AF: Small: Algorithms for Inference

本奖项的第一个焦点是因果关系的推断问题。这个问题对于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，这种实验可能是不可能的，或者在实际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，或在预测人类活动对整个生态系统的影响方面。舒尔曼博士在这一领域的研究起点是珀尔的结构化因果模型理论（SCM），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）。首先，拟议的研究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，扩展上述特殊类型的情况，从而使理论更广泛地适用。松弛的概念更健壮，即使假定的SCM稍微不准确，也能进行有效的推断。其次，研究旨在为从经验数据中进行弱识别提供高效且数值稳定的算法。这个奖项的第二个重点，同样是在计算统计学领域，是用一个小得多的数据集来表示一个大数据集（被认为是一个经验测度），以这样一种方式，对于一个特定的积分族，该测度的所有积分都是近似保留的。这项工作包括两个独立的应用领域。第一部分涉及聚类和相关的高维数据分析问题。在这里，压缩的数据集被称为输入度量的近似或核心集。一个特别的焦点是“欠聚类”，即在规范被指定之前，在赋范空间中为聚类准备核心集。此应用程序所需的技术工具必须与最近发展的关于积分族的“总灵敏度”的想法有关，以及在算法方面的双准则近似。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征理论、经典的矩问题理论和凸几何。该奖项将用于培养研究生和博士后在算法、统计学和代数和几何基础数学主题方面的研究。

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.