This proposal develops new algebro-geometric tools for data analysis, and in particular, for the investigation and understanding of massive data sets arising in computer science, where both the ambient dimension and number of samples may be large. Particular examples of interest to the team include brain-computer interface (BCI) problems and computer-vision and recognition problems. Specifically, the team is relating secant varieties (of arbitrary order) to the determination of optimal projections of data sets; they inquire what order of variety is appropriate and are investigating the relationship between varieties of different orders. Secant varieties also play a role in understanding canonical forms and decompositions of higher-order tensors, which in turn are of fundamental interest in signal-processing applications such as the BCI problem. Algebraic geometry provides a framework for developing algorithms for higher-order tensors. Finally, very recent developments in the Schubert calculus on Grassmannians have dramatically increased the potential applicability of this theory. The investigators are studying how to exploit these ideas to develop efficient algorithms for finding near-optimal projectors subject to constraints.
The research, which this team is carrying out, has distinctive impacts on the mathematical and computer science communities. By bringing together groups of researchers from seemingly disparate areas of expertise, it aims for cross-fertilization among these fields. Computer science provides the problems for which the team is developing new algorithms. Conversely, the team expects new mathematical conjectures to arise from the practical problems being addressed by this research. Furthermore, the graduate students (all working towards the Ph.D.) will receive innovative training during the course of this project that will prepare them for jobs in either academia or industry where they will have tools and preparation essentially unlike any of their peers. This should greatly enhance their ability to contribute new and innovative ideas to the team research environment. Beyond this effect on the professional communities, the applications driving the proposed research have clear and immediate impact on aspects of such disparate issues as national security and the broadening of the ability of disabled individuals to participate more fully in society.
