AF: Small: Algorithms for Inference

AF：小：推理算法

• 批准号: 1319745
• 负责人: Leonard Schulman
• 金额: $ 47.39万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2017-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1319745&HistoricalAwards=false
• 关键词: AF; Small; Algorithms; Inference

The first focus of this award is the problem of inferring causal relationships. This problem is essential to many statistical applications. Generally speaking, causation can be inferred only by active intervention, through controlled experiment. However such experiments may be impossible or else practically or morally infeasible: for instance, in predicting potential effects regulations or laws on medical, educational and economic outcomes, or, in predicting whole-ecosystem effects of human activity. The starting point for Dr. Schulman's research in this area is Pearl's theory of Structured Causal Models (SCM) which, allows, in special circumstances, "identification" of a causal relationship (as opposed to a statistical correlation) from passive observation. The proposed research aims, in the first place, to expand the above special class of circumstances---and thereby make the theory more widely applicable---by using a relaxed but still useful notion of "weak identification" of causal relationships. The relaxed notion is more robust, and enables valid inference even if the posited SCM is slightly inaccurate. In the second place, the research aims to provide efficient and numerically stable algorithms for weak identification from empirical data.The second focus of this award, again in computational statistics, is the representation of a large data set (considered as an empirical measure) by a much smaller data set, in such a way that for a specific family of integrals, all integrals of the measure are approximately preserved. This work encompasses two separate application areas. The first concerns clustering and related high dimensional data analysis problems. Here the compressed data set is known as an epsilon-approximation or core-set of the input measure. A particular focus is on "underclustering", namely, preparation of core-sets for clustering in a normed space, before the norm has been specified. The technical tools needed in this application have to do with recently developed ideas about the "total sensitivity" of the family of integrals, as well as with, on the algorithmic side, bicriteria approximations. The second application area concerns signal processing (or approximation theory) on compact groups. Here the methods draw on representation theory, the classical theory of the moment problem, and convex geometry.This award will be used to train graduate students and postdoctoral fellows in research in algorithms, statistics, and underlying mathematical topics in algebra and geometry.

该奖项的第一个焦点是因果关系推断问题。这个问题对于许多统计应用来说是至关重要的。一般来说，因果关系只能通过主动干预、通过受控实验来推断。然而，此类实验可能是不可能的，或者实际上或道德上不可行：例如，预测法规或法律对医疗、教育和经济成果的潜在影响，或者预测人类活动对整个生态系统的影响。 Schulman 博士在这一领域的研究起点是 Pearl 的结构化因果模型 (SCM) 理论，该理论允许在特殊情况下从被动观察中“识别”因果关系（而不是统计相关性）。拟议的研究首先旨在通过使用因果关系的“弱识别”这一宽松但仍然有用的概念来扩展上述特殊情况，从而使该理论更广泛地适用。宽松的概念更加稳健，即使假设的 SCM 稍微不准确，也可以进行有效的推理。其次，该研究的目的是提供有效且数值稳定的算法，用于从经验数据中进行弱识别。该奖项的第二个重点，同样是在计算统计学中，是通过以下方式表示大型数据集（被视为经验测量）：一个小得多的数据集，这样对于特定的积分族，测度的所有积分都被近似保留。这项工作涵盖两个独立的应用领域。第一个涉及聚类和相关的高维数据分析问题。这里，压缩数据集被称为输入测量的 epsilon 近似或核心集。特别关注的是“聚类不足”，即在指定范数之前准备用于在规范空间中聚类的核心集。该应用程序所需的技术工具与最近开发的关于积分族“总灵敏度”的想法有关，以及在算法方面与双标准近似有关。第二个应用领域涉及紧群上的信号处理（或近似理论）。这里的方法借鉴了表示论、矩问题的经典理论和凸几何。该奖项将用于培训研究生和博士后研究员进行算法、统计学以及代数和几何中的基础数学主题的研究。