Variational data fitting (Q1125515)

Let \(\Omega\subset{\mathbb R}^n\) be a given closed, smoothly bounded region which supports at least one \(\Omega\)-regular embedding. For the probability densities considered here, it is assumed that their support is contained in \(\Omega\). Starting with a jet bundle formulation to the theory of \(\Omega\)-regular embeddings, the author generalises the results on critical curves proved by \textit{T. Duchamp} and \textit{W. Stuetzle} [Ann. Stat. 24, No. 4, 1511-1520 (1996; Zbl 0867.62025)] by proving that the distance functional does not have local minima within the class of \( \Omega\)-regular embeddings. Conditions under which a critical embedding is a local minimum are also studied. It is also shown that under certain conditions, critical curves depend smoothly on both the probability density and the spring constant.