A space quantization method for numerical integration

Abstract

We propose a new method (SQM) for numerical integration of C α functions ( α ∈ (0,2]) defined on a convex subset C of R d with respect to a continuous distribution μ. It relies on a space quantization of C by a n-tuple x:=( x 1,…, x n ) ∈ C n . ∫ fdμ is approximated by a weighted sum of the f( x i )'s. The integration error bound depends on the distortion E n α, μ ( x) of the Voronoï tessellation of x. This notion comes from Information Theoretists. Its main properties (existence of a minimizing n-tuple in C n , asymptotics of min C n E n α, 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.