Adversarial Estimation of Topological Dimension with Harmonic Score Maps
Eric Yeats Cameron Darwin Frank Liu Hai Li
Quantification of the number of variables needed to locally explain complex
data is often the first step to better understanding it. Existing techniques
from intrinsic dimension estimation leverage statistical models to glean this
information from samples within a neighborhood. However, existing methods often
rely on well-picked hyperparameters and ample data as manifold dimension and
curvature increases. Leveraging insight into the fixed point of the score
matching objective as the score map is regularized by its Dirichlet energy, we
show that it is possible to retrieve the topological dimension of the manifold
learned by the score map. We then introduce a novel method to measure the
learned manifold's topological dimension (i.e., local intrinsic dimension)
using adversarial attacks, thereby generating useful interpretations of the
learned manifold.