论文标题
KNN密度估计分析
Analysis of KNN Density Estimation
论文作者
论文摘要
我们分析了K最近的邻居密度估计方法的$ \ ell_1 $和$ \ ell_ \ eld_ \ infty $收敛率。我们的分析包括两个不同的案例，具体取决于支持集有限的情况。在第一种情况下，概率密度函数具有有界的支持，并远离零。我们表明，如果已知支持集，则KNN密度估计在$ \ ell_1 $和$ \ ell_ \ infty $标准下都是最佳的。如果支持集未知，则$ \ ell_1 $错误的收敛速率不会受到影响，而$ \ ell_ \ infty $ error不会收敛。在第二种情况下，概率密度函数可以接近零，并且到处都是光滑的。此外，假定黑森与密度值衰减。在这种情况下，我们的结果表明，KNN密度估计的$ \ ell_ \ infty $误差几乎是最小的。 $ \ ell_1 $错误未达到minimax下限，而是比内核密度估计更好。
We analyze the $\ell_1$ and $\ell_\infty$ convergence rates of k nearest neighbor density estimation method. Our analysis includes two different cases depending on whether the support set is bounded or not. In the first case, the probability density function has a bounded support and is bounded away from zero. We show that kNN density estimation is minimax optimal under both $\ell_1$ and $\ell_\infty$ criteria, if the support set is known. If the support set is unknown, then the convergence rate of $\ell_1$ error is not affected, while $\ell_\infty$ error does not converge. In the second case, the probability density function can approach zero and is smooth everywhere. Moreover, the Hessian is assumed to decay with the density values. For this case, our result shows that the $\ell_\infty$ error of kNN density estimation is nearly minimax optimal. The $\ell_1$ error does not reach the minimax lower bound, but is better than kernel density estimation.
