Application of Greedy Approximations in Numerical Integration and Learning Theory

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TemlyakovDMS-0906260 The goal of the project is to design and study greedy type approximation methods that are practically implementable in numerical integration and learning theory. One of the biggest challenges of contemporary applied mathematics is high-dimensional problems. High-dimensional problems with dimensions of hundreds, and even thousands, arise naturally in finance, quantum chemistry, biology, medicine, and other areas. Typical problems of this kind are numerical integration and statistical estimation. The fundamental question is how to construct good methods of numerical integration (cubature formulas) and statistical estimation. Recent investigations show that greedy type approximation methods are good for different high-dimensional problems, including problems from numerical integration and learning theory. The investigator and his colleagues study application of greedy approximations in numerical integration and learning theory. Preliminary investigations show that greedy type approximation methods work well in high dimensions and can be considered as a constructive deterministic alternative to some powerful probabilistic methods. The greedy approximation has potential to become a transformative concept of numerical integration andlearning theory. Understanding intelligence and how it learns and assimilates information is one of the great scientific challenges of this decade. It is key to designing systems to efficiently analyze data and extract essential information. The scientific discipline that studies this aspect of intelligence is called learning theory. It has a myriad of existing and potential applications in both the defense and civilian sectors. For instance, managing large data bases such as security data bases obtained through surveillance requires classification of the data sets in order to speed up extraction of significant features or specific information. Learning theory discovers rules that allow the classification of new data from past data that have already been classified. A prototypical application is the search through a large data base (for example emails) to determine which of these have possible links to terrorist activities. It is the goal of this project to utilize fundamental concepts in approximation and statistics to clearly define and quantify the learning challenge and design new, more efficient techniques (greedy algorithms).