The goal of this thesis to contribute towards a computational complexity theory of statistical inference problems. In recent years, researchers have built evidence in favor of an emerging hypothesis that the class of ...
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The goal of this thesis to contribute towards a computational complexity theory of statistical inference problems. In recent years, researchers have built evidence in favor of an emerging hypothesis that the class of semi-definite programming (SDP) algorithms is optimal among for computationally efficient algorithms for a certain family of estimation problems. In this thesis, we present four main research efforts that refine this hypothesis and initiate preliminary efforts to go beyond it: • Optimal algorithms for private and robust estimation: We give the first polynomial-time algorithms for privately and robustly estimating a Gaussian distribution with optimal dependence on the dimension in the sample complexity. This adds the fundamental problem of private statistical estimation to a growing list of problems for which SDPs are optimal among polynomial-time algorithms. • Limitations of SDPs: Given independent standard Gaussian points in dimension d, for what values of (n, d) does there exist with high probability an origin-symmetric ellipsoid that simultaneously passes through all of the points? Based on strong numerical evidence, it was conjectured that the ellipsoid fitting problem transitions from feasible to infeasible as the number of points n increases, with a sharp threshold at n ∼ d 2/4; we resolve this conjecture up to logarithmic factors. A corollary of this result is that a canonical SDP-based algorithm fails to successfully solve inference problems involving low-rank matrix decompositions, independent component analysis, and principal component analysis. • New algorithms for discrepancy certification: We initiate the study of the algorithmic problem of certifying lower bounds on the discrepancy of random matrices, which has connections to conjecturally-hard average-case problems such as negatively-spiked PCA, the number-balancing problem and refuting random constraint satisfaction problems. We give the first polynomial-time algorithms with non-trivial
With the development of more sophisticated robots that are increasingly aimed to venture outside of the lab, the higher dimensionality, complexity and nonlinearities of the underlying robotic systems as well as enviro...
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With the development of more sophisticated robots that are increasingly aimed to venture outside of the lab, the higher dimensionality, complexity and nonlinearities of the underlying robotic systems as well as environmental uncertainty pose challenges to reliance on nominal models of robots obtained from first principles. Koopman operator theory, a tool first introduced for data-driven model extraction and global linearization, and its various extensions have been widely implemented in robot modeling and control. This dissertation contributes fundamental theory and practical algorithms for the implementation of the Koopman operator theory across distinctive types of robots. Specific contributions of this dissertation span over four key sub-phases. These include data collection, model extraction, controller design, and physical implementation. First, the practical scenario of working with noisy data for modeling is considered. Prediction errors because of noisy measurements when estimating the model via the data-driven Koopman operator-based approaches for control are derived. Then, the explicit quantification of the error is embedded into an existing Koopman-based data-driven robot modeling and control architecture to enhance its robustness without making significant changes to current parts of the underlying structure. Second, considering the importance of the space-lifting process in the Koopman theory, we propose a general and analytical algorithm to formalize the construction of the lifting functions based on characteristic properties of robots - namely their configuration space or, in the case of soft robots, their workspace. The resulting design of the lifting functions is proven to be complete and leads to an approximated Koopman operator with provable guarantees of convergence to the true one. Finally, we develop and present Koopman-based controllers and implement them to drive and/or improve the performance of robots. The first design is an online modeling
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