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

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