Multidimensional partitioning and bi-partitioning: analysis and application to gene expression datasets

Eigenvectors and, more generally, singular vectors, have proved to be useful tools for data mining and dimension reduction. Spectral clustering and reordering algorithms have been designed and implemented in many disciplines, and they can be motivated from several dierent standpoints. Here we give a general, unied, derivation from an applied linear algebra perspective. We use a variational approach that has the benet of (a) naturally introducing an appropriate scaling, (b) allowing for a solution in any desired dimension, and (c) dealing with both the clustering and bi-clustering issues in the same framework. The motivation and analysis is then backed up with examples involving two large data sets from modern, high-throughput, experimental cell biology. Here, the objects of interest are genes and tissue samples, and the experimental data represents gene activity. We show that looking beyond the dominant, or Fiedler, direction reveals important information.