[论文解读] On Universal Equivariant Set Networks

Nimrod Segol, Yaron Lipman|arXiv (Cornell University)|Oct 6, 2019

Generative Adversarial Networks and Image Synthesis参考文献 16被引用 6

一句话总结

本文证明了PointNet并非等变通用的，但通过添加单个线性传输层（PointNetST），可使其对所有连续排列等变函数实现普遍逼近。其关键理论贡献是利用幂和完全对称多项式对排列等变多项式层进行表征，从而支持证明，并确立PointNetST为目前已知最简的通用等变集合网络。

ABSTRACT

Using deep neural networks that are either invariant or equivariant to permutations in order to learn functions on unordered sets has become prevalent. The most popular, basic models are DeepSets [Zaheer et al. 2017] and PointNet [Qi et al. 2017]. While known to be universal for approximating invariant functions, DeepSets and PointNet are not known to be universal when approximating \emph{equivariant} set functions. On the other hand, several recent equivariant set architectures have been proven equivariant universal [Sannai et al. 2019], [Keriven et al. 2019], however these models either use layers that are not permutation equivariant (in the standard sense) and/or use higher order tensor variables which are less practical. There is, therefore, a gap in understanding the universality of popular equivariant set models versus theoretical ones. In this paper we close this gap by proving that: (i) PointNet is not equivariant universal; and (ii) adding a single linear transmission layer makes PointNet universal. We call this architecture PointNetST and argue it is the simplest permutation equivariant universal model known to date. Another consequence is that DeepSets is universal, and also PointNetSeg, a popular point cloud segmentation network (used eg, in [Qi et al. 2017]) is universal. The key theoretical tool used to prove the above results is an explicit characterization of all permutation equivariant polynomial layers. Lastly, we provide numerical experiments validating the theoretical results and comparing different permutation equivariant models.

研究动机与目标

• 解决实际排列等变网络（如PointNet）与理论通用模型之间的差距。
• 确定PointNet及其相关架构是否能对连续排列等变函数实现普遍逼近。
• 表征所有排列等变多项式层，以支持对普遍逼近的理论分析。
• 建立一个最小且实用的架构，使其对等变集合函数具有普遍性。

提出的方法

• 通过在标准PointNet中增加单个线性传输层（X ↦ 11ᵀX）提出PointNetST。
• 利用幂和多重对称多项式，推导出排列等变多项式映射的显式表征。
• 将任意等变多项式表示为逐元素函数与单个线性传输层的复合。
• 利用该表征证明PointNetST对等变函数具有普遍性。
• 将该理论框架应用于证明DeepSets和PointNetSeg也具有普遍性。
• 在集合分类与回归任务上开展数值实验，以验证理论结论。

实验结果

研究问题

• RQ1 PointNet是否能对连续排列等变函数实现普遍逼近？
• RQ2 对PointNet进行最小的架构修改，能否实现等变函数的普遍逼近？
• RQ3 排列等变多项式层的完整代数结构是什么？
• RQ4 PointNetST的理论普遍性与在真实任务上的实际性能相比如何？
• RQ5 如DeepSets和PointNetSeg等广泛使用的模型是否对等变函数具有普遍逼近能力？

主要发现

• PointNet并非等变通用，因其无法逼近所有连续排列等变函数。
• 添加单个线性传输层（PointNetST）后，PointNet可对所有连续排列等变函数实现普遍逼近。
• 实现排列等变通用网络的最小宽度受限于 ω ≤ k_out + k_in + (n + k_in choose k_in)。
• 证明了DeepSets和PointNetSeg对等变函数具有普遍性，扩展了其理论依据。
• 数值实验表明，PointNetST、DeepSets和PointNetSeg在分类与回归任务上表现相当，而非通用的PointNet表现较差。
• 尽管参数量相近，非等变的MLP基线模型表现较差，凸显了等变性对泛化能力的重要性。