Stochastic Neural Network Symmetrisation in Markov Categories (2406.11814v5)

Abstract: 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.

• The paper proposes a general framework for symmetrising neural networks using Markov categories, enabling the conversion of H-equivariant nets into G-equivariant ones.
• It employs Markov categories to formally represent neural networks as morphisms, abstracting over deterministic and stochastic behaviors for use in probabilistic models.
• The methodology refines symmetrisation techniques and is empirically validated, showing potential for designing robust, symmetry-aware models in domains like computer vision and physics.

Stochastic Neural Network Symmetrisation in Markov Categories

The paper "Stochastic Neural Network Symmetrisation in Markov Categories" presents a novel methodology for introducing symmetries into neural networks. The primary objective is to transform neural networks that exhibit equivariance with respect to one group into networks equivariant under a larger group. The discussion is framed within the context of Markov categories, which offer a high-level abstraction for reasoning about stochastic maps or Markov kernels.

Key Contributions

1. Symmetrisation Framework: The authors propose a general framework for symmetrising neural networks using Markov categories. This framework allows the conversion of an $H$-equivariant neural network into a $G$-equivariant one, where $H$ and $G$ are groups within the category. The transformation relies on finding a map from the $H$-equivariant to the $G$-equivariant domain using orbit maps and an induced action on orbits.
2. Markov Category Formalism: The paper employs Markov categories to abstract over potential stochastic behaviors in neural networks. By representing neural networks as morphisms in a Markov category, the authors can encompass both deterministic and stochastic networks within the same formalism. This abstraction is instrumental for tasks involving probabilistic models, such as those encountered in machine learning applications.
3. Enhanced Symmetrisation Techniques: Leveraging the concepts of Markov categories, the authors refine existing symmetrisation techniques. They provide a systematic approach to using a left adjoint to a restriction functor, which allows for constructing transformations from $H$-equivariant to $G$-equivariant layers systematically.
4. Implementation of Orbit and Coset Concepts: The paper formalizes notions from group theory, such as orbits and cosets, within Markov categories. This enables reasoning about symmetries in a mathematically rigorous framework, extending classical group notions to the stochastic setting.

Implications and Future Directions

The methodology outlined in the paper holds significant potential for theoretical advancements and practical applications. By enabling neural networks to be symmetrised efficiently, this approach facilitates the design of more robust and symmetry-aware models. This is particularly relevant in domains such as computer vision and physics, where data often possesses inherent symmetrical properties.

Future developments may explore extending the results to other types of categorical formalisms, potentially broadening the applicability to more complex symmetries and higher-dimensional data structures. The integration with existent machine learning frameworks could foster the development of new algorithms that automatically exploit symmetry properties, leading to improvements in efficiency and generalisation capability of neural models.

Empirical Results

The paper includes empirical validation through a synthetic example involving matrix inversion, demonstrating that the proposed stochastic symmetrisation can outperform traditional deterministic methods. This suggests that the additional flexibility afforded by the approach can be beneficial in practice, not just theoretically.

Conclusion

Overall, the work presented provides a comprehensive framework for symmetrising neural networks using a sophisticated categorical approach. Its integration of stochastic symmetrisation within Markov categories is positioned as a significant augmentation to the traditional methods, offering both theoretical elegance and practical utility. The insights gained could propel further exploration in AI fields where symmetry and equivariance play crucial roles.