  Biclustering is a method for detecting homogeneous submatrices in a given
observed matrix, and it is an effective tool for relational data analysis.
Although there are many studies that estimate the underlying bicluster
structure of a matrix, few have enabled us to determine the appropriate number
of biclusters in an observed matrix. Recently, a statistical test on the number
of biclusters has been proposed for a regular-grid bicluster structure, where
we assume that the latent bicluster structure can be represented by row-column
clustering. However, when the latent bicluster structure does not satisfy such
regular-grid assumption, the previous test requires a larger number of
biclusters than necessary (i.e., a finer bicluster structure than necessary)
for the null hypothesis to be accepted, which is not desirable in terms of
interpreting the accepted bicluster structure. In this study, we propose a new
statistical test on the number of biclusters that does not require the
regular-grid assumption and derive the asymptotic behavior of the proposed test
statistic in both null and alternative cases. We illustrate the effectiveness
of the proposed method by applying it to both synthetic and practical
relational data matrices.
