论文标题
内核条件平均嵌入的一种测量理论方法
A Measure-Theoretic Approach to Kernel Conditional Mean Embeddings
论文作者
论文摘要
我们为条件平均嵌入（CME）提出了一种无操作的，测量理论的方法，作为一个随机变量，在繁殖的核希尔伯特空间中采用值。尽管严格定义了无条件分布的内核平均值嵌入，但现有的条件版本的现有方法取决于严格的假设，阻碍了其分析。我们通过对CME的理论处理来克服这一局限性。我们得出自然回归解释以获得经验估计，并提供了彻底的理论分析，包括普遍的一致性。作为天然副产品，我们获得了最大平均差异和希尔伯特 - 奇特独立标准的条件类似物，并通过模拟证明了它们的行为。
We present an operator-free, measure-theoretic approach to the conditional mean embedding (CME) as a random variable taking values in a reproducing kernel Hilbert space. While the kernel mean embedding of unconditional distributions has been defined rigorously, the existing operator-based approach of the conditional version depends on stringent assumptions that hinder its analysis. We overcome this limitation via a measure-theoretic treatment of CMEs. We derive a natural regression interpretation to obtain empirical estimates, and provide a thorough theoretical analysis thereof, including universal consistency. As natural by-products, we obtain the conditional analogues of the maximum mean discrepancy and Hilbert-Schmidt independence criterion, and demonstrate their behaviour via simulations.
