Graph coarsening: From scientific computing to machine learning

The general method of graph coarsening or graph reduction has been a remarkably useful and ubiquitous tool in scientific computing and it is now just starting to have a similar impact in machine learning. The goal of this paper is to take a broad look into coarsening techniques that have been successfully deployed in scientific computing and see how similar principles are finding their way in more recent applications related to machine learning. In scientific computing, coarsening plays a central role in algebraic multigrid methods as well as the related class of multilevel incomplete LU factorizations. In machine learning, graph coarsening goes under various names, e.g., graph downsampling or graph reduction. Its goal in most cases is to replace some original graph by one which has fewer nodes, but whose structure and characteristics are similar to those of the original graph. As will be seen, a common strategy in these methods is to rely on spectral properties to define the coarse graph

Cerebral white matter analysis using diffusion imaging

Thesis (Ph. D.)--Harvard-MIT Division of Health Sciences and Technology, 2006.Includes bibliographical references (p. 183-198).In this thesis we address the whole-brain tractography segmentation problem. Diffusion magnetic resonance imaging can be used to create a representation of white matter tracts in the brain via a process called tractography. Whole brain tractography outputs thousands of trajectories that each approximate a white matter fiber pathway. Our method performs automatic organization, or segmention, of these trajectories into anatomical regions and gives automatic region correspondence across subjects. Our method enables both the automatic group comparison of white matter anatomy and of its regional diffusion properties, and the creation of consistent white matter visualizations across subjects. We learn a model of common white matter structures by analyzing many registered tractography datasets simultaneously. Each trajectory is represented as a point in a high-dimensional spectral embedding space, and common structures are found by clustering in this space. By annotating the clusters with anatomical labels, we create a model that we call a high-dimensional white matter atlas.(cont.) Our atlas creation method discovers structures corresponding to expected white matter anatomy, such as the corpus callosum, uncinate fasciculus, cingulum bundles, arcuate fasciculus, etc. We show how to extend the spectral clustering solution, stored in the atlas, using the Nystrom method to perform automatic segmentation of tractography from novel subjects. This automatic tractography segmentation gives an automatic region correspondence across subjects when all subjects are labeled using the atlas. We show the resulting automatic region correspondences, demonstrate that our clustering method is reproducible, and show that the automatically segmented regions can be used for robust measurement of fractional anisotropy.by Lauren Jean O'Donnell.Ph.D