# Highdimensional manifold reconstruction

1. Sep 6, 2011

### wurzel77

Suppose I have a highdimensional space $\mathbb{R}^N$ that is sparesely populated by a finite set of samples $\{ \mathbf{x} \}_{1 \le i \le k}$, for example N = 500, k = 100. I assume the points x to be sampled from a n-manifold embedded in $\mathbb{R}^N$, where $n << N$. From a mathematical point of view, would it be legitimate to reconstruct the manifold between samples, e.g. by interpolating the points Hessians? I am aware that therefore the Nyquist criterion must be fulfilled.