Abstract: It has long been thought that high-dimensional data encountered in many
practical machine learning tasks have low-dimensional structure, i.e., the
manifold hypothesis holds. A natural question, thus, is to estimate the
intrinsic dimension of a given population distribution from a finite sample. We
introduce a new estimator of the intrinsic dimension and provide finite sample,
non-asymptotic guarantees. We then apply our techniques to get new sample
complexity bounds for Generative Adversarial Networks (GANs) depending only on
the intrinsic dimension of the data.