Title: Application of fiber bundle structure in data analysis --

alternating diffusion and vector nonlocal Euclidean median

Abstract: Understanding the intrinsic structure, which is often nonlinear and complex, from a given massive dataset is a common challenge shared in almost all scientific fields. The manifold model is a flexible framework that has led to several convincing results. We will discuss a generalization of the manifold model by taking the bundle structure into account, and show two current progresses in data analysis. The first one is the common manifold model and alternating diffusion, which aim to deal with the sensor fusion problem. Its application to the sleep dynamics analysis will be shown. The second one is the patch space model and vector nonlocal median for image denoising problem, which is a generalization of the nonlocal median. Theoretical justifications will be provided, particularly the asymptotical analysis of the alternating diffusion, the finite dimensional spectral embedding theory, and the statistical arguments for the patch space.