论文标题
模型的普遍性与对称性的可学习性之间的权衡
A tradeoff between universality of equivariant models and learnability of symmetries
论文作者
论文摘要
我们证明了一个不可能的结果，在功能学习的背景下，它说了以下内容：在某些条件下，不可能使用由epoivariant函数组成的ANSATZ同时学习对称性和功能在它们下面。为了正式化这一说法，我们仔细研究了小组和半群的近似概念。我们分析了某些神经网络的家庭是否满足了不可能结果的条件：我们所说的````线性arteraliant''网络''网络和群体跨革命网络。关于线性的等效网络，可以精确地说很多事情，从而使它们在理论上有用。从实际方面来说，我们对群体横向神经网络的分析使我们概括了众所周知的``卷积就是您需要的''定理'定理到非均匀的空间。我们还发现小组卷积和半群卷积之间的重要区别。
We prove an impossibility result, which in the context of function learning says the following: under certain conditions, it is impossible to simultaneously learn symmetries and functions equivariant under them using an ansatz consisting of equivariant functions. To formalize this statement, we carefully study notions of approximation for groups and semigroups. We analyze certain families of neural networks for whether they satisfy the conditions of the impossibility result: what we call ``linearly equivariant'' networks, and group-convolutional networks. A lot can be said precisely about linearly equivariant networks, making them theoretically useful. On the practical side, our analysis of group-convolutional neural networks allows us generalize the well-known ``convolution is all you need'' theorem to non-homogeneous spaces. We additionally find an important difference between group convolution and semigroup convolution.
