Symbolic Density Estimation: A Decompositional Approach

TL;DR

AI-Kolmogorov is a new framework for symbolic density estimation that decomposes problems, estimates densities, and uses symbolic regression to interpret complex distributions.

Contribution

It introduces a multi-stage pipeline combining clustering, density estimation, support estimation, and symbolic regression for density modeling.

Findings

Successfully applied to synthetic mixture models and normal distributions.

Able to discover underlying distributions and interpret complex mathematical expressions.

Demonstrated effectiveness on distributions motivated by high-energy physics.

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.