The kernel thinning (KT) algorithm of Dwivedi and Mackey (2021) compresses a probability distribution more effectively than independent sampling by targeting a reproducing kernel Hilbert space (RKHS) and leveraging a less smooth square-root kernel. Here we provide four improvements. First, we show that KT applied directly to the target RKHS yields tighter, dimension-free guarantees for any kernel, any distribution, and any fixed function in the RKHS. Second, we show that, for analytic kernels like Gaussian, inverse multiquadric, and sinc, target KT admits maximum mean discrepancy (MMD) guarantees comparable to or better than those of square-root KT without making explicit use of a square-root kernel. Third, we prove that KT with a fractional power kernel yields better-than-Monte-Carlo MMD guarantees for non-smooth kernels, like Laplace and Mat\'ern, that do not have square-roots. Fourth, we establish that KT applied to a sum of the target and power kernels (a procedure we call KT+) simultaneously inherits the improved MMD guarantees of power KT and the tighter individual function guarantees of target KT. In our experiments with target KT and KT+, we witness significant improvements in integration error even in $100$ dimensions and when compressing challenging differential equation posteriors.

翻译：Dwivedi 和 Mackey (2021年) 的内核稀释算法(KT) 压缩概率分布比独立取样更有效,方法是针对复制的内核Hilbert 空间(RKHS),利用不光滑的平方根内核。 我们在这里提供四个改进。 首先, 我们显示, KT直接应用到目标的RKHS, 对任何内核、任何分布和任何在RKHS的固定功能提供较紧的、无维度的保障。 其次, 我们显示, 对于Gausian、反多方形和辛肯等分析核心, 目标的概率分布比独立取样法(MMDD) 承认最大平均值差异(MMDD) 与平方根内空空间(RKT) 的类似或更好的保障。 第三, 我们证明, 拥有小核核内核核核部分内核(MMD) 的保证对非薄核内核(Laplet 和 Mat\ ) 的保证, 平底底, 我们确定KT 目标和更精确的KMC 的精确的保证, 我们的不断的K- 的K- T 和K- m) 的精确的K- m) 的保证。