  Biclustering is a powerful data mining technique that allows simultaneously
clustering rows (observations) and columns (features) in a matrix-format data
set, which can provide results in a checkerboard-like pattern for visualization
and exploratory analysis in a wide array of domains. Multiple biclustering
algorithms have been developed in the past two decades, among which the convex
biclustering can guarantee a global optimum by formulating in as a convex
optimization problem. On the other hand, the application of biclustering has
not progressed in parallel with the algorithm techniques. For example,
biclustering for increasingly popular microbiome research data is under-applied
possibly due to its compositional constraints for each sample. In this
manuscript, we propose a new convex biclustering algorithm, called the bi-ADMM,
under general setups based on the ADMM algorithm, which is free of extra
smoothing steps to visualize informative biclusters required by existing convex
biclustering algorithms. Furthermore, we tailor it to the algorithm named
biC-ADMM specifically to tackle compositional constraints confronted in
microbiome data. The key step of our methods utilizes the Sylvester Equation to
derive the ADMM algorithm, which is new to the clustering research. The
effectiveness of the proposed methods is examined through a variety of
numerical experiments and a microbiome data application.
