Feasibility of Multivariate Density Estimates
本文重新审视了维数诅咒对核密度估计的影响，比较了不同误差准则下的最优窗宽和样本量需求，并给出了一个10维核密度估计的例子，表明维数诅咒的本质更多是数据缺乏满秩而非稀疏性。
The ‘curse of dimensionality’ has been interpreted as suggesting that kernel methods have limited applicability in more than several dimensions. In this note, qualitative and quantitative performance measures for multivariate density estimates are examined. Optimal pointwise and global window widths for mean absolute and mean squared errors are compared for multivariate data. One result is that the optimal pointwise absolute and squared error window widths are nearly equal for all dimensions. We also show that sample size requirements predicted by absolute rather than squared error criterion are substantially less. Further reductions are realized by using a coefficient of variation criterion. Finally, an example of a 10-dimensional kernel density estimate is given. It is suggested that the true nature of the curse of dimensionality is as much the lack of full rank as sparseness of the data.