CAREER: High Dimensional Statistics -- Adaptive Networks, Structure and Robustness

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Data-driven adaptation has emerged as a powerful paradigm for algorithm design in engineered, social, and biological large-scale complex systems. Many phenomena in complex dynamical networks are naturally high dimensional: a model?s dimensionality may equal or exceed the number of data points one can collect or experiments one can perform. This novel regime poses severe algorithmic, computational and analytical challenges.Intellectual Merit: Low-dimensional structure ? often hidden but prevalent in many complex systems ? offers a way forward. We propose an essentially complete rethinking of Robust Optimization: imagining fictitious parameter uncertainty we design a new algorithmic framework for finding and exploiting structure. This greatly broadens the scope of problems where structure can be exploited, unifying results that previously seemed disconnected, and opening the door for the design of new efficient and provably effectivealgorithms. Then, marrying essential ideas of robust optimization with tools from high-dimensional statistics, we explore robustness to potentially severe data corruption in high-dimensions ? a problem that classical robust statistics has largely been unable to address.Broader Impacts: Curriculum initiatives include a vertical and horizontal integration of data-driven techniques in new curriculum. The work will influence and be motivated by strong connections to industry partners. High-dimensional data will become increasingly pervasive (the length of genomes sequenced increases; the number of patients carrying a genetic disease does not). Many questions critical to society, science and our future depend fundamentally on successful analysis and efficient, robust algorithms for the high dimensional regime; the impact to real applications promises to be immense.