Sampling on locally defined principal manifolds

We start with a locally defined principal curve definition for a given probability density function (pdf) and define a pairwise manifold score based on local derivatives of the pdf. Proposed manifold score can be used to check if data pairs lie on the same manifold. We use this score to (i) cluster nonlinear manifolds having irregular shapes, and (ii) (down)sample a selected principal curve with sufficient accuracy sparsely. Our goal is to provide a heuristic-free formulation for principal graph generation and curve parametrization in order to form a basis for a principled principal manifold unwrapping method.