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A grand challenge to solve a large-scale linear inverse problem (LIP) is to retain computational efficiency and accuracy regardless of the growth of the problem size. Despite the plenitude of methods available for solving LIPs, various challenges have emerged in recent times due to the sheer volume of data, inadequate computational resources to handle an oversized problem, security and privacy concerns, and the interest in the associated incremental or decremental problems. Removing these barriers requires a holistic upgrade of the existing methods to be computationally efficient, tractable, and equipped with scalable features. We, therefore, develop the parallel residual projection (PRP), a parallel computational framework involving the decomposition of a large-scale LIP into sub-problems of low complexity and the fusion of the sub-problem solutions to form the solution to the original LIP. We analyze the convergence properties of the PRP and accentuate its benefits through its application to complex problems of network inference and gravimetric survey. We show that any existing algorithm for solving an LIP can be integrated into the PRP framework and used to solve the sub-problems while handling the prevailing challenges.

 

Knowledge discovery and information extraction of large and complex datasets has attracted great attention in wide-ranging areas from statistics and biology to medicine. Tools from machine learning, data mining, and neurocomputing have been extensively explored and utilized to accomplish such compelling data analytics tasks. However, for time-series data presenting active dynamic characteristics, many of the state-of-the-art techniques may not perform well in capturing the inherited temporal structures in these data. In this paper, integrating the Koopman operator and linear dynamical systems theory with support vector machines (SVMs), we develop a novel dynamic data mining framework to construct low-dimensional linear models that approximate the nonlinear flow of high-dimensional time-series data generated by unknown nonlinear dynamical systems. This framework then immediately enables pattern recognition, e.g., classification, of complex time-series data to distinguish their dynamic behaviors by using the trajectories generated by the reduced linear systems. Moreover, we demonstrate the applicability and efficiency of this framework through the problems of time-series classification in bioinformatics and healthcare, including cognitive classification and seizure detection with fMRI and EEG data, respectively. The developed Koopman dynamic learning framework then lays a solid foundation for effective dynamic data mining and promises a mathematically justified method for extracting the dynamics and significant temporal structures of nonlinear dynamical systems.