A varifold-type estimation for data sampled on a rectifiable set

We investigate the inference of varifold structures in a statistical framework: assuming that we have access to i.i.d. samples in ${\mathbb{R}}^n$ obtained from an underlying $d$–dimensional shape $S$ endowed with a possibly non uniform density $\theta$, we propose and analyse an estimator of the varifold structure associated to $S$. The shape $S$ is assumed to be piecewise $C^{1,a}$ in a sense that allows for a singular set whose small enlargements are of small $d$–dimensional measure. The estimators are kernel–based both for infering the density and the tangent spaces and the convergence result holds for the bounded Lipschitz distance between varifolds, in expectation and in a noiseless model. The mean convergence rate involves the dimension $d$ of $S$, its regularity through $a\in (0,1]$ and the regularity of the density $\theta$.