Nonparametric classification with missing data
提出一种新的非参数分类框架，处理特征可能非随机缺失的数据，推导出超额风险的极小极大速率，并设计了一种基于k近邻和阈值化的HAM分类器，其收敛速度不受数据维度影响。
We introduce a new nonparametric framework for classification problems in the presence of missing data. The key aspect of our framework is that the regression function decomposes into an anova-type sum of orthogonal functions, of which some (or even many) may be zero. Working under a general missingness setting, which allows features to be missing not at random, our main goal is to derive the minimax rate for the excess risk in this problem. In addition to the decomposition property, the rate depends on parameters that control the tail behaviour of the marginal feature distributions, the smoothness of the regression function and a margin condition. The ambient data dimension does not appear in the minimax rate, which can therefore be faster than in the classical nonparametric setting. We further propose a new method, called the Hard-thresholding Anova Missing data (HAM) classifier, based on a careful combination of a k-nearest neighbour algorithm and a thresholding step. The HAM classifier attains the minimax rate up to polylogarithmic factors and numerical experiments further illustrate its utility.