  Data vectors are obtained from multiple domains. They are feature vectors of
images or vector representations of words. Domains may have different numbers
of data vectors with different dimensions. These data vectors from multiple
domains are projected to a common space by linear transformations in order to
search closely related vectors across domains. We would like to find projection
matrices to minimize distances between closely related data vectors. This
formulation of cross-domain matching is regarded as an extension of the
spectral graph embedding to multi-domain setting, and it includes several
multivariate analysis methods of statistics such as multiset canonical
correlation analysis, correspondence analysis, and principal component
analysis. Similar approaches are very popular recently in pattern recognition
and vision. In this paper, instead of proposing a novel method, we will
introduce an embarrassingly simple idea of coding the data vectors for
explaining all the above mentioned approaches. A data vector is concatenated
with zero vectors from all other domains to make an augmented vector. The
cross-domain matching is solved by applying the single-domain version of
spectral graph embedding to these augmented vectors of all the domains. An
interesting connection to the classical associative memory model of neural
networks is also discussed by noticing a coding for association. A
cross-validation method for choosing the dimension of the common space and a
regularization parameter will be discussed in an illustrative numerical
example.
