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Data vectors are obtained from multiple domains. They are feature vectors ofimages or vector representations of words. Domains may have different numbersof data vectors with different dimensions. These data vectors from multipledomains are projected to a common space by linear transformations in order tosearch closely related vectors across domains. We would like to find projectionmatrices to minimize distances between closely related data vectors. Thisformulation of cross-domain matching is regarded as an extension of thespectral graph embedding to multi-domain setting, and it includes severalmultivariate analysis methods of statistics such as multiset canonicalcorrelation analysis, correspondence analysis, and principal componentanalysis. Similar approaches are very popular recently in pattern recognitionand vision. In this paper, instead of proposing a novel method, we willintroduce an embarrassingly simple idea of coding the data vectors forexplaining all the above mentioned approaches. A data vector is concatenatedwith zero vectors from all other domains to make an augmented vector. Thecross-domain matching is solved by applying the single-domain version ofspectral graph embedding to these augmented vectors of all the domains. Aninteresting connection to the classical associative memory model of neuralnetworks is also discussed by noticing a coding for association. Across-validation method for choosing the dimension of the common space and aregularization parameter will be discussed in an illustrative numericalexample.

A simple coding for cross-domain matching with dimension reduction via spectral graph embedding