Manifold Reconstruction

Years and Authors of Summarized Original Work

2005; Cheng, Dey, Ramos2008; Niyogi, Smale, Weinberger2014; Boissonnat, Ghosh2014; Cheng, Chiu

Problem Definition

With the widespread of sensing and Internet technologies, a large number of numeric attributes for a physical or cyber phenomenon can now be collected. If each attribute is viewed as a coordinate, an instance in the collection can be viewed as a point in $\mathbb{R}^{d}$ for some large d. When the physical or cyber phenomenon is governed by only a few latent parameters, it is often postulated that the data points lie on some unknown smooth compact manifold $\mathcal{M}$ of dimension k, where k ≪ d. The goal is to reconstruct a faithful representation of $\mathcal{M}$from the data points. Reconstruction problem are ill-posed in general. Therefore, the data points are assumed to be dense enough so that it becomes theoretically possible to obtain a faithful reconstruction. The quality of the reconstruction is measured in...