We consider the problem of symmetrising a neural network along a group homomorphism: given a homomorphism $\varphi : H \to G$, we would like a procedure that converts $H$-equivariant neural networks to $G$-equivariant ones. We formulate this in terms of Markov categories, which allows us to consider neural networks whose outputs may be stochastic, but with measure-theoretic details abstracted away. We obtain a flexible and compositional framework for symmetrisation that relies on minimal assumptions about the structure of the group and the underlying neural network architecture. Our approach recovers existing canonicalisation and averaging techniques for symmetrising deterministic models, and extends to provide a novel methodology for symmetrising stochastic models also. Beyond this, our findings also demonstrate the utility of Markov categories for addressing complex problems in machine learning in a conceptually clear yet mathematically precise way.

翻译：我们研究沿群同态对称化神经网络的问题：给定同态 $\varphi : H \to G$，我们希望找到一种将 $H$-等变神经网络转换为 $G$-等变网络的方法。我们在马尔可夫范畴的框架下形式化这一问题，这使得我们可以处理输出可能具有随机性的神经网络，同时将测度论的细节抽象化。我们获得了一个灵活且可组合的对称化框架，该框架仅需关于群结构和底层神经网络架构的最小假设。我们的方法恢复了现有用于对称化确定性模型的典型化与平均技术，并进一步扩展为对称化随机模型提供了新颖的方法论。此外，我们的研究结果也展示了马尔可夫范畴以概念清晰且数学精确的方式处理机器学习中复杂问题的实用性。