论文标题
Kullback-Leibler和Renyi Diverencencence在复制内核Hilbert Space和Gaussian Process设置中
Kullback-Leibler and Renyi divergences in reproducing kernel Hilbert space and Gaussian process settings
论文作者
论文摘要
In this work, we present formulations for regularized Kullback-Leibler and Rényi divergences via the Alpha Log-Determinant (Log-Det) divergences between positive Hilbert-Schmidt operators on Hilbert spaces in two different settings, namely (i) covariance operators and Gaussian measures defined on reproducing kernel Hilbert spaces (RKHS); （ii）具有平方集成样品路径的高斯工艺。对于特征内核，第一个设置导致在完整的，可分开的度量空间上进行任意鲍尔概率度量之间的差异。我们表明，Hilbert-Schmidt Norm中的Alpha Log-Det差异是连续的，这使我们能够将大量定律应用于希尔伯特太空值的随机变量。因此，我们表明，在这两种情况下，都可以使用有限维的gram矩阵/高斯矩阵和有限样本数据从其有限维版本中始终有效地估算无限差异，并具有{\ IT dimension-Intepention-Intepention-Intiperention-Intiperention-Intiperention}样品复杂性。 RKHS方法论在两种情况下的理论分析中都起着核心作用。数值实验说明了数学公式。
In this work, we present formulations for regularized Kullback-Leibler and Rényi divergences via the Alpha Log-Determinant (Log-Det) divergences between positive Hilbert-Schmidt operators on Hilbert spaces in two different settings, namely (i) covariance operators and Gaussian measures defined on reproducing kernel Hilbert spaces (RKHS); and (ii) Gaussian processes with squared integrable sample paths. For characteristic kernels, the first setting leads to divergences between arbitrary Borel probability measures on a complete, separable metric space. We show that the Alpha Log-Det divergences are continuous in the Hilbert-Schmidt norm, which enables us to apply laws of large numbers for Hilbert space-valued random variables. As a consequence of this, we show that, in both settings, the infinite-dimensional divergences can be consistently and efficiently estimated from their finite-dimensional versions, using finite-dimensional Gram matrices/Gaussian measures and finite sample data, with {\it dimension-independent} sample complexities in all cases. RKHS methodology plays a central role in the theoretical analysis in both settings. The mathematical formulation is illustrated by numerical experiments.
