Estimation of Hybrid Models as Algebraic Sets

随着信息技术的飞速发展，各种类型的新数据图像、视频、音频、文本、经济数据和基因组数据以前所未有的速度和规模产生。如何有效地表示、分类和分析大量具有复杂几何或统计结构的高维数据成为现代信息技术时代最基本和最具挑战性的问题之一。对复杂数据建模的主要困难之一是，给定的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。在不同的背景下，这样的数据被称为混合，或多模态或分段，或异构，或hybrid.The建议的项目的目标是研究一种新的方法，建模混合数据作为代数集。这很容易将混合模型估计问题与代数几何中的代数集研究联系起来。它提供了不同于传统统计学习方法的全新技术工具，可以帮助解决混合模型估计中明显的鸡和蛋问题，并为混合模型的估计和分解提供高效、鲁棒的算法。近年来，混合模型估计已成为许多重要应用中的基本问题，从计算机视觉中的图像/视频/运动分割、图像处理中的稀疏图像表示和近似，到系统理论中的混合系统识别。拟议的跨学科研究项目为数学和工程专业的学生提供了充分的机会相互交流。传统上，工程专业的学生在抽象代数方面的训练比分析、几何和统计方面的训练要弱。这个项目将显着提高代数几何的可见性，工程专业的学生和研究人员。它将有助于开发数学和工程之间的新的跨学科研究项目。此外，从这个项目开发的可视化的例子和演示也将提高抽象的代数概念的教学本科数学学生，甚至向公众。

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.