AF: Small: Algorithms for Inference

这个奖项的第一个焦点是因果关系的推断问题。这个问题对许多统计应用来说是必不可少的。一般来说，因果关系只能通过主动干预，通过控制实验来推断。然而，这样的实验可能是不可能的，或者实际上或道德上是不可行的：例如，在预测法规或法律对医疗、教育和经济成果的潜在影响方面，或者在预测人类活动对整个生态系统的影响方面。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）。建议的研究的目的，首先，扩大上述特殊类别的情况-从而使理论更广泛地适用-通过使用一个宽松的，但仍然有用的概念，“弱识别”的因果关系。放松的概念更健壮，即使假设的SCM稍微不准确，也能进行有效的推理。其次，该研究旨在提供有效的和数值稳定的算法，从经验数据中识别弱。该奖项的第二个重点，再次在计算统计，是一个大的数据集（被认为是一个经验措施）的表示由一个小得多的数据集，在这样一种方式，对于一个特定的家庭的积分，所有积分的措施是近似保存。这项工作包括两个独立的应用领域。第一个是关于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度量的ε近似或核心集。一个特别的重点是“下聚类”，即准备核心集聚类在一个规范的空间，规范之前已被指定。在这个应用程序中所需的技术工具与最近开发的想法有关的“总灵敏度”的家庭的积分，以及与，在算法方面，双准则近似。第二个应用领域涉及信号处理（或近似理论）的紧凑组。该奖项将用于培训研究生和博士后研究员，研究算法、统计学以及代数和几何中的基础数学主题。

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.