoï tessellation of x. This notion comes from Information Theoretists. Its main properties (existence of a minimizing n-tuple in Cn, asymptotics of minCn Enα,μ as n → +∞) are presented for a wide class of measures μ. A simple stochastic optimization procedure is proposed to compute, in any dimension d, x∗ and the characteristics of its Voronoï tessellation. Some new results on the Competitive Learning Vector Quantization algorithm (when α = 2) are obtained as a by-product. Some tests, simulations and provisional remarks are proposed as a conclusion.