Estimation of Hybrid Models as Algebraic Sets

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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.