In domain adaptation, maximum mean discrepancy (MMD) has been widely adopted as a discrepancy metric between the distributions of source and target domains. However, existing MMD-based domain adaptation methods generally ignore the changes of class prior distributions, i.e., class weight bias across domains. This remains an open problem but ubiquitous for domain adaptation, which can be caused by changes in sample selection criteria and application scenarios. We show that MMD cannot account for class weight bias and results in degraded domain adaptation performance. To address this issue, a weighted MMD model is proposed in this paper. Specifically, we introduce class-specific auxiliary weights into the original MMD for exploiting the class prior probability on source and target domains, whose challenge lies in the fact that the class label in target domain is unavailable. To account for it, our proposed weighted MMD model is defined by introducing an auxiliary weight for each class in the source domain, and a classification EM algorithm is suggested by alternating between assigning the pseudo-labels, estimating auxiliary weights and updating model parameters. Extensive experiments demonstrate the superiority of our weighted MMD over conventional MMD for domain adaptation.
Due to the unavailability of labels in the target domain, one commonly used strategy of UDA is to learn domain invariant representation via minimizing the domain distribution discrepancy. Maximum Mean Discrepancy (MMD) is an effective non-parametric metric for comparing the distributions based on two sets of data @cite_2 . Given two distributions @math and @math , by mapping the data to a reproducing kernel Hilbert space (RKHS) using function @math , the MMD between @math and @math is defined as, where @math denotes the expectation with regard to the distribution , and @math defines a set of functions in the unit ball of a RKHS @math . Based on the statistical tests defined by MMD, we have @math iff @math . Denote by @math and @math two sets of samples drawn from the distributions @math and @math , respectively. An empirical estimate of MMD can be given by @cite_33 , where @math denotes the feature map associated with the kernel map @math . @math is usually defined as the convex combination of @math basis kernels @math @cite_9 ,
@cite_9 @cite_33 @cite_2
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Recent studies reveal that a deep neural network can learn transferable features which generalize well to novel tasks for domain adaptation. However, as deep features eventually transition from general to specific along the network, the feature transferability drops significantly in higher layers with increasing domain discrepancy. Hence, it is important to formally reduce the dataset bias and enhance the transferability in task-specific layers. In this paper, we propose a new Deep Adaptation Network (DAN) architecture, which generalizes deep convolutional neural network to the domain adaptation scenario. In DAN, hidden representations of all task-specific layers are embedded in a reproducing kernel Hilbert space where the mean embeddings of different domain distributions can be explicitly matched. The domain discrepancy is further reduced using an optimal multi-kernel selection method for mean embedding matching. DAN can learn transferable features with statistical guarantees, and can scale linearly by unbiased estimate of kernel embedding. Extensive empirical evidence shows that the proposed architecture yields state-of-the-art image classification error rates on standard domain adaptation benchmarks. We propose a framework for analyzing and comparing distributions, which we use to construct statistical tests to determine if two samples are drawn from different distributions. Our test statistic is the largest difference in expectations over functions in the unit ball of a reproducing kernel Hilbert space (RKHS), and is called the maximum mean discrepancy (MMD).We present two distribution free tests based on large deviation bounds for the MMD, and a third test based on the asymptotic distribution of this statistic. The MMD can be computed in quadratic time, although efficient linear time approximations are available. Our statistic is an instance of an integral probability metric, and various classical metrics on distributions are obtained when alternative function classes are used in place of an RKHS. We apply our two-sample tests to a variety of problems, including attribute matching for databases using the Hungarian marriage method, where they perform strongly. Excellent performance is also obtained when comparing distributions over graphs, for which these are the first such tests. Motivation: Many problems in data integration in bioinformatics can be posed as one common question: Are two sets of observations generated by the same distribution? We propose a kernel-based statistical test for this problem, based on the fact that two distributions are different if and only if there exists at least one function having different expectation on the two distributions. Consequently we use the maximum discrepancy between function means as the basis of a test statistic. The Maximum Mean Discrepancy (MMD) can take advantage of the kernel trick, which allows us to apply it not only to vectors, but strings, sequences, graphs, and other common structured data types arising in molecular biology. Results: We study the practical feasibility of an MMD-based test on three central data integration tasks: Testing cross-platform comparability of microarray data, cancer diagnosis, and data-content based schema matching for two different protein function classification schemas. In all of these experiments, including high-dimensional ones, MMD is very accurate in finding samples that were generated from the same distribution, and outperforms its best competitors. Conclusions: We have defined a novel statistical test of whether two samples are from the same distribution, compatible with both multivariate and structured data, that is fast, easy to implement, and works well, as confirmed by our experiments. Availability: Contact: [email protected]