Mixture and separation of log-concave measures

For a two-component log-concave mixture model, we investigate the extent to which the weight, mean, and covariance of each component distribution can be accurately recovered when given sufficient samples of the mixture distributions.

One fundamental obstruction is that the mixture distribution itself could sometimes be log-concave, and in this case, accurate recovery is impossible.

In this paper, we identify two regimes, one where the mixture distribution itself could be log-concave and another one where the mixture distribution is never log-concave, and one can always separate the two component distributions using a quadratic classifier.