# Diffusion Secant Alignment for Score-Based Density Ratio Estimation

arXiv: https://arxiv.org/abs/2509.04852 · HTML: https://mlsanity.com/paper/2509.04852/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 48 references that are in the map.
- First posted on arXiv 5 Sep 2025.

## Builds on (most cited first, 30 of 32)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [Improved Techniques for Training Consistency Models](https://mlsanity.com/paper/2310.14189/) (2023)
- [Guiding a Diffusion Model with a Bad Version of Itself](https://mlsanity.com/paper/2406.02507/) (2024)
- [Telescoping Density-Ratio Estimation](https://mlsanity.com/paper/2006.12204/) (2020)
- [Minimizing Trajectory Curvature of ODE-based Generative Models](https://mlsanity.com/paper/2301.12003/) (2023)
- [Density Ratio Estimation via Infinitesimal Classification](https://mlsanity.com/paper/2111.11010/) (2021)
- [Transfer Adaptation Learning: A Decade Survey](https://mlsanity.com/paper/1903.04687/) (2019)
- [Beyond Normal: On the Evaluation of Mutual Information Estimators](https://mlsanity.com/paper/2306.11078/) (2023)
- [Featurized Density Ratio Estimation](https://mlsanity.com/paper/2107.02212/) (2021)
- [On Numerical Integration in Neural Ordinary Differential Equations](https://mlsanity.com/paper/2206.07335/) (2022)
- [Variational Representations and Neural Network Estimation of R'enyi Divergences](https://mlsanity.com/paper/2007.03814/) (2020)
- [Adaptive Multi-stage Density Ratio Estimation for Learning Latent Space Energy-based Model](https://mlsanity.com/paper/2209.08739/) (2022)
- [Estimating the Density Ratio between Distributions with High Discrepancy using Multinomial Logistic Regression](https://mlsanity.com/paper/2305.00869/) (2023)
- [Trimmed Density Ratio Estimation](https://mlsanity.com/paper/1703.03216/) (2017)
- [Copula Density Neural Estimation](https://mlsanity.com/paper/2211.15353/) (2022)
- [Density Ratio Estimation with Conditional Probability Paths](https://mlsanity.com/paper/2502.02300/) (2025)
- [Dequantified Diffusion-Schr"odinger Bridge for Density Ratio Estimation](https://mlsanity.com/paper/2505.05034/) (2025)
- [Knowledge-augmented Deep Learning and Its Applications: A Survey](https://mlsanity.com/paper/2212.00017/) (2022)
- [On regularized Radon-Nikodym differentiation](https://mlsanity.com/paper/2308.07887/) (2023)
- [Learning to Integrate Diffusion ODEs by Averaging the Derivatives](https://mlsanity.com/paper/2505.14502/) (2025)
- [Accelerating Neural ODEs Using Model Order Reduction](https://mlsanity.com/paper/2105.14070/) (2021)
- [Adaptive learning of density ratios in RKHS](https://mlsanity.com/paper/2307.16164/) (2023)
- [Density Ratio Estimation via Sampling along Generalized Geodesics on Statistical Manifolds](https://mlsanity.com/paper/2406.18806/) (2024)
- [Robust Multi-Agent Reinforcement Learning by Mutual Information Regularization](https://mlsanity.com/paper/2310.09833/) (2023)
- [Overcoming Saturation in Density Ratio Estimation by Iterated Regularization](https://mlsanity.com/paper/2402.13891/) (2024)
- [Projection Pursuit Density Ratio Estimation](https://mlsanity.com/paper/2506.00866/) (2025)

## Built on by (most cited first, 3 of 3)

- [A Minimum Variance Path Principle for Accurate and Stable Score-Based Density Ratio Estimation](https://mlsanity.com/paper/2602.00834/) (2026)
- [Learning Energy-Based Models from Stochastic Interpolants using Spatiotemporal Differences](https://mlsanity.com/paper/2605.26850/) (2026)
- [One-Step Score-Based Density Ratio Estimation](https://mlsanity.com/paper/2604.10672/) (2026)

## Abstract (arXiv)

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.