Symbolic Density Estimation: A Decompositional Approach

2026-03-29

AI Summary

This work proposes the AI-Kolmogorov framework, which introduces symbolic regression systematically into probability density estimation to address the Symbolic Density Estimation (SymDE) problem. The method decomposes complex distributions through clustering or probabilistic graphical models in a multi-stage process, sequentially integrating support set estimation, nonparametric density estimation, and symbolic regression to construct interpretable analytic expressions of probability densities. Evaluated on synthetic mixture models, multivariate normal distributions, and exotic distributions from high-energy physics, the framework successfully recovers or uncovers their underlying mathematical structures, enabling both interpretable modeling and structural discovery for complex probability distributions.

Technology Category

Machine Learning: Probabilistic Circuits and Graphical ModelsReasoning under Uncertainty: Relational Probabilistic ModelsCognitive Modeling & Cognitive Systems: Symbolic Representations

Application Category

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 semanticsGraph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphsUser Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

We introduce AI-Kolmogorov, a novel framework for Symbolic Density Estimation (SymDE). Symbolic regression (SR) has been effectively used to produce interpretable models in standard regression settings but its applicability to density estimation tasks has largely been unexplored. To address the SymDE task we introduce a multi-stage pipeline: (i) problem decomposition through clustering and/or probabilistic graphical model structure learning; (ii) nonparametric density estimation; (iii) support estimation; and finally (iv) SR on the density estimate. We demonstrate the efficacy of AI-Kolmogorov on synthetic mixture models, multivariate normal distributions, and three exotic distributions, two of which are motivated by applications in high-energy physics. We show that AI-Kolmogorov can discover underlying distributions or otherwise provide valuable insight into the mathematical expressions describing them.

Problem

• Symbolic Density Estimation
• Symbolic Regression
• Density Estimation
• Interpretable Models
• Probabilistic Modeling

Innovation

• Symbolic Density Estimation
• Symbolic Regression
• Problem Decomposition
• Nonparametric Density Estimation
• Interpretable Modeling