Title: Detection of low dimensionality and data denoising via set estimation techniques,
Abstract: This work is closely related to the theories of set estimation and manifold
estimation.
Our object of interest is a, possibly lower-dimensional, compact set $S
\subset {\mathbb R}^d$.
The general aim is to identify (via stochastic procedures) some qualitative
or quantitative features of $S$, of geometric or topological character. The
available information is just a random sample of points drawn on $S$.
The term "to identify" means here to achieve a correct answer almost surely
(a.s.) when the sample size tends to infinity. More specifically the paper aims
at giving some partial answers to the following questions: is $S$ full
dimensional? Is $S$ "close to a lower dimensional set" $\mathcal{M}$? If so,
can we estimate $\mathcal{M}$ or some functionals of $\mathcal{M}$ (in
particular, the Minkowski content of $\mathcal{M}$)? As an important auxiliary
tool in the answers of these questions, a denoising procedure is proposed in
order to partially remove the noise in the original data. The theoretical
results are complemented with some simulations and graphical illustrations.