| This thesis deals with the rigorous application of nonlinear dimension reduction and data organization techniques to biomedical data analysis. The Laplacian Eigenmaps algorithm is representative of these methods and has been widely applied in manifold learning and related areas. While their asymptotic manifold recovery behavior has been well-characterized, the clustering properties of Laplacian embeddings with finite data are largely motivated by heuristic arguments. We develop a precise bound, characterizing cluster structure preservation under Laplacian embeddings. From this foundation, we introduce flexible and mathematically well-founded approaches for information fusion and feature representation. These methods are applied to three substantial case studies in bioinformatics, illustrating their capacity to extract scientifically valuable information from complex data. |
