Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data
Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data
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发表时间：
2020-02
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通讯作者：
Marc Finzi;S. Stanton;Pavel Izmailov;A. Wilson
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作者：
Marc Finzi;S. Stanton;Pavel Izmailov;A. Wilson
卷积层的平移等变性使得卷积神经网络能够很好地推广到图像问题上。虽然平移等差为图像提供了强大的归纳偏差，但我们通常还希望与其他变换等同，例如旋转，特别是对于非图像数据。我们提出了一种构造卷积层的一般方法，该卷积层与任何指定的李群的变换等变，并且具有满足性指数映射。将等方差合并到一个新的组中只需要实现组指数和对数图，从而实现快速原型制作。为了展示我们方法的简单性和通用性，我们将相同的模型架构应用于图像、球棒分子数据和哈密顿动力学系统。对于哈密顿系统，我们的模型的等变性是特别重要的，它导致了线动量和角动量的精确守恒。
The translation equivariance of convolutional layers enables convolutional neural networks to generalize well on image problems. While translation equivariance provides a powerful inductive bias for images, we often additionally desire equivariance to other transformations, such as rotations, especially for non-image data. We propose a general method to construct a convolutional layer that is equivariant to transformations from any specified Lie group with a surjective exponential map. Incorporating equivariance to a new group requires implementing only the group exponential and logarithm maps, enabling rapid prototyping. Showcasing the simplicity and generality of our method, we apply the same model architecture to images, ball-and-stick molecular data, and Hamiltonian dynamical systems. For Hamiltonian systems, the equivariance of our models is especially impactful, leading to exact conservation of linear and angular momentum.