Learning Broken Symmetries with Approximate Invariance (2412.18773v2)

Abstract: Recognizing symmetries in data allows for significant boosts in neural network training, which is especially important where training data are limited. In many cases, however, the exact underlying symmetry is present only in an idealized dataset, and is broken in actual data, due to asymmetries in the detector, or varying response resolution as a function of particle momentum. Standard approaches, such as data augmentation or equivariant networks fail to represent the nature of the full, broken symmetry, effectively overconstraining the response of the neural network. We propose a learning model which balances the generality and asymptotic performance of unconstrained networks with the rapid learning of constrained networks. This is achieved through a dual-subnet structure, where one network is constrained by the symmetry and the other is not, along with a learned symmetry factor. In a simplified toy example that demonstrates violation of Lorentz invariance, our model learns as rapidly as symmetry-constrained networks but escapes its performance limitations.

• The paper introduces a dual-subnetwork architecture that combines invariant and general models with a learned symmetry factor to address approximate data symmetries.
• It demonstrates improved training efficiency on high-dimensional particle physics datasets by effectively managing symmetry violations.
• Empirical results reveal rapid convergence and enhanced performance compared to strictly invariant or unconstrained models.

Insights into Learning Broken Symmetries with Approximate Invariance

The paper "Learning Broken Symmetries with Approximate Invariance" presents a novel approach to enhancing neural network training in the context of high-dimensional datasets, where inherent data symmetries are often violated due to real-world measurement inconsistencies. This research explores the field of approximating invariance within machine learning models, particularly when traditional data augmentation methods fall short due to broken symmetries in the datasets.

Problem Definition and Proposed Solution

In experiments such as those conducted at the Large Hadron Collider, data symmetries play a critical role. However, these symmetries are frequently disrupted due to detector resolution variations and other anomalies. The paper addresses the challenge of training neural networks efficiently when symmetries are approximate rather than absolute. A primary example discussed is the Lorentz invariance in particle physics datasets, which is inadequately represented when relying solely on standard methods like data augmentation or equivariant networks.

The authors introduce a hybrid neural network architecture that combines the strengths of two subnetworks: one that is invariant and the other that is general and unconstrained. Through this dual-subnet structure, they integrate a learned symmetry factor, enabling the network to judiciously balance and benefit from both symmetry-unbiased and symmetry-constrained learning paradigms. Notably, this architecture is tested on a simplified example involving Lorentz symmetry violations, revealing the capacity to maintain rapid learning while also overcoming performance limitations commonly experienced by strictly constrained models.

Experimental Validation

The paper provides empirical evidence via experiments involving particle interactions, specifically focusing on events where a $Z$ boson decays into two muons. The datasets simulate momentum-dependent resolution effects, highlighting the inadequacies of purely invariant or general models. By integrating a dual-subnet approach with a learned symmetry factor, the hybrid model achieves a statistically significant improvement in classification tasks, particularly under conditions of limited training data. The findings indicate that while invariant networks learn rapidly, their performance plateaus, failing to surpass the ceiling imposed by adherence to strict symmetry assumptions. Conversely, the proposed model circumvents this limitation, achieving both rapid convergence and high asymptotic performance.

Implications and Future Research Directions

This research suggests a pathway to improved neural network training techniques by embracing approximation in symmetry constraints, thereby aligning models more closely with the realities of empirical data. From a practical standpoint, this method holds promise not only in high-energy physics but potentially in other domains where data are subject to systemic measurement variations. The proposal to employ a flexible symmetry factor enables a dynamic response to varying degrees of symmetry violation within datasets.

Looking ahead, there are numerous opportunities for expansion. Future work could explore the applicability of this method to more complex datasets and innovative architectural designs that incorporate additional forms of invariance. The approach could be extended to image recognition systems, where pixelization and edge effects break translational symmetries. Moreover, the learned weighting mechanism offers potential for applications in diverse machine learning frameworks, fostering enhanced data efficiency in training neural networks across different scientific and engineering fields.

The paper contributes to the ongoing dialogue around optimizing neural networks' learning processes by advocating for adaptive methodologies that accommodate the realities of imperfect data symmetries. This advancement underscores the significance of nuanced model design in the context of evolving data complexities and the quest for computational efficiency.