Dictionary-based Manifold Learning
This work addresses the need for interpretable manifold learning in scientific data analysis, offering a domain-specific approach that leverages expert knowledge.
The authors tackled the problem of interpretable manifold learning for scientific data by proposing a paradigm that parametrizes manifolds with domain-specific functions from a scientist-provided dictionary, and they provided an algorithm based on sparse non-linear regression to find such parameterizations, demonstrating it with experimental results from a real scientific domain.
We propose a paradigm for interpretable Manifold Learning for scientific data analysis, whereby we parametrize a manifold with $d$ smooth functions from a scientist-provided dictionary of meaningful, domain-related functions. When such a parametrization exists, we provide an algorithm for finding it based on sparse non-linear regression in the manifold tangent bundle, bypassing more standard manifold learning algorithms. We also discuss conditions for the existence of such parameterizations in function space and for successful recovery from finite samples. We demonstrate our method with experimental results from a real scientific domain.