论文标题
与歧管上的一般内核的内核密度估计器的强度均匀一致性
Strong Uniform Consistency with Rates for Kernel Density Estimators with General Kernels on Manifolds
论文作者
论文摘要
在分析现代机器学习算法时，我们可能需要使用不受用户设计的复杂内核来处理内核密度估计（KDE），甚至可能是不规则和不对称的。为了应对这一新出现的挑战，我们提供了强大的统一一致性结果，其中包括Riemannian歧管的$ l^\ infty $融合率，并带有Riemann Antignable -ablectable内核（在环境Euclidean Space中）。我们还提供了与Lebesgue Antegsable bernels对Riemannian歧管的内核密度估算的$ l^1 $一致性结果。本文考虑的各向同性核与统计社会中经常考虑的Vapnik-Chervonenkis类中的内核不同。当我们应用它们以估计概率密度函数时，我们说明了差异。此外，当内核在内在的歧管上设计时和环境欧几里得空间时，我们会阐述微妙的差异，这两者在实践中都可能遇到。最后，我们证明了各向同性核可以在欧几里得空间中的子曼群上集成的必要条件。
When analyzing modern machine learning algorithms, we may need to handle kernel density estimation (KDE) with intricate kernels that are not designed by the user and might even be irregular and asymmetric. To handle this emerging challenge, we provide a strong uniform consistency result with the $L^\infty$ convergence rate for KDE on Riemannian manifolds with Riemann integrable kernels (in the ambient Euclidean space). We also provide an $L^1$ consistency result for kernel density estimation on Riemannian manifolds with Lebesgue integrable kernels. The isotropic kernels considered in this paper are different from the kernels in the Vapnik-Chervonenkis class that are frequently considered in statistics society. We illustrate the difference when we apply them to estimate the probability density function. Moreover, we elaborate the delicate difference when the kernel is designed on the intrinsic manifold and on the ambient Euclidian space, both might be encountered in practice. At last, we prove the necessary and sufficient condition for an isotropic kernel to be Riemann integrable on a submanifold in the Euclidean space.
