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

本提案的目标是开发鲁棒算法，用于基于从训练数据集中学习的低维表示来重建或合成高度结构化的高维数据，即流形学习中的插值和外推问题。该项目将解决在流形学习的各种插值和外推问题的设置中计算通常不明确的低维参数化的难以捉摸的问题，强调物理上有意义的参数化的概念。它将开发创新的计算方法，以便在无监督和半监督学习以及特别是主动学习设置的背景下，灵活地学习低维参数化以及其他物理重要变量，用于动态数据的学习和综合，以及基于迁移学习的流形外推。该项目包括开发一个公开可用的软件包，该软件包将传播研究结果并促进非线性降维方法在实际问题中的应用。这项研究的发现有望影响广泛的应用领域。计算高维数据的紧凑表示是一个非常具有挑战性的统计学习问题，流形学习已经成为一个非常活跃的研究领域，旨在从许多高维数据中固有的统计和几何规则中发现隐藏的结构。在插值和外推的背景下重建和合成高维数据将在图像和视频处理、计算机视觉、国土安全视频监控、计算生物学和科学可视化方面具有重要应用。所提出的理论工具和计算方法有望显著扩展现有和新的流形学习方法的适用性和功能性，从而推动非线性降维研究的发展。本研究是应用数学、计算科学和机器学习应用的交叉领域，为跨学科研究提供了一个理想的研究环境，同时也为跨学科研究的研究生培养提供了一个理想的环境。

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.