Local continuity of log-concave projection, with applications to estimation under model misspecification

Abstract

The log-concave projection is an operator that maps a d-dimensional distribution P to an approximating log-concave density. It is known that, with suitable metrics on the underlying spaces, this projection is continuous, but not uniformly continuous. In this work, we prove a local uniform continuity result for log-concave projection – in particular, establishing that this map is locally Hölder-(1/4) continuous. A matching lower bound verifies that this exponent cannot be improved. We also examine the implications of this continuity result for the empirical setting – given a sample drawn from a distribution P, we bound the squared Hellinger distance between the log-concave projection of the empirical distribution of the sample, and the log-concave projection of P. In particular, this yields interesting statistical results for the misspecified setting, where P is not itself log-concave.