用蛋糕波函数实现方向积分的最小不确定性，提升模型可解释性
Orientation Scores should be a Piece of Cake
- 基于位置-方向空间的最小不确定性原理构造蛋糕波函数
- 实验显示蛋糕波函数不确定性差距小于1.1，极限趋近理论下限1
- 无需训练即可替代传统提升层，显著降低网络复杂度
我们从公理出发推导出一类方向积分波函数，将二维位置空间 ℝ² 提升至位置-方向空间 ℝ²×S¹，具备快速重建能力且最小化位置-方向不确定性。随后证明，这些最小不确定性态可用蛋糕波函数良好逼近：在标准参数下，不确定性差距小于1.1；极限情况下，该差距趋于理论最小值1。我们完善了此前关于PDE-G-CNN中无需训练提升层的理论论证，表明可直接使用蛋糕波函数。最后，实验表明，采用此方法可显著降低网络复杂度并增强模型可解释性，仅带来轻微性能损失。
原文摘要 · Abstract (English)
We axiomatically derive a family of wavelets for an orientation score, lifting from position space $\mathbb{R}^2$ to position and orientation space $\mathbb{R}^2\times S^1$, with fast reconstruction property, that minimise position-orientation uncertainty. We subsequently show that these minimum uncertainty states are well-approximated by cake wavelets: for standard parameters, the uncertainty gap of cake wavelets is less than 1.1, and in the limit, we prove the uncertainty gap tends to the minimum of 1. Next, we complete a previous theoretical argument that one does not have to train the lifting layer in (PDE-)G-CNNs, but can instead use cake wavelets. Finally, we show experimentally that in this way we can reduce the network complexity and improve the interpretability of (PDE-)G-CNNs, with only a slight impact on the model's performance.
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