Ensure invertibility of term from Wendland in MLS oracle

Orecchia-Research-Group / manifold_learning

Learning topological manifold representations from point clouds

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Ensure invertibility of term from Wendland in MLS oracle #3

Open robbobbinett opened 4 years ago

robbobbinett commented 4 years ago

Thee matrix 'D' from the Wendland chapter is of shape '# x Q', where '#' is the number of points within suitable radius of the input value and 'Q' is, for our purposes, equal to 'm+1', where 'm' is the degree of the MLS approximator. We need to make sure that delta and 'm+1' always behave such that the term 'P^T DP' is always invertible.

We should also have a test case that guarantees we catch this mode of failure.

ideclue commented 4 years ago

https://github.com/Orecchia-Research-Group/manifold_learning/commit/cb0a34eed26f361ec825bf488ba9c4a6051e9e23

robbobbinett commented 4 years ago

What your change currently does is cause the function to return a string. This itself causes issues.

According to Wendland, we can guarantee the invertibility of the term 'P^T DP' by making sure there are sufficiently many data points represented in D as rows. Happy to meet and talk about this if need be.