用于可解释分类的单元划分神经网络及显式类别区域
Partition of Unity Neural Networks for Interpretable Classification with Explicit Class Regions
AI总结：
提出单元划分神经网络（PUNN），通过直接学习满足和为1的非负函数来定义类别概率，无需softmax层，实现可解释分类并证明其稠密性，实验表明在保持精度的同时大幅减少参数。
摘要：
尽管神经网络分类器在经验上取得了成功，但它们仍然难以解释。在基于softmax的模型中，类别区域被隐式定义为logits之间不等式系统的解，这使得它们难以提取和可视化。我们引入了单元划分神经网络（PUNN），这是一种架构，其中类别概率直接来自学习到的单元划分，无需softmax层。PUNN构造了$k$个非负函数$h_1, \ldots, h_k$，满足$\sum_i h_i(x) = 1$，其中每个$h_i(x)$直接表示$P(\text{类别 } i \mid x)$。与softmax不同，其中类别区域通过logits之间的耦合不等式隐式定义，每个PUNN划分函数$h_i$直接定义类别$i$的概率作为$x$的独立函数。我们证明了PUNN在紧致域上的连续概率映射空间中是稠密的。定义划分的门函数$g_i$可以使用各种激活函数（sigmoid、高斯、bump）和参数化方法，从灵活的MLP到参数高效、形状感知的设计（球壳、椭球、球谐函数）。在合成数据、UCI基准和MNIST上的实验表明，基于MLP门的PUNN在精度上达到标准多层感知机的0.3-0.6%以内。当几何先验与数据结构匹配时，形状感知的门在参数减少多达300倍的情况下实现了相当的精度。这些结果表明，可解释性设计架构可以与黑盒模型竞争，同时提供透明的类别概率分配。
英文摘要：
We introduce \emph{Partition of Unity Neural Networks} (PUNNs), a neural-network architecture for multiclass classification based on the classical mathematical notion of a partition of unity. The starting point is the observation that the characteristic functions of ideal class regions form a partition of unity. PUNNs replace these discontinuous indicators by learned continuous functions \[ h_1,\ldots,h_C:\mathcal X\to[0,1] \] whose sum is identically one and whose values are interpreted directly as class probabilities. The partition functions are generated through a recursive family of input-dependent gates. This construction guarantees nonnegative class probabilities summing to one without a separate normalization layer such as softmax, while providing an explicit ordered factorization of each probability in terms of the gate values. The resulting gate trace gives an interpretable representation of how individual class probabilities are formed. The framework also allows multiple partition functions to represent a single class and admits both neural-network and geometry-informed realizations of the gates. We prove that PUNNs are dense in the space of continuous probability maps from compact subsets of $\mathbb R^d$ into the probability simplex. Thus, the recursive partition-of-unity structure retains universal approximation of continuous probabilistic classifiers, including maps whose components may vanish. Numerical experiments on synthetic datasets, MNIST, and CIFAR-100 illustrate the learned partitions, the effect of class ordering, and the use of multiple partition components per class. We also develop shape-informed gates that incorporate geometric information directly; when the chosen geometry is well matched to the class regions, these models achieve comparable accuracy with substantially fewer trainable parameters.
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