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 into $G$-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, compositional, and generic framework for symmetrisation that relies on minimal assumptions about the structure of the group and the underlying neural network architecture. Our approach recovers existing methods for deterministic symmetrisation as special cases, and extends directly to provide a novel methodology for stochastic symmetrisation also. Beyond this, we believe our findings also demonstrate the utility of Markov categories for addressing problems in machine learning in a conceptual yet mathematically rigorous 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 $\varphi : H \to G$，我们希望获得一种将 $H$-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表述该问题，这使得我们能够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。我们建立了一个灵活、可组合且通用的对称化框架，该框架仅需关于群结构及底层神经网络架构的最小假设。我们的方法将现有确定性对称化方法作为特例包含其中，并可直接扩展为一种新颖的随机对称化方法。此外，我们认为本研究结果也证明了马尔可夫范畴能以概念清晰且数学严谨的方式处理机器学习问题的实用性。