Data Analytics for Nonlinear Problems: Theory and Algorithms
Abstract
This project is motivated by the computational challenges arising in the operation and control of many complex real~world systems, w""here decisions need to be made based on unreliable data and measurements. Data is abundant in many areas such as health, transportat""ion, energy and social networks, but it is difficult to use them due to the lack of reliable tools and techniques for data processin"g and decision making as well as the scalability of the existing computational methods. The objective of this project is to develop" efficient computational methods for analyzing data, extracting information and making reliable decisions. This project is interdisc"iplinary and combines modern optimization theory and numerical algorithms with machine learning and statistics. The proposed approac"h is based on convex analysis, nonlinear optimization, low~rank optimization, statistics, stochastic systems, graph theory, algebrai""c geometry, and numerical algorithms, among others.Some of the objectives of this project are as follows:Data analytics: Two maj""or challenges in data science are: (i) how to learn models from data, and (ii) how to make decisions based on unreliable and partial""ly observable data. Although optimization theory has been incorporated in the operation of many real~world systems, the tools in dat"a science have not yet had such a large~scale impact in many fields. Part of the reason is that optimal statistical learning algorit"hms either work under strong assumptions or are normally too complex (e.g., they are in the form of nonconvex continuous problems du"e to the way probability distributions appear in the problem). One objective of this project is to use modern optimization methods to study different problems on data analysis and learning. We will also investigate how the amount of data or its type affects the le"arning process. Unlike most of the techniques in statistics and machine learning that are based on probabilistic approaches, our stu"dy has deep roots in conic optimization that is beyond the classical optimization theory and resonates with advanced graph~theoretic" concepts (such as tree decomposition), low~rank matrices, algebraic geometry, manifolds, and sparsity~inducing techniques.Numeric""al algorithms for big data problems: Due to staggering computing advances in recent years, it is possible to efficiently solve extre""mely largescalelinear and certain convex problems. In contrast, conic optimization is very expensive to solve using the existingal""gorithms, and this is a major bottleneck for their real~world implementation. The design of simple and cheapalgorithms for large~sc""ale conic problems (especially in the context of big data) has attracted a lot of attention, due in part to the fact that many recen"t techniques in data science and machine learning rely on conic optimization problems. One objective of this project is to develop highly efficient numerical algorithms for both sparse and dense conic optimization as well as big~data problems. To achieve this goal", we will study the geometry of the feasible set of conic problems and use advanced techniques related to projection on manifolds an"d matrix analysis. We will study the tradeoff between the complexity of each algorithm and its convergence rate.Applications: We w"ill apply the results developed in this project to various cast studies, especially those with DoDapplications.The results of thi""s project would have significant impacts on real~world applications, as well as Applied Mathematics and Engineering. It would greatl""y advance the areas of optimization theory, algorithms, data science, and machine learning.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Sep 29, 2017
- Source ID
- N000141712933
Entities
People
- Javad Lavaei
Organizations
- Office of Naval Research
- United States Navy
- University of California Regents