The Kernel Mean Matching (KMM) is an elegant algorithm that produces density ratios between training and test data by minimizing their maximum mean discrepancy in a kernel space. The applicability of KMM to large-scale problems is however hindered by the quadratic complexity of calculating and storing the kernel matrices over training and test data. To address this problem, this paper proposes a novel ensemble algorithm for KMM, which divides test samples into smaller partitions, estimates a density ratio for each partition and then fuses these local estimates with a weighted sum. Our theoretical analysis shows that the ensemble KMM has a lower error bound than the centralized KMM, which uses all the test data at once to estimate the density ratio. Considering its suitability for distributed implementation, the proposed algorithm is also favorable in terms of time and space complexities. Experiments on benchmark datasets confirm the superiority of the proposed algorithm in terms of estimation accuracy and running time.

Covariate shift is a challenging problem in supervised learning that results from the discrepancy between the training and test distributions. An effective approach which recently drew a considerable attention in the research community is to reweight the training samples to minimize that discrepancy. In specific, many methods are based on developing Density-ratio (DR) estimation techniques that apply to both regression and classification problems. Although these methods work well for regression problems, their performance on classification problems is not satisfactory. This is due to a key observation that these methods focus on matching the sample marginal distributions without paying attention to preserving the separation between classes in the reweighted space. In this paper, we propose a novel method for Discriminative Density-ratio (DDR) estimation that addresses the aforementioned problem and aims at estimating the density ratio of joint distributions in a class-wise manner. The proposed algorithm is an iterative procedure that alternates between estimating the class information for the test data and estimating a new density ratio for each class. To incorporate the estimated class information of the test data, a soft matching technique is proposed. In addition, we employ an effective criterion which adopts mutual information as an indicator to stop the iterative procedure while resulting in a decision boundary that lies in a sparse region. Experiments on synthetic and benchmark datasets demonstrate the superiority of the proposed method.

Even though facial expressions have universal meaning in communications, their appearances show a large amount of variation due to many factors, such as different image acquisition setups, different ages, genders, and cultural backgrounds etc. Collecting enough amounts of annotated samples for each target domain is impractical, this paper investigates the problem of facial expression recognition in the more challenging situation, where the training and testing samples are taken from different domains. To address this problem, after observing the fact of unsatisfactory performance of the Kernel Mean Matching (KMM) algorithm, we propose a supervised extension that matches the distributions in a class-to-class manner, called Supervised Kernel Mean Matching (SKMM). The new approach stands out by taking into consideration both matching the distributions and preserving the discriminative information between classes at the same time. The extensive experimental studies on four cross-dataset facial expression recognition tasks show promising improvements of the proposed method, in which a small number of labeled samples guide the matching process.