Statistical analysis of kernel-based least-squares density-ratio estimation

The ratio of two probability densities can be used for solving various machine learning tasks such as covariate shift adaptation (importance sampling), outlier detection (likelihood-ratio test), feature selection (mutual information), and conditional probability estimation. Several methods of directly estimating the density ratio have recently been developed, e.g., moment matching estimation, maximum-likelihood density-ratio estimation, and least-squares density-ratio fitting. In this paper, we propose a kernelized variant of the least-squares method for density-ratio estimation, which is called kernel unconstrained least-squares importance fitting (KuLSIF). We investigate its fundamental statistical properties including a non-parametric convergence rate, an analytic-form solution, and a leave-one-out cross-validation score. We further study its relation to other kernel-based density-ratio estimators. In experiments, we numerically compare various kernel-based density-ratio estimati...

• Type: Journal
• Year: 2012
• Where: ML
• Authors: Takafumi Kanamori, Taiji Suzuki, Masashi Sugiyama
• Keywords: Fundamental Statistical Properties, Least Squares Method, Likelihood Ratio Test, Machine Learning, ML 2012