AF: Small: Algorithms for Inference

这个奖项的第一个焦点是推断因果关系的问题。这个问题对于许多统计应用来说是必不可少的。一般来说，只有通过积极干预，通过对照实验才能推断因果关系。然而，这样的实验可能是不可能的，或者实际上或在道德上是不可行的：例如，在预测潜在影响方面，例如，在预测关于医疗、教育和经济结果的法规或法律方面，或者在预测人类活动的整个生态系统影响方面。舒尔曼博士在这一领域研究的起点是珀尔的结构化因果模型(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)。首先，建议的研究目的是扩大上述特殊情况的类别-从而使该理论更广泛地适用-通过使用一个宽松但仍然有用的因果关系的“弱识别”概念。放松的概念更稳健，即使假设的SCM略有不准确，也能进行有效的推理。第二，在计算统计学中，该奖项的第二个重点是用小得多的数据集来表示大数据集(被认为是经验度量)，使得对于特定的积分族，该度量的所有积分都被近似保持。这项工作包括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度量的epsilon近似或核心集。一个特别的重点是“欠聚类”，即在范数被指定之前，为在范数空间中聚类而准备核心集。这个应用程序所需要的技术工具与最近发展起来的关于积分族的“总敏感性”的想法有关，以及在算法方面与双准则近似有关。第二个应用领域涉及紧群上的信号处理(或逼近理论)。这个奖项将用来训练研究生和博士后研究员在算法、统计学以及代数和几何的基本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.