CIF: Small: Statistical Inference via Convex Optimization

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In a variety of applications in modern science and technology, there is a strong need in accurate statistical inferences from massive sets of high-dimensional data. To meet this need, it is a must to develop novel methods combining provably (nearly) optimal statistical performance with computational efficiency and scalability. The project aims at designing innovative Convex Optimization based inference techniques meeting the above requirements and utilizing these techniques in important applications (Positron Emission Tomography, Nanoscale Fluorescent Microscopy, Quantum Statistics,...). Challenges to be addressed combined with clear "applied appeal" make the project a good training ground for Ph.D. students. Project?s outcomes could make a valuable contribution to the computational tools of "Big Data."The approach is based on designing statistical tests with near-optimal risk for multiple composite hypotheses in a class of statistical models where observation is: (a) affine image of unknown vector ("signal") corrupted by Gaussian noise; (b) random vector with independent Poisson entries, the underlying parameters being affine functions of the signal; (c) random variable taking finitely many values with probabilities affinely depending on the signal; (d) direct products of models (a) - (c). While restrictive with respect to the allowed models, the approach is highly permissive with respect to the number and the structure of the hypotheses. The proposed efficiently computable and scalable tests and their risks stem from optimal solutions to explicit convex programs, and can be used as building blocks in more complicated inferential problems. The research agenda includes the design of sequential tests and dynamical tests; change point detection; estimating functionals of a signal; "sparsity-oriented" testing/estimation; applications to Poisson Imaging, Active Learning, and Quantum Statistics.