Adversarial Estimation of Topological Dimension with Harmonic Score Maps

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.