Moments of Clarity: Streamlining Latent Spaces in Machine Learning using Moment Pooling (2403.08854v2)

Abstract: Many machine learning applications involve learning a latent representation of data, which is often high-dimensional and difficult to directly interpret. In this work, we propose "Moment Pooling", a natural extension of Deep Sets networks which drastically decrease latent space dimensionality of these networks while maintaining or even improving performance. Moment Pooling generalizes the summation in Deep Sets to arbitrary multivariate moments, which enables the model to achieve a much higher effective latent dimensionality for a fixed latent dimension. We demonstrate Moment Pooling on the collider physics task of quark/gluon jet classification by extending Energy Flow Networks (EFNs) to Moment EFNs. We find that Moment EFNs with latent dimensions as small as 1 perform similarly to ordinary EFNs with higher latent dimension. This small latent dimension allows for the internal representation to be directly visualized and interpreted, which in turn enables the learned internal jet representation to be extracted in closed form.

• The paper introduces Moment Pooling to reduce latent space dimensionality while preserving model performance.
• It extends Deep Sets with higher-order moment aggregation, significantly improving quark/gluon jet classification in EFNs.
• The method enhances interpretability by linking low-dimensional latent representations to known physical observables.

Streamlining Latent Spaces in Machine Learning with Moment Pooling

Introduction

In the field of machine learning, especially within applications to collider physics, the concept of learning latent representations of data presents both a challenge and an opportunity. Traditional methods involve leveraging high-dimensional latent spaces that, while effective, pose serious challenges in terms of interpretation and computational efficiency. In the quest for more streamlined and interpretable models, this paper introduces a novel advancement: Moment Pooling.

Moment Pooling: A Natural Extension to Deep Sets

Deep Sets, and by extension Energy Flow Networks (EFNs), offer a powerful framework for handling unordered sets of data. However, they inherently involve high-dimensional latent spaces. Moment Pooling emerges as a pivotal generalization of these architectures, enabling drastic reductions in latent space dimensionality without compromising on model performance. By extending the summation operation in Deep Sets to capture arbitrary multivariate moments, Moment Pooling leverages the inherent structure in the data, allowing for much more efficient learning and representation.

Moment Pooling's effectiveness is demonstrated through its application to EFNs in collider physics tasks, where it is shown to significantly outperform traditional methods in terms of efficiency and interpretability. Specifically, the implementation of Moment Pooling within EFNs, dubbed Moment EFNs, yields models that are not only equally or more accurate but also far less complex and easier to interpret.

Empirical Validation: Quark/Gluon Jet Classification

The efficacy of Moment Pooling is empirically validated through its application to the quark/gluon jet classification task. A series of Moment EFN models, with varying orders of moments and latent dimensions, were trained and evaluated. The results underscore the superiority of Moment EFNs; for a given latent dimension, increasing the order of moments (and thus, the effective latent dimension) consistently improves model performance. Remarkably, Moment EFNs with just a single latent dimension, when extended to higher-order moments, perform on par with traditional EFNs that employ significantly higher latent dimensions. This not only highlights the efficiency of Moment Pooling but also its potential in reducing computational costs and enhancing model interpretability.

Interpretability: Beyond Performance Metrics

A key advantage of Moment EFNs, and by extension, Moment Pooling, is their interpretability. For models with low latent dimensions, it becomes feasible to directly analyze and understand the learned latent representations. Such analyses reveal that Moment EFNs learn representations that bear a strong resemblance to known physical observables, thereby providing not just a powerful classification tool but also insights into the underlying physical processes.

Future Directions

Moment Pooling and Moment EFNs mark a significant step forward in machine learning applications, particularly in fields where interpretability and computational efficiency are paramount. Moving forward, the applicability of Moment Pooling extends beyond collider physics to any domain involving set-based data. Further exploration into the theoretical underpinnings of Moment Pooling, as well as its potential extensions and applications, remains a promising avenue for future research.

Conclusion

This paper introduced Moment Pooling, a powerful extension to Deep Sets architectures, and demonstrated its efficacy through the development and analysis of Moment EFNs. The results highlight Moment Pooling's ability to drastically reduce the complexity of machine learning models without sacrificing performance, thereby opening new doors for efficient and interpretable machine learning across various domains.