A Spectral Framework for Closed-Form Relative Density Estimation

Francis Bach

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arXiv · 2605.10668

A Spectral Framework for Closed-Form Relative Density Estimation

Francis Bach

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Abstract

We propose a closed-form spectral framework for relative log-density estimation in linearly parameterized probabilistic models, including unnormalized and conditional models. This is achieved by representing the Kullback-Leibler (KL) divergence as an integral of weighted chi-squared divergences, converting KL estimation into a family of least-squares problems. We derive an explicit spectral formula based only on first- and second-order feature moments, yielding closed-form estimators of both divergences and log-density potentials for fixed features. The framework extends to a broad class of f-divergences and can be combined with kernelization or feature learning with neural networks. We prove convergence guarantees for the resulting estimators and empirically compare them on synthetic data with optimization-based variational formulations, including logistic and softmax regression for normalized conditional models.

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Francis Bach. 2026-05-11. A Spectral Framework for Closed-Form Relative Density Estimation. https://arxiv.org/abs/2605.10668

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