# Strong Uniform Consistency with Rates for Kernel Density Estimators with General Kernels on Manifolds

arXiv: https://arxiv.org/abs/2007.06408 · HTML: https://mlsanity.com/paper/2007.06408/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Jul 2020.

## Built on by (most cited first, 6 of 6)

- [On the minimax optimality of Flow Matching through the connection to kernel density estimation](https://mlsanity.com/paper/2504.13336/) (2025)
- [Robust Inference of Manifold Density and Geometry by Doubly Stochastic Scaling](https://mlsanity.com/paper/2209.08004/) (2022)
- [Kernel-Gradient Drifting Models](https://mlsanity.com/paper/2605.10727/) (2026)
- [Conformal inference for regression on Riemannian Manifolds](https://mlsanity.com/paper/2310.08209/) (2023)
- [Boundary Detection Algorithm Inspired by Locally Linear Embedding](https://mlsanity.com/paper/2406.18456/) (2024)
- [Convergence of graph Dirichlet energies and graph Laplacians on intersecting manifolds of varying dimensions](https://mlsanity.com/paper/2509.24458/) (2025)

## Abstract (arXiv)

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^∞ convergence rate for KDE on Riemannian manifolds with Riemann integrable kernels (in the ambient Euclidean space). We also provide an L1 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.