论文标题
Nyström内核平均嵌入
Nyström Kernel Mean Embeddings
论文作者
论文摘要
内核平均嵌入是一种强大的工具，可以用作希尔伯特空间中的单个点上的概率分布。但是，计算和存储此类嵌入的成本禁止在大规模设置中直接使用。我们提出了一个基于NyStröm方法的有效近似过程，该过程利用了数据集的一个小组随机子集。我们的主要结果是对此过程的近似误差的上限。它在子样本大小上产生足够的条件，以获得标准的$ n^{ - 1/2} $速率，同时降低计算成本。我们讨论了此结果的应用，以逼近最大平均差异和正交规则，并通过数值实验说明了我们的理论发现。
Kernel mean embeddings are a powerful tool to represent probability distributions over arbitrary spaces as single points in a Hilbert space. Yet, the cost of computing and storing such embeddings prohibits their direct use in large-scale settings. We propose an efficient approximation procedure based on the Nyström method, which exploits a small random subset of the dataset. Our main result is an upper bound on the approximation error of this procedure. It yields sufficient conditions on the subsample size to obtain the standard $n^{-1/2}$ rate while reducing computational costs. We discuss applications of this result for the approximation of the maximum mean discrepancy and quadrature rules, and illustrate our theoretical findings with numerical experiments.
