Contrastive Moments: Unsupervised Halfspace Learning in Polynomial Time

We give a polynomial-time algorithm for learning high-dimensional halfspaces with margins in d -dimensional space to within desired Total Variation (TV) distance when the ambient distribution is an unknown affine transformation of the d -fold product of an (unknown) symmetric one-dimensional logconcave distribution, and the halfspace is introduced by deleting at least an \epsilon fraction of the data in one of the component distributions. Notably, our algorithm does not need labels and establishes the unique (and efficient) identifiability of the hidden halfspace under this distributional assumption. The sample and time complexity of the algorithm are polynomial in the dimension and 1/\epsilon . The algorithm uses only the first two moments of *suitable re-weightings* of the empirical distribution, which we call *contrastive moments*; its analysis uses classical facts about generalized Dirichlet polynomials and relies crucially on a new monotonicity property of the moment ratio of truncations of logconcave distributions. Such algorithms, based only on first and second moments were suggested in earlier work, but hitherto eluded rigorous guarantees.Prior work addressed the special case when the underlying distribution is Gaussian via Non-Gaussian Component Analysis.