分布族的多项式学习
Polynomial Learning of Distribution Families
摘要：
概率分布的多项式可学习性问题，特别是高斯混合分布，最近在理论计算机科学和机器学习中受到了极大的关注。然而，尽管取得了重大进展，高斯混合分布的多项式可学习性的一般问题仍然是开放的。目前的工作解决的问题多项式学习高斯混合在高维任意固定数量的组件。学习高斯混合的结果依赖于对属于我们所说的低维“多项式族”的分布的分析。这些族的特征在于它们的矩是参数的多项式，并且包括几乎所有常见的概率分布以及它们的混合和乘积。使用工具，从真正的代数几何，我们表明，属于这样一个家庭的任何分布的参数可以在多项式时间和使用多项式数量的样本点。学习多项式族的结果是相当普遍的，是独立的利益。为了估计高维高斯混合分布的参数，我们提供了一个确定性降维算法。这使我们能够将高维混合的学习减少到低维参数估计的多项式数量。将这种减少与多项式族的结果相结合，得到了我们在高维中学习任意高斯混合的结果。
英文摘要：
The question of polynomial learnability of probability distributions, particularly Gaussian mixture distributions, has recently received significant attention in theoretical computer science and machine learning. However, despite major progress, the general question of polynomial learnability of Gaussian mixture distributions still remained open. The current work resolves the question of polynomial learnability for Gaussian mixtures in high dimension with an arbitrary fixed number of components. The result on learning Gaussian mixtures relies on an analysis of distributions belonging to what we call "polynomial families" in low dimension. These families are characterized by their moments being polynomial in parameters and include almost all common probability distributions as well as their mixtures and products. Using tools from real algebraic geometry, we show that parameters of any distribution belonging to such a family can be learned in polynomial time and using a polynomial number of sample points. The result on learning polynomial families is quite general and is of independent interest. To estimate parameters of a Gaussian mixture distribution in high dimensions, we provide a deterministic algorithm for dimensionality reduction. This allows us to reduce learning a high-dimensional mixture to a polynomial number of parameter estimations in low dimension. Combining this reduction with the results on polynomial families yields our result on learning arbitrary Gaussian mixtures in high dimensions.
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