# Detection of low dimensionality and data denoising via set estimation techniques

## TL;DR

This paper investigates set and manifold estimation from random samples, focusing on identifying lower-dimensional structures and denoising data, with theoretical guarantees and practical illustrations.

## Contribution

It introduces methods for determining the dimensionality of sets, estimating lower-dimensional manifolds, and denoising data based on set estimation theories.

## Key findings

- Proposes procedures to identify if a set is full-dimensional or lower-dimensional.
- Develops algorithms to estimate lower-dimensional manifolds from noisy data.
- Provides theoretical guarantees and simulation results demonstrating effectiveness.

## 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.