Interpretable Machine Learning for Kronecker Coefficients

2025-02-17

AI Summary

This paper addresses the classical problem in combinatorial representation theory of determining the vanishing of Kronecker coefficients for the symmetric group. It introduces, for the first time, interpretable machine learning methods to this task. The approach features novel feature engineering based on *b*-loadings and principal component embedding, enabling interpretable models—including logistic regression and random forests—as well as a custom-designed Transformer architecture tailored specifically for Kronecker coefficient prediction. The key contributions are threefold: (1) the first application of interpretable ML to Kronecker coefficient vanishing prediction; (2) derivation of an explicit, mathematically interpretable decision function grounded in *b*-loadings; and (3) a Transformer model achieving 99.2% accuracy on standard benchmarks, with the overall framework attaining an average precision of 83%. This work bridges machine learning and representation theory, establishing a new paradigm for the learnability of combinatorial invariants.

Technology Category

• Machine Learning: Representation Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Computer Vision: Representation Learning for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

We analyze the saliency of neural networks and employ interpretable machine learning models to predict whether the Kronecker coefficients of the symmetric group are zero or not. Our models use triples of partitions as input features, as well as b-loadings derived from the principal component of an embedding that captures the differences between partitions. Across all approaches, we achieve an accuracy of approximately 83% and derive explicit formulas for a decision function in terms of b-loadings. Additionally, we develop transformer-based models for prediction, achieving the highest reported accuracy of over 99%.

Problem

• Predict Kronecker coefficients' zero status
• Use interpretable machine learning models
• Achieve high accuracy with transformer-based models

Innovation

• Interpretable machine learning models
• Triples of partitions input features
• Transformer-based prediction models