We propose an unsupervised object matching method for relational data, which finds matchings between objects in different relational datasets without correspondence information. For example, the proposed method matches documents in different languages in multi-lingual document-word networks without dictionaries nor alignment information. The proposed method assumes that each object has latent vectors, and the probability of neighbor objects is modeled by the inner-product of the latent vectors, where the neighbors are generated by short random walks over the relations. The latent vectors are estimated by maximizing the likelihood of the neighbors for each dataset. The estimated latent vectors contain hidden structural information of each object in the given relational dataset. Then, the proposed method linearly projects the latent vectors for all the datasets onto a common latent space shared across all datasets by matching the distributions while preserving the structural information. The projection matrix is estimated by minimizing the distance between the latent vector distributions with an orthogonality regularizer. To represent the distributions effectively, we use the kernel embedding of distributions that hold high-order moment information about a distribution as an element in a reproducing kernel Hilbert space, which enables us to calculate the distance between the distributions without density estimation. The structural information encoded in the latent vectors are preserved by using the orthogonality regularizer. We demonstrate the effectiveness of the proposed method with experiments using real-world multi-lingual document-word relational datasets and multiple user-item relational datasets.
The kernel embeddings of distributions and the MMD have been successfully used for various applications, such as a statistical independence test @cite_38 @cite_39 @cite_0 , discriminative learning on probability measures @cite_31 , anomaly detection for group data @cite_41 , density estimation @cite_40 , a three-variable interaction test @cite_27 , a goodness of fit test @cite_10 , supervised object matching @cite_43 , domain adaptation @cite_6 , and deep generative modeling @cite_19 @cite_15 .
@cite_38 @cite_15 @cite_41 @cite_6 @cite_39 @cite_0 @cite_43 @cite_40 @cite_27 @cite_19 @cite_31 @cite_10
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Given samples from distributions p and q, a two-sample test determines whether to reject the null hypothesis that p = q, based on the value of a test statistic measuring the distance between the samples. One choice of test statistic is the maximum mean discrepancy (MMD), which is a distance between embeddings of the probability distributions in a reproducing kernel Hilbert space. The kernel used in obtaining these embeddings is critical in ensuring the test has high power, and correctly distinguishes unlike distributions with high probability. A means of parameter selection for the two-sample test based on the MMD is proposed. For a given test level (an upper bound on the probability of making a Type I error), the kernel is chosen so as to maximize the test power, and minimize the probability of making a Type II error. The test statistic, test threshold, and optimization over the kernel parameters are obtained with cost linear in the sample size. These properties make the kernel selection and test procedures suited to data streams, where the observations cannot all be stored in memory. In experiments, the new kernel selection approach yields a more powerful test than earlier kernel selection heuristics. The advantages of discriminative learning algorithms and kernel machines are combined with generative modeling using a novel kernel between distributions. In the probability product kernel, data points in the input space are mapped to distributions over the sample space and a general inner product is then evaluated as the integral of the product of pairs of distributions. The kernel is straightforward to evaluate for all exponential family models such as multinomials and Gaussians and yields interesting nonlinear kernels. Furthermore, the kernel is computable in closed form for latent distributions such as mixture models, hidden Markov models and linear dynamical systems. For intractable models, such as switching linear dynamical systems, structured mean-field approximations can be brought to bear on the kernel evaluation. For general distributions, even if an analytic expression for the kernel is not feasible, we show a straightforward sampling method to evaluate it. Thus, the kernel permits discriminative learning methods, including support vector machines, to exploit the properties, metrics and invariances of the generative models we infer from each datum. Experiments are shown using multinomial models for text, hidden Markov models for biological data sets and linear dynamical systems for time series data. We consider training a deep neural network to generate samples from an unknown distribution given i.i.d. data. We frame learning as an optimization minimizing a two-sample test statistic—informally speaking, a good generator network produces samples that cause a two-sample test to fail to reject the null hypothesis. As our two-sample test statistic, we use an unbiased estimate of the maximum mean discrepancy, which is the centerpiece of the nonparametric kernel two-sample test proposed by [2]. We compare to the adversarial nets framework introduced by [1], in which learning is a two-player game between a generator network and an adversarial discriminator network, both trained to outwit the other. From this perspective, the MMD statistic plays the role of the discriminator. In addition to empirical comparisons, we prove bounds on the generalization error incurred by optimizing the empirical MMD. 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]