证明了分段线性雅诺西池化无法实现多重集的单射，挑战了简化模型的可行性。
On the (Non) Injectivity of Piecewise Linear Janossy Pooling
- 研究分段线性雅诺西池化在多重集上的单射性限制
- 证明任意阶数的分段线性雅诺西池化均非单射
- 适用于关注图神经网络表示性质的研究者
多重集函数是构建多重集与图神经网络的基础工具。为确保多重集向量表示的忠实性，通常期望映射具有单射性和双利普希茨性。目前已有多种构造同时满足这两项性质，虽提升了部分任务性能，但计算开销较高。因此自然提出：能否设计更简单的、仍满足上述性质的多重集函数？本文对一类广泛使用的k-ary Janossy pooling进行研究，证明不存在任何分段线性形式的Janossy pooling可实现单射。另一方面，在无重复元素的多重集上，简单的DeepSets模型即可实现单射性和双利普希茨性。
原文摘要 · Abstract (English)
Multiset functions, which are functions that map multisets to vectors, are a fundamental tool in the construction of neural networks for multisets and graphs. To guarantee that the vector representation of the multiset is faithful, it is often desirable to have multiset mappings that are both injective and bi-Lipschitz. Currently, there are several constructions of multiset functions achieving both these guarantees, leading to improved performance in some tasks but often also to higher compute time than standard constructions. Accordingly, it is natural to inquire whether simpler multiset functions achieving the same guarantees are available. In this paper, we make a large step towards giving a negative answer to this question. We consider the family of k-ary Janossy pooling, which includes many of the most popular multiset models, and prove that no piecewise linear Janossy pooling function can be injective. On the positive side, we show that when restricted to multisets without multiplicities, even simple deep-sets models suffice for injectivity and bi-Lipschitzness.
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