Lower Bounds on the Total Variation Distance Between Mixtures of Two Gaussians
This work addresses a fundamental issue in statistics and learning theory for researchers studying distribution learning, offering incremental improvements in bounding techniques.
The paper tackles the problem of deriving tight lower bounds on the total variation distance between mixtures of two Gaussians, which is analytically challenging, and provides new bounds using functional approximations based on characteristic functions.
Mixtures of high dimensional Gaussian distributions have been studied extensively in statistics and learning theory. While the total variation distance appears naturally in the sample complexity of distribution learning, it is analytically difficult to obtain tight lower bounds for mixtures. Exploiting a connection between total variation distance and the characteristic function of the mixture, we provide fairly tight functional approximations. This enables us to derive new lower bounds on the total variation distance between pairs of two-component Gaussian mixtures that have a shared covariance matrix.