CIF: Small: Statistical Inference via Convex Optimization

在现代科学技术的各种应用中，迫切需要对海量高维数据进行精确的统计推断。为了满足这一需求，它是一个必须开发新的方法相结合的可证明（接近）最佳的统计性能与计算效率和可扩展性。该项目旨在设计创新的基于凸优化的推理技术，以满足上述要求，并在重要应用中利用这些技术（正电子发射断层扫描，纳米荧光显微镜，量子统计等）。要解决的挑战与明确的“应用吸引力”相结合，使该项目成为博士的良好培训基地。学生项目？的结果可以为“大数据”的计算工具做出有价值的贡献。and The results show that the mathematical model is a mathematical model for the mathematical model.“（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，其具有取决于信号的概率的多个值;（d）模型（a）-（c）的直积。虽然对允许的模型有限制性，但这种方法对假设的数量和结构是高度宽容的。所提出的有效的可计算和可扩展的测试和他们的风险源于显式凸规划的最优解，并可以作为构建块在更复杂的推理问题。研究议程包括序贯测试和动态测试的设计;变点检测;估计信号的泛函;“稀疏导向”测试/估计;泊松成像，主动学习和量子统计的应用。

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.