论文标题
内核平均嵌入von Neumann-Algebra值措施的嵌入
Kernel Mean Embeddings of Von Neumann-Algebra-Valued Measures
论文作者
论文摘要
内核平均嵌入（KME）是分析数据概率度量的强大工具，其中这些措施通常嵌入了繁殖的内核Hilbert Space（RKHS）中。在本文中，我们将KME推广到von Neumann-Algebra var的措施中，以重现内核希尔伯特模块（RKHMS），该模块（RKHMS）提供了内部产物和von Neumann-Algebra价值措施之间的距离。冯·诺伊曼·阿尔格拉（Von Neumann-Algebra）估计的措施可以例如，在多变量分布或阳性操作员对量子力学的阳性算法中编码变量对之间的关系。因此，这使我们能够进行概率分析，并通过变量之间的高阶相互作用明确反映，并提供了将机器学习框架应用于量子力学中问题的方法。我们还表明，现有KME的注射率和RKHS的普遍性被推广到RKHM，这证实了现有KME的许多有用特征保留在我们的广义KME中。而且，我们使用合成和现实世界数据研究了我们方法的经验性能。
Kernel mean embedding (KME) is a powerful tool to analyze probability measures for data, where the measures are conventionally embedded into a reproducing kernel Hilbert space (RKHS). In this paper, we generalize KME to that of von Neumann-algebra-valued measures into reproducing kernel Hilbert modules (RKHMs), which provides an inner product and distance between von Neumann-algebra-valued measures. Von Neumann-algebra-valued measures can, for example, encode relations between arbitrary pairs of variables in a multivariate distribution or positive operator-valued measures for quantum mechanics. Thus, this allows us to perform probabilistic analyses explicitly reflected with higher-order interactions among variables, and provides a way of applying machine learning frameworks to problems in quantum mechanics. We also show that the injectivity of the existing KME and the universality of RKHS are generalized to RKHM, which confirms many useful features of the existing KME remain in our generalized KME. And, we investigate the empirical performance of our methods using synthetic and real-world data.
