Assume a set of $N$ points in $\mathbb{R}^d$ for wich we would like to find it's convex hull.
If $N$ or $d$ is large the problem gets computationally intractable. In that case we may resort to an approximation.
One kind of approximation that came to my mind is doing a low rank approximation to reduce the dimensionality of either $N$, $d$ or both.
For illustration consider a set of $N$ points that indeed do not span the space of either $\mathbb{R}^d$ or $N$. We could then take the set of points as a Matrix $X \in \mathbb{R}^{N \times d}$ and project them to a lower dimensional space (e.g. using PCA). A decision for wether an out of set point is inside the convex hull or not, could then be carried out by projecting the query point to the subspace and compare it to the hull in that lower dimensional representation.
The example of a linear projection to lower dimensions seems however like a rather adhoc choice.
Are there principled approaches (possibly nonlinear) to map to a lower dimensional representation that approximately preserve the structure of the hull?