From Kernels to Attention: A Transformer Framework for Density and Score Estimation

• URL: http://arxiv.org/abs/2511.05924v1
• Date: Sat, 08 Nov 2025 08:38:37 GMT
• Title: From Kernels to Attention: A Transformer Framework for Density and Score Estimation
• Authors: Vasily Ilin, Peter Sushko,
• Abstract summary: We introduce a unified attention-based framework for joint score and density estimation.<n>We develop a permutation- and affine-equivariant transformer that estimates both the probability density $f(x)$ and its score $nabla_x log f(x)$ directly from i.i.d. samples.
• Score: 0.47745223151611654
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We introduce a unified attention-based framework for joint score and density estimation. Framing the problem as a sequence-to-sequence task, we develop a permutation- and affine-equivariant transformer that estimates both the probability density $f(x)$ and its score $\nabla_x \log f(x)$ directly from i.i.d. samples. Unlike traditional score-matching methods that require training a separate model for each distribution, our approach learns a single distribution-agnostic operator that generalizes across densities and sample sizes. The architecture employs cross-attention to connect observed samples with arbitrary query points, enabling generalization beyond the training data, while built-in symmetry constraints ensure equivariance to permutation and affine transformations. Analytically, we show that the attention weights can recover classical kernel density estimation (KDE), and verify it empirically, establishing a principled link between classical KDE and the transformer architecture. Empirically, the model achieves substantially lower error and better scaling than KDE and score-debiased KDE (SD-KDE), while exhibiting better runtime scaling. Together, these results establish transformers as general-purpose, data-adaptive operators for nonparametric density and score estimation.

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