Constructive Setting of the Density Ratio Estimation Problem and its Rigorous Solution
This work addresses a fundamental problem in machine learning for applications like domain adaptation and anomaly detection, but appears incremental as it builds on existing density ratio estimation frameworks.
The paper tackles the density ratio estimation problem by formulating it as a solution to a multidimensional integral equation with approximate components, and obtains a rigorous closed-form solution using a novel V-matrix that captures sample geometry, showing good potential in experiments compared to existing methods.
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.