Parameter Estimation for Unnormalized Discrete Models via Empirically Localized Deformed Bregman Divergence

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.

We propose a continuous-time formulation of a noisy persistent contrastive divergence (PCD)-like method for maximum likelihood estimation (MLE) of unnormalised densities. Our approach couples parameter updates and sampling of the parametrised density in a multiscale system of stochastic differential equations (SDEs). From this formulation, we derive non-asymptotic bounds for weak test-function errors between the resulting numerical schemes and the MLE point target. The error is decomposed into numerical discretisation, slow-fast averaging, and finite-temperature concentration terms. We also introduce an efficient implementation based on explicit stabilized integrators and establish corresponding long-time error estimates. This leads to a novel method for training energy-based models (EBMs) with quantitative error guarantees.

Density estimation is a central primitive in probabilistic modeling, yet continuous, discrete, and mixed-variable domains are often treated by separate objectives, limiting the ability to exploit a common statistical structure across data types. Continuous score-based methods rely on log-density gradients, while discrete extensions typically use concrete score whose unbounded targets become unstable near low-probability states. We introduce Constant-Target Energy Matching (CTEM), a unified energy-based framework for density estimation on general state spaces. CTEM replaces ordinary density-ratio regression with a bounded energy-difference transform and derives from it a sample-only training objective with the constant target 1. The learned scalar potential recovers log p without partition-function estimation or explicit unbounded ratio regression. Across continuous, discrete, and mixed-variable benchmarks, CTEM substantially improves density estimation over competitive baselines and yields higher-quality samples under standard sampling procedures.