# Density Ratio Estimation via Infinitesimal Classification

arXiv: https://arxiv.org/abs/2111.11010 · HTML: https://mlsanity.com/paper/2111.11010/

## Key facts

- Cited by 61 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 66 references that are in the map.
- First posted on arXiv 22 Nov 2021.

## Builds on (most cited first, 30 of 31)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [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)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Group Normalization](https://mlsanity.com/paper/1803.08494/) (2018)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [MINE: Mutual Information Neural Estimation](https://mlsanity.com/paper/1801.04062/) (2018)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [On Variational Bounds of Mutual Information](https://mlsanity.com/paper/1905.06922/) (2019)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation](https://mlsanity.com/paper/1602.03253/) (2016)
- [Formal Limitations on the Measurement of Mutual Information](https://mlsanity.com/paper/1811.04251/) (2018)
- [Understanding the Limitations of Variational Mutual Information Estimators](https://mlsanity.com/paper/1910.06222/) (2019)
- [Telescoping Density-Ratio Estimation](https://mlsanity.com/paper/2006.12204/) (2020)
- [NeuTra-lizing Bad Geometry in Hamiltonian Monte Carlo Using Neural Transport](https://mlsanity.com/paper/1903.03704/) (2019)
- [Learning Weighted Representations for Generalization Across Designs](https://mlsanity.com/paper/1802.08598/) (2018)
- [Neural Bridge Sampling for Evaluating Safety-Critical Autonomous Systems](https://mlsanity.com/paper/2008.10581/) (2020)
- [Featurized Density Ratio Estimation](https://mlsanity.com/paper/2107.02212/) (2021)
- [Conditional Noise-Contrastive Estimation of Unnormalised Models](https://mlsanity.com/paper/1806.03664/) (2018)
- [Accurate and Conservative Estimates of MRF Log-likelihood using Reverse Annealing](https://mlsanity.com/paper/1412.8566/) (2014)

## Built on by (most cited first, 30 of 61)

- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Data Selection for Language Models via Importance Resampling](https://mlsanity.com/paper/2302.03169/) (2023)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Concrete Score Matching: Generalized Score Matching for Discrete Data](https://mlsanity.com/paper/2211.00802/) (2022)
- [Action Matching: Learning Stochastic Dynamics from Samples](https://mlsanity.com/paper/2210.06662/) (2022)
- [Improved Techniques for Maximum Likelihood Estimation for Diffusion ODEs](https://mlsanity.com/paper/2305.03935/) (2023)
- [Compositional Score Modeling for Simulation-based Inference](https://mlsanity.com/paper/2209.14249/) (2022)
- [Learning normalized image densities via dual score matching](https://mlsanity.com/paper/2506.05310/) (2025)
- [Estimating the Density Ratio between Distributions with High Discrepancy using Multinomial Logistic Regression](https://mlsanity.com/paper/2305.00869/) (2023)
- [The Data Addition Dilemma](https://mlsanity.com/paper/2408.04154/) (2024)
- [Density Ratio Estimation with Conditional Probability Paths](https://mlsanity.com/paper/2502.02300/) (2025)
- [Constraining Generative Models for Engineering Design with Negative Data](https://mlsanity.com/paper/2306.15166/) (2023)
- [Annealing Flow Generative Models Towards Sampling High-Dimensional and Multi-Modal Distributions](https://mlsanity.com/paper/2409.20547/) (2024)
- [Dequantified Diffusion-Schr"odinger Bridge for Density Ratio Estimation](https://mlsanity.com/paper/2505.05034/) (2025)
- [Data Retrieval with Importance Weights for Few-Shot Imitation Learning](https://mlsanity.com/paper/2509.01657/) (2025)
- [Neural Sinkhorn Gradient Flow](https://mlsanity.com/paper/2401.14069/) (2024)
- [Computing high-dimensional optimal transport by flow neural networks](https://mlsanity.com/paper/2305.11857/) (2023)
- [Density Ratio Estimation via Sampling along Generalized Geodesics on Statistical Manifolds](https://mlsanity.com/paper/2406.18806/) (2024)
- [Your copula is a classifier in disguise: classification-based copula density estimation](https://mlsanity.com/paper/2411.03014/) (2024)
- [Energy based diffusion generator for efficient sampling of Boltzmann distributions](https://mlsanity.com/paper/2401.02080/) (2024)
- [Synthetic Tabular Data Validation: A Divergence-Based Approach](https://mlsanity.com/paper/2405.07822/) (2024)
- [BoltzNCE: Learning Likelihoods for Boltzmann Generation with Stochastic Interpolants and Noise Contrastive Estimation](https://mlsanity.com/paper/2507.00846/) (2025)
- [Statistical applications of contrastive learning](https://mlsanity.com/paper/2204.13999/) (2022)
- [Your Diffusion Model is Secretly a Noise Classifier and Benefits from Contrastive Training](https://mlsanity.com/paper/2407.08946/) (2024)
- [Projection Pursuit Density Ratio Estimation](https://mlsanity.com/paper/2506.00866/) (2025)
- [Approximate Data Deletion in Generative Models](https://mlsanity.com/paper/2206.14439/) (2022)
- [Hindsight-DICE: Stable Credit Assignment for Deep Reinforcement Learning](https://mlsanity.com/paper/2307.11897/) (2023)
- [Optimal Aggregation of Prediction Intervals under Unsupervised Domain Shift](https://mlsanity.com/paper/2405.10302/) (2024)
- [General Frameworks for Conditional Two-Sample Testing](https://mlsanity.com/paper/2410.16636/) (2024)
- [FMMI: Flow Matching Mutual Information Estimation](https://mlsanity.com/paper/2511.08552/) (2025)

## Abstract (arXiv)

Density ratio estimation (DRE) is a fundamental machine learning technique for comparing two probability distributions. However, existing methods struggle in high-dimensional settings, as it is difficult to accurately compare probability distributions based on finite samples. In this work we propose DRE-∞, a divide-and-conquer approach to reduce DRE to a series of easier subproblems. Inspired by Monte Carlo methods, we smoothly interpolate between the two distributions via an infinite continuum of intermediate bridge distributions. We then estimate the instantaneous rate of change of the bridge distributions indexed by time (the "time score") -- a quantity defined analogously to data (Stein) scores -- with a novel time score matching objective. Crucially, the learned time scores can then be integrated to compute the desired density ratio. In addition, we show that traditional (Stein) scores can be used to obtain integration paths that connect regions of high density in both distributions, improving performance in practice. Empirically, we demonstrate that our approach performs well on downstream tasks such as mutual information estimation and energy-based modeling on complex, high-dimensional datasets.