# Constructive Setting of the Density Ratio Estimation Problem and its Rigorous Solution

arXiv: https://arxiv.org/abs/1306.0407 · HTML: https://mlsanity.com/paper/1306.0407/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Jun 2013.

## Built on by (most cited first, 4 of 4)

- [Featurized Density Ratio Estimation](https://mlsanity.com/paper/2107.02212/) (2021)
- [LNL+K: Enhancing Learning with Noisy Labels Through Noise Source Knowledge Integration](https://mlsanity.com/paper/2306.11911/) (2023)
- [Noise-Aware Generalization: Robustness to In-Domain Noise and Out-of-Domain Generalization](https://mlsanity.com/paper/2504.02996/) (2025)
- [Unsupervised Change Detection using DRE-CUSUM](https://mlsanity.com/paper/2201.11678/) (2022)

## Abstract (arXiv)

We introduce a general constructive setting of the density ratio estimation problem as a solution of a (multidimensional) integral equation. In this equation, not only its right hand side is known approximately, but also the integral operator is defined approximately. We show that this ill-posed problem has a rigorous solution and obtain the solution in a closed form. The key element of this solution is the novel V-matrix, which captures the geometry of the observed samples. We compare our method with three well-known previously proposed ones. Our experimental results demonstrate the good potential of the new approach.