Recently, the ratio of probability density functions was demonstrated to be useful in solving various machine learning tasks such as outlier detection, non-stationarity adaptation, feature selection, and clustering. The key idea of this density ratio approach is that the ratio is directly estimated so that difficult density estimation is avoided. So far, parametric and non-parametric direct density ratio estimators with various loss functions have been developed, and the kernel least-squares method was demonstrated to be highly useful both in terms of accuracy and computational efficiency. On the other hand, recent study in pattern recognition exhibited that deep architectures such as a convolutional neural network can significantly outperform kernel methods. In this paper, we propose to use the convolutional neural network in density ratio estimation, and experimentally show that the proposed method tends to outperform the kernel-based method in outlying image detection.

Squared-loss mutual information (SMI) is a robust measure of the statistical dependence between random variables. The sample-based SMI approximator called least-squares mutual information (LSMI) was demonstrated to be useful in performing various machine learning tasks such as dimension reduction, clustering, and causal inference. The original LSMI approximates the pointwise mutual information by using the kernel model, which is a linear combination of kernel basis functions located on paired data samples. Although LSMI was proved to achieve the optimal approximation accuracy asymptotically, its approximation capability is limited when the sample size is small due to an insufficient number of kernel basis functions. Increasing the number of kernel basis functions can mitigate this weakness, but a naive implementation of this idea significantly increases the computation costs. In this article, we show that the computational complexity of LSMI with the multiplicative kernel model, which locates kernel basis functions on unpaired data samples and thus the number of kernel basis functions is the sample size squared, is the same as that for the plain kernel model. We experimentally demonstrate that LSMI with the multiplicative kernel model is more accurate than that with plain kernel models in small sample cases, with only mild increase in computation time.

Identifying the statistical independence of random variables is one of the important tasks in statistical data analysis. In this paper, we propose a novel non-parametric independence test based on a least-squares density ratio estimator. Our method, called least-squares independence test (LSIT), is distribution-free, and thus it is more flexible than parametric approaches. Furthermore, it is equipped with a model selection procedure based on cross-validation. This is a significant advantage over existing non-parametric approaches which often require manual parameter tuning. The usefulness of the proposed method is shown through numerical experiments.

Density ratio estimation has gathered a great deal of attention recently since it can be used for various data processing tasks. In this paper, we consider three methods of density ratio estimation: (A) the numerator and denominator densities are separately estimated and then the ratio of the estimated densities is computed, (B) a logistic regression classifier discriminating denominator samples from numerator samples is learned and then the ratio of the posterior probabilities is computed, and (C) the density ratio function is directly modeled and learned by minimizing the empirical Kullback-Leibler divergence. We first prove that when the numerator and denominator densities are known to be members of the exponential family, (A) is better than (B) and (B) is better than (C). Then we show that once the model assumption is violated, (C) is better than (A) and (B). Thus in practical situations where no exact model is available, (C) would be the most promising approach to density ratio estimation.

Estimating the conditional mean of an input-output relation is the goal of regression. However, regression analysis is not sufficiently informative if the conditional distribution has multi-modality, is highly asymmetric, or contains heteroscedastic noise. In such scenarios, estimating the conditional distribution itself would be more useful. In this paper, we propose a novel method of conditional density estimation that is suitable for multi-dimensional continuous variables. The basic idea of the proposed method is to express the conditional density in terms of the density ratio and the ratio is directly estimated without going through density estimation. Experiments using benchmark and robot transition datasets illustrate the usefulness of the proposed approach.