Collaborative Research: Statistical Estimation with Algebraic Structure

从结构生物学到计算机视觉的科学和工程学科都依赖于数据收集和分析来指导科学发现。至关重要的是，在这样的应用程序中，所研究的系统约束和管理收集的数据中的信息结构。该研究项目的目标是开发一系列统计模型，通过汇集统计和优化的跨学科概念，从这些数据集中系统地提取相关统计信息。正在开发的方法旨在提供一套新的统计工具，适用于这类问题，并可能对若干科学学科产生变革性影响，该项目围绕一套核心技术进行阐述，以便在潜在代数结构的背景下分析数据集，这种结构往往产生于数据收集过程所依据的物理定律。与更传统的统计问题不同，在传统的统计问题中，线性基础结构通常被构建到模型中，数据驱动的科学产生了代数但通常非线性结构的问题。该项目侧重于各种科学和工程学科中的核心重要性问题，包括信号处理，结构生物学和计算机视觉，这些问题具有相似的功能：需要利用代数结构从数据中提取信息。该项目旨在开发一种系统的方法来分析这类问题，以及使用低秩张量分解构建计算效率高的算法的一般程序。重要的是，这些方法可以被证明是统计上最优的，因此可以最有效地利用收集到的数据。

Scientific and engineering disciplines ranging from structural biology to computer vision rely on data collection and analysis to guide scientific discovery. Critically, in such applications, the systems under study constrain and govern the structure of information in collected data. The goal of this research project is to develop a family of statistical models that enables a systematic extraction of relevant statistical information from these datasets by bringing together interdisciplinary concepts from statistics and optimization. The approach under development aims to provide a new set of statistical tools that is adapted to this class of problems and that could have a transformative impact on several scientific disciplines.This project is articulated around a core set of techniques to analyze datasets in the context of a latent algebraic structure, often arising from the physical laws underlying the data collection processes. Unlike more traditional statistical problems where a linear underlying structure is often built into the model, data-driven science generates problems with algebraic but often non-linear structure. The project focuses on problems of central importance in a variety of scientific and engineering disciplines, including signal processing, structural biology, and computer vision, that share a similar feature: the need to leverage algebraic structure in order to extract information from data. The project aims at developing a systematic approach to analyze this family of problems, together with a general procedure to construct computationally efficient algorithms using low-rank tensor decomposition. Importantly, these methods can be proved to be statistically optimal and therefore make the most efficient use of collected data.