We give a polynomial-time algorithm for learning high-dimensional halfspaces with margins in $d$-dimensional space to within desired 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. We improve on this by providing polytime guarantees based on Total Variation (TV) distance, in place of existing moment-bound guarantees that can be super-polynomial. Our work is also the first to go beyond Gaussians in this setting.

翻译：我们提出一种多项式时间算法，用于在$d$维空间中学习具有间隔的高维半空间，达到期望的TV距离，前提是环境分布是$d$重乘积（未知的对称一维对数凹分布）的未知仿射变换，且半空间通过删除至少一个分量分布中$\epsilon$比例的数据引入。值得注意的是，我们的算法无需标签，并在此分布假设下确立了隐藏半空间的唯一（且高效）可辨识性。算法的样本复杂度和时间复杂度在维度及$1/\epsilon$上是多项式的。该算法仅利用经验分布适当重加权的第一、二阶矩（我们称之为对比时刻）；其分析借鉴了广义狄利克雷多项式的经典结论，并关键依赖于对数凹分布截断矩比的新单调性性质。此类仅基于一阶和二阶矩的算法在早期工作中曾被提出，但此前缺乏严格的理论保证。先前工作通过非高斯成分分析处理了基础分布为高斯分布的特殊情形。我们通过基于总变差（TV）距离的多项式时间保证改进了这一结果，取代了可能为超多项式的现有矩界保证。我们的工作也是该领域中首个超越高斯分布假设的研究。