Fractal-based measure approximation with entropy maximization and sparsity constraints

Abstract

In this paper, we focus on the inverse problem associated with IFSP: Given a target measure μ, find an IFSP (w,p), such that its invariantmeasure μ̄ is sufficiently close to μ in Monge-Kantorovich distance, i.e., dMK(μ,μ̄) < ε. We extend themoment-basedmethod developed by Forte and Vrscay (1995) along two different directions. First, we search for a set of probabilities pi that not only minimizes the collage error but also maximizes the entropy of the iterated function system. Second, we include an extra term in the minimization process which takes into account the sparsity of the set of probabilities. In our new formulations, the minimization of collage error (for moments) can be understood as amulti-criteria problem: At least three different and conflicting criteriamust be considered, i.e., collage error, entropy and sparsity. In this paper, we employ scalarization to reduce the multi-criteria program to a single-criteria program by combining all objective functions with different trade-off weights. The results of some numerical computations are presented. Preliminarly numerical studies indicate that a "MaxEnt principle" exists for this approximation problem, i.e., that the suboptimal solutions produced by collage coding can be improved at least slightly by adding a maximum entropy criterion.