CAREER: Statistical Analysis of Massive Data Sets under Low-Complexity Constraints

近年来，随着在气象学、基因组学或金融等广泛应用中收集数据的急剧增加，大规模数据集的统计推断方法的发展已成为统计学和机器学习研究的焦点。这些新数据集的共同点是它们满足某些低复杂性的结构条件。在特定情况下，这可能意味着，例如，高维回归中向量的稀疏性或高维矩阵的低秩属性。本研究旨在提供新的有效的解决方案，利用这些特殊的结构，以执行准确的推理。未来的研究结果应该会引起数据科学界广大读者的兴趣。研究者还打算将他的贡献整合到教育活动中，如课程开发和指导本科生和研究生。该项目的研究目标是在大规模数据集中识别新的信息低复杂性结构，并提出适应这些结构的新方法，计算效率高，在各种模型中统计最优，特别是矢量回归，跟踪回归，标准和功能数据的主成分分析。新的程序将基于惩罚性的经验风险最小化和指数权重混合，有利于低复杂性的解决方案。调查人员需要(a)确定控制这些程序执行的问题的基本特征；(b)建立新的高维oracle不等式结果来评估这些过程的统计性能；(c)研究这些程序在不同条件下对模型的计算性能。

The development of methods of statistical inference for massive data sets has become a focal point of research in statistics and machine learning in recent years with the dramatic increase of collected data in a wide range of applications such as meteorology, genomics, or finance. A common denominator of these new data sets is that they satisfy certain low-complexity structure conditions. In specific settings, this could mean, for instance, sparsity of a vector in high-dimensional regression or low-rank properties of a high-dimensional matrix. This research aims at providing new efficient solutions exploiting these particular structures in order to perform accurate inference. The future findings should be of interest to a broad audience in the data science community. The investigator also intends to integrate his contributions into educational activities such as course development and mentoring of undergraduate and graduate students.The research objectives of this project are to identify new informative low-complexity structures in large scale data sets and propose new methods that are adaptive to these structures, computationally efficient, and statistically optimal in a variety of models, including in particular vector regression, trace regression, principal component analysis for standard and functional data. The new procedures will be based on penalized empirical risk minimization and exponential weights mixing that favor low-complexity solutions. The investigator plains to (a) determine fundamental characteristics of the problems that govern the performance of these procedures; (b) establish new oracle inequalities results in high dimension to assess the statistical performances of these procedures; (c) investigate the computational performance of these procedures under various conditions on the models.