In this paper we consider high-dimensional multiclass classification by sparse multinomial logistic regression. We propose first a feature selection procedure based on penalized maximum likelihood with a complexity penalty on the model size and derive the nonasymptotic bounds for misclassification excess risk of the resulting classifier. We establish also their tightness by deriving the corresponding minimax lower bounds. In particular, we show that there is a phase transition between small and large number of classes. The bounds can be reduced under the additional low noise condition. To find a penalized maximum likelihood solution with a complexity penalty requires, however, a combinatorial search over all possible models. To design a feature selection procedure computationally feasible for high-dimensional data, we propose multinomial logistic group Lasso and Slope classifiers and show that they also achieve the minimax order.

中文翻译：

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基于稀疏多项 Logistic 回归的多类分类

在本文中，我们通过稀疏多项逻辑回归考虑高维多类分类。我们首先提出了一个基于惩罚最大似然的特征选择程序，对模型大小进行复杂性惩罚，并推导出所得分类器误分类过度风险的非渐近界限。我们还通过推导相应的极小极大下界来建立它们的紧密度。特别是，我们表明小类和大量类之间存在相变。在额外的低噪声条件下可以减少界限。然而，要找到具有复杂性惩罚的惩罚最大似然解决方案，需要对所有可能的模型进行组合搜索。为了设计一个对高维数据在计算上可行的特征选择程序，