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arXiv 2311.01435cs.LGmath.PRstat.ML
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对比矩:多项式时间内的无监督半空间学习

Contrastive Moments: Unsupervised Halfspace Learning in Polynomial Time

Georgia Tech(佐治亚理工学院)

Xinyuan Cao, Santosh S. Vempala

AI总结:

本文提出仅利用经验分布再加权前两阶矩的无监督多项式时间算法,在未知仿射变换的对称对数凹乘积分布下唯一辨识并学习隐藏半空间。

摘要:

我们给出一种多项式时间算法,用于在 d 维空间中学习带间隔的高维半空间,并达到所需的 TV 距离;其环境分布是某个(未知)对称一维对数凹分布的 d 重乘积经过未知仿射变换后的分布,而半空间是通过在其中一个分量分布中删除至少 ε 比例的数据而引入的。值得注意的是,我们的算法不需要标签,并在该分布假设下建立了隐藏半空间的唯一(且高效的)可辨识性。该算法的样本复杂度和时间复杂度均为维度及 1/ε 的多项式。算法仅使用对经验分布进行适当再加权后的前两阶矩,我们称之为对比矩;其分析使用了关于广义 Dirichlet 多项式的经典事实,并关键依赖于对数凹分布截断的矩比的一个新单调性。这类仅基于一阶和二阶矩的算法曾在早期工作中被提出,但迄今缺乏严格保证。先前工作通过 Non-Gaussian Component Analysis 处理了底层分布为 Gaussian 的特殊情形。我们对此作出改进:基于 Total Variation(TV)距离提供多项式时间保证,以取代现有的、可能为超多项式的矩界保证。我们的工作也是该设置下首次超越 Gaussian 情形的研究。

英文摘要:

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 $ε$ 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 $1/ε$. 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.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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