Collaborative Research: CDS&E-MSS: Robust Algorithms for Interpolation and Extrapolation in Manifold Learning

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The objective of this proposal is to develop robust algorithms for reconstructing or synthesizing highly structured high-dimensional data based on a low-dimensional representation learned from a training dataset, i.e., the interpolation and extrapolation problems in manifold learning. The project will address the elusive issue of computing a usually not well-defined low-dimensional parametrization in the setting of various interpolation and extrapolation problems for manifold learning, emphasizing the notion of physically meaningful paramterizations. It will develop innovative computational methodology for flexibly learning a low-dimensional parametrization together with other physically important variables in the context of both unsupervised and semi-supervised learning and especially active learning settings, for learning and synthesis of dynamic data, and for manifold extrapolation based on transfer learning. Included in the project is a development of a publicly available software package which will disseminate the research results and promote applications of nonlinear dimension reduction methodology to real-world problems.The discoveries from this proposed research are expected to impact a wide range of areas of applications. Computing compact representation of high-dimensional data represents a very challenging statistical learning problem, and manifold learning has become a very active research field aiming at discovering hidden structures from the statistical and geometric regularity inherent in many high-dimensional data. Reconstruction and synthesis of high-dimensional data in the context of interpolation and extrapolation will have significant applications in image and video processing, computer vision, video surveillance for homeland security, computational biology, and scientific visualization. The proposed theoretical tools and computational methods have the promise of significantly expanding the applicability and functionality of existing and new manifold learning methods and thus advancing the state of the art in nonlinear dimension reduction research. The proposed research lies at the interface between applied mathematics, computational science, and machine learning applications and provides an ideal setting for research cross-fertilization and collaboration as well as training of graduate students in interdisciplinary research.