Diffusion Secant Alignment for Score-Based Density Ratio Estimation

Estimating density ratios has become increasingly important with the recent rise of score-based and diffusion-inspired methods. However, current tangent-based approaches rely on a high-variance learning objective, which leads to unstable training and costly numerical integration during inference. We propose textitInterval-annealed Secant Alignment Density Ratio Estimation (ISA-DRE), a score-based framework along diffusion interpolants that replaces the instantaneous tangent with its interval integral, the secant, as the learning target. We show theoretically that the secant is a provably lower variance and smoother target for neural approximation, and also a strictly more general representation that contains the tangent as the infinitesimal limit. To make secant learning feasible, we introduce the textitSecant Alignment Identity (SAI) to enforce self consistency between secant and tangent representations, and textitContraction Interval Annealing (CIA) to ensure stable convergence. Empirically, this stability-first formulation produces high efficiency and accuracy. ISA-DRE achieves comparable or superior results with fewer function evaluations, demonstrating robustness under large distribution discrepancies and effectively mitigating the density-chasm problem.