Highdimensional manifold reconstruction

[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.

Thanks in advance

 

[pat_connell]

for your help.Yes, it is possible to reconstruct the manifold between samples by interpolating the points Hessians, provided that the Nyquist criterion is satisfied. However, this method can be very computationally intensive and may not be practical in high dimensional spaces. It may be more efficient to use other methods such as principal component analysis (PCA) or singular value decomposition (SVD) to reduce the dimensionality of the space and then use a non-linear interpolation technique to reconstruct the manifold.