Anisotropy

[kuberry]

Deal with n-dimensional anisotropy:

• Boolean as to whether to worry about anisotropy
• Detect anisotropy in neighbor search results
  • Determine principal directions
  • Continue neighbor search, dealing with these principal direction (ensuring unisolvency)
• Let weighting kernel depend on these principal directions

[kuberry]

Some thoughts on a solution for this:

• start with normal search knn, or radius
• from final point cloud, put together manifold problem only for the coarse approximation
• from eigenvalues / eigenvectors to this solution, we get principal direction
• what can be metrics for how these skew/ratio?
  • can provide max skew
  • can calculate this quantity for a number of existing point clouds to see what is reasonable
• if eig ratios much large for one than another, come up with ratio in principal direction to recover some reasonable skew
• assume that distribution outside of initial search has same type of distribution
• do search for max ratio in L2 ball (so all possible neighbors are found)
• for each point, calculate inverse skew * eig with each principal direction to get some altered distance so for distance, instead of mapping onto V subspace, map onto three directions
• distance will increase for oversampled directions, decrease for undersampled
• need to provide kernel and get all neighbors with non-zero weight