DiScoFormer: Plug-In Density and Score Estimation with Transformers

TL;DR

DiScoFormer is a versatile Transformer model that estimates probability densities and scores from samples, generalizing across distributions and sample sizes, and outperforms traditional kernel methods in accuracy and convergence speed.

Contribution

It introduces a train-once, infer-anywhere Transformer that unifies density and score estimation, with theoretical and empirical evidence of its effectiveness and generalization capabilities.

Findings

Outperforms KDE in density estimation accuracy

Faster convergence compared to KDE

Provides high-fidelity score estimates for various applications

Abstract

Estimating probability density and its score from samples remains a core problem in generative modeling, Bayesian inference, and kinetic theory. Existing methods are bifurcated: classical kernel density estimators (KDE) generalize across distributions but suffer from the curse of dimensionality, while modern neural score models achieve high precision but require retraining for every target distribution. We introduce DiScoFormer (Density and Score Transformer), a ``train-once, infer-anywhere" equivariant Transformer that maps i.i.d. samples to both density values and score vectors, generalizing across distributions and sample sizes. Analytically, we prove that self-attention can recover normalized KDE, establishing it as a functional generalization of kernel methods; empirically, individual attention heads learn multi-scale, kernel-like behaviors. The model converges faster and achieves higher precision than KDE for density estimation, and provides a high-fidelity plug-in score oracle for score-debiased KDE, Fisher information computation, and Fokker-Planck-type PDEs.