  With the proliferation of generative AI and the increasing volume of
generative data (also called as synthetic data), assessing the fidelity of
generative data has become a critical concern. In this paper, we propose a
discriminative approach to estimate the total variation (TV) distance between
two distributions as an effective measure of generative data fidelity. Our
method quantitatively characterizes the relation between the Bayes risk in
classifying two distributions and their TV distance. Therefore, the estimation
of total variation distance reduces to that of the Bayes risk. In particular,
this paper establishes theoretical results regarding the convergence rate of
the estimation error of TV distance between two Gaussian distributions. We
demonstrate that, with a specific choice of hypothesis class in classification,
a fast convergence rate in estimating the TV distance can be achieved.
Specifically, the estimation accuracy of the TV distance is proven to
inherently depend on the separation of two Gaussian distributions: smaller
estimation errors are achieved when the two Gaussian distributions are farther
apart. This phenomenon is also validated empirically through extensive
simulations. In the end, we apply this discriminative estimation method to rank
fidelity of synthetic image data using the MNIST dataset.
