Sparse Reduced-Rank Regression for Simultaneous Dimension Reduction and Variable Selection

We have determined how most of the transcriptional regulators encoded in the eukaryote Saccharomyces cerevisiae associate with genes across the genome in living cells. Just as maps of metabolic networks describe the potential pathways that may be used by a cell to accomplish metabolic processes, this network of regulator-gene interactions describes potential pathways yeast cells can use to regulate global gene expression programs. We use this information to identify network motifs, the simplest units of network architecture, and demonstrate that an automated process can use motifs to assemble a transcriptional regulatory network structure. Our results reveal that eukaryotic cellular functions are highly connected through networks of transcriptional regulators that regulate other transcriptional regulators.

In this paper we develop new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds. These manifolds represent the constraints that arise in such areas as the symmetric eigenvalue problem, nonlinear eigenvalue problems, electronic structures computations, and signal processing. In addition to the new algorithms, we show how the geometrical framework gives penetrating new insights allowing us to create, understand, and compare algorithms. The theory proposed here provides a taxonomy for numerical linear algebra algorithms that provide a top level mathematical view of previously unrelated algorithms. It is our hope that developers of new algorithms and perturbation theories will benefit from the theory, methods, and examples in this paper.