# A Spectral Framework for Closed-Form Relative Density Estimation

arXiv: https://arxiv.org/abs/2605.10668 · HTML: https://mlsanity.com/paper/2605.10668/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 0 of its 76 references that are in the map.
- First posted on arXiv 11 May 2026.

## Built on by (most cited first, 1 of 1)

- [Regularized Variational and Spectral Log-Density-Ratio Estimation in the Gaussian Location Model](https://mlsanity.com/paper/2607.01895/) (2026)

## Abstract (arXiv)

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.