CIF: Small: Statistical Inference via Convex Optimization

在现代科学技术的各种应用中，强烈需要从海量高维数据中进行准确的统计推断。为了满足这一需求，必须开发将可证明（接近）最优的统计性能与计算效率和可扩展性相结合的新方法。该项目旨在设计满足上述要求的创新的基于凸优化的推理技术，并将这些技术用于重要应用（正电子发射断层扫描、纳米级荧光显微镜、量子统计等）。需要解决的挑战与明确的“应用吸引力”相结合，使该项目成为博士生的良好培训基地。学生。项目的成果可以对“大数据”的计算工具做出有价值的贡献。该方法基于在一类统计模型中设计具有接近最佳风险的多个复合假设的统计测试，其中观察是：（a）被高斯噪声破坏的未知向量（“信号”）的仿射图像； (b) 具有独立泊松项的随机向量，基本参数是信号的仿射函数； (c) 随机变量取有限多个值，其概率密切依赖于信号； (d) 模型(a) - (c) 的直接乘积。虽然对允许的模型有限制，但该方法对假设的数量和结构是高度宽松的。所提出的有效可计算和可扩展的测试及其风险源于显式凸程序的最优解决方案，并且可以用作更复杂的推理问题的构建块。研究议程包括顺序测试和动态测试的设计；变化点检测；估计信号的泛函； “面向稀疏性”的测试/估计；在泊松成像、主动学习和量子统计中的应用。

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.