Estimation of Hybrid Models as Algebraic Sets

摘要：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、分类和分析具有复杂几何或统计结构的大量高维数据，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（子集）可能具有不同的拓扑、几何或统计结构。在不同的上下文中，这样的数据被称为混合的、多模态的、分段的、异构的或混合的。该项目的目标是研究一种新的方法，将混合数据建模为代数集。这很容易将混合模型估计问题与代数几何中代数集的研究联系起来。与传统的统计学习方法不同，它提供了令人耳目一新的技术工具，可以帮助解决混合模型估计中明显的先有鸡还是先有蛋的问题，并产生高效、鲁棒的混合模型估计和分解算法。在过去的几年中，混合模型估计已经成为许多重要应用中的一个基本问题，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逼近，到系统理论中的混合系统识别。提议的跨学科研究项目为数学系和工程系的学生提供了充分的相互交流的机会。传统上，工科学生在抽象代数方面的训练要弱于分析、几何和统计学。这个项目将显著提高代数几何对工程学生和研究人员的可见度。它将有助于在数学和工程之间发展新的跨学科研究项目。此外，从这个项目开发的视觉例子和演示也将加强对数学本科生甚至公众的抽象代数概念的教学。

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.