Estimation of Hybrid Models as Algebraic Sets

作为代数集的混合模型的估计

• 批准号: 0514955
• 负责人: Yi Ma
• 金额: $ 23万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-08-01 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0514955&HistoricalAwards=false
• 关键词: Estimation; Hybrid; Models; Algebraic; Sets

Abstract0514955Yi MaU of Illinois @ UrbanaEstimation of Hybrid Models as Algebraic SetsWith the rapid advance in information technologies, all types of new data images, videos, audios, texts, economic data, and genomic data have been produced at an unprecedented rate and scale. How efficiently to represent, classify, and analyze a large quantity of high-dimensional data that have complex geometric or statistical structures becomes one of the most fundamental and challenging problems in our modern information-technology era. One of the main difficulties with modeling complex data is that a given data set may consist of many different components (subsets), each of which may have different topological, geometric or statistical structures. In different contexts, such data are called mixed, or multi-modal or piecewise, or heterogeneous,or hybrid.The goal of the proposed project is to investigate a novel approach that models hybrid data as algebraic sets instead. This readily relates the hybrid-model estimation problem with the study of algebraic sets in algebraic geometry. It offers rather refreshing technical tools, different from the conventional statistical learning methods, which can help resolve the apparent chicken-and-egg problem in the estimation of hybrid models as well as lead to efficient and robust algorithms for estimating and decomposing hybrid models.In the past few years, hybrid model estimation has become a fundamental problem in many important applications, ranging from image/video/motion segmentation in computer vision, sparse image representation and approximation in image processing, to hybrid system identification in systems theory. The proposed interdisciplinary research project provides ample opportunities for students from both mathematics and engineering to interact with each other. Engineering students are traditionally weaker in training in abstract algebra than in analysis, geometry, and statistics. This project will significantly improve the visibility of algebraic geometry to engineering students and researchers. It will help to develop new cross-discipline research programs between mathematics and engineering. Furthermore, visual examples and demonstrations developed from this project will also enhance the teaching of abstract algebraic concepts to undergraduate mathematical students or even to the general public.

伊利诺伊州的Abstract051495555yi mau @ urbanaestimation作为代数SetSwith的信息技术，所有类型的新数据图像，视频，音频，文本，经济数据和基因组数据的快速进步是在前所未有的。在我们现代信息技术时代，具有复杂几何或统计结构的大量高维数据的表现，分类和分析的效率如何成为大量具有复杂几何或统计结构的高维数据。建模复杂数据的主要困难之一是给定数据集可能由许多不同的组件（子集）组成，每个组件可能具有不同的拓扑，几何或统计结构。在不同的情况下，此类数据称为混合，多模式或分段，或异质性或混合动力。拟议项目的目的是研究一种新颖的方法，该方法将混合数据建模为代数集。这很容易将杂种模型估计问题与代数几何形状的代数集的研究联系起来。 It offers rather refreshing technical tools, different from the conventional statistical learning methods, which can help resolve the apparent chicken-and-egg problem in the estimation of hybrid models as well as lead to efficient and robust algorithms for estimating and decomposing hybrid models.In the past few years, hybrid model estimation has become a fundamental problem in many important applications, ranging from image/video/motion segmentation in computer vision, sparse image representation and图像处理中的近似值，系统理论中的混合系统识别。拟议的跨学科研究项目为来自数学和工程学的学生提供了足够的机会，可以相互互动。传统上，工程专业的学生在抽象代数的培训中比分析，几何和统计数据要弱。该项目将显着提高代数几何学对工程学生和研究人员的可见性。这将有助于在数学和工程学之间制定新的跨学科研究计划。此外，该项目开发的视觉示例和演示还将增强对本科数学学生甚至公众的抽象代数概念的教学。