CIF: RI: Small: Information-theoretic measures of dependencies and novel sample-based estimators

依赖性的度量在发现导致科学发现的变量之间的关联方面起着核心作用。在实践中，分析师需要从数据中计算这些指标，这可能具有挑战性。例如，当数据混合了连续变量和离散变量，或者当数据位于具有丰富边界的复杂空间时，标准估计器可能会失效。这个项目的目的是解决评估依赖度量的实际问题，并提供新的评估器来克服这些挑战。所建议的工作的成功将产生用于发现数据新方面的新颖估计器。直接的影响是在两个特定的背景下：发现生物数据集的相关性和分析深度神经网络的内部工作；这将对包括基因组学、生物学、机器学习和人工智能在内的多个领域产生持久影响。该项目还通过开设统计学习研究生课程，将研究与教育结合起来。此外，该项目将为本科生提供参与研究的机会。该建议解决了两个基本问题：为信息理论测度设计新的估计器和为现代相关测度设计新的估计器，这被定义为优化问题的解决方案。在前者中，解决了两个主要挑战：混合类型变量（连续和离散）和边界偏差。借鉴局部对数似然密度估计器、最近邻方法和有序统计的技术，这导致了一个新的估计器，它可以以一种有原则的方式适应分布的局部几何，这比现有的估计器有了显著的改进。在现代数据分析中，一些相关性度量自然地被定义为优化问题的解决方案，这使得它们难以估计。这一建议旨在提供一个原则性的方法，并提出了一个新的估计借鉴重要抽样和最近邻方法的见解。该框架被用于估计超收缩比，超收缩比是一种信息理论量，它捕获了数据中隐藏的相关性，并且自然地被定义为无限维优化的解。在典型的合成示例和真实数据集中，所提出的超收缩性度量显示出其他标准度量无法发现的潜在相关性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Measures of dependencies play central roles in discovering associations between variables that leads to scientific discoveries. In practice, analysts need to compute these measures from data, which can be challenging. The standard estimators can fail when, for example, the data has a mixture of continuous and discrete variables, or when the data lies on a complex space with abundant boundaries. The aim of this project is to address practical issues in estimating measures of dependencies, and provide novel estimators to overcome these challenges. The success of the proposed work will result in novel estimators for discovering new aspects of data. The immediate impact is in two specific contexts: discovering correlations in biological datasets and analyzing the inner-workings of deep neural networks; the lasting impact will be in diverse fields including genomic, biology, machine learning, and artificial intelligence. This project also integrates research with education through the creation of a graduate course on statistical learning. In addition, the project will offer undergraduates the opportunity to be involved in research.This proposal addresses two fundamental questions: designing novel estimators for information theoretic measures and designing novel estimators for modern measures of correlation that is defined as a solution of optimization problems. In the former, two major challenges are addressed: variables of mixed type (continuous and discrete) and boundary biases. Borrowing techniques from local log-likelihood density estimators, nearest neighbor methods, and order statistics, this leads to a new estimator that can adapt to the local geometry of the distributions in a principled way, that improves significantly over existing estimators. In modern data analysis, several measures of correlations are naturally defined as solutions of optimization problems, making them challenging to estimate. This proposal aims to provide a principled approach and propose a new estimator borrowing insights from importance sampling and nearest neighbor methods. The proposed framework is applied to estimate hypercontractivity ratio, an information theoretic quantity that captures hidden correlations in the data and is naturally defined as a solution of an infinite dimensional optimization. The proposed measure of hypercontractivity is shown to discover potential correlations that other standard measures are not able to, in canonical synthetic examples and real datasets.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.