The Numerical Recovery of High Dimensional Functions
Abstract
Statement of Work: The Principal investigator, Ronald DeVore, will develop theory and algorithms for learning and recovering functions of many variables. Two settings will be studied: (i) Optimal recovery of functions by deterministic querying, (ii) classification and regression from data. Novel techniques for (i) will be developed using high dimensional concepts such as hashing, discrepancy, variable reduction, and scarcity. A rate/distortion theory, which gives a priori performance, will be established. Regarding (ii), new algorithms for model classes based on sparsity and variable reduction will be developed using the concept of sparse occupancy trees. Rate performance bounds will be established for regression and classification for the new model classes. Objective: The main goal of this research is to break the so-called "curse of dimensionality" when facing high dimensional problems. The curse says that traditional methods cannot succeed in the high dimensional setting. This means that new concepts must be found and studied which describe the high dimensional functions occurring in real world applications such as data assimilation and imaging. Approach: The proposed approach will introduce new model classes for functions in high dimensions using concepts such as sparsity, variable reduction, and tensor formats. The PI will then establish the optimal performance bounds for optimal recovery of functions in these model classes. Algorithms that reach the optimal performance will be developed using highly nonlinear methods such as greedy algorithms and adaptivity. Overall Merit and ONR Mission/Relevance: The security of the United States requires the ability to analyze large data sets with many possible features or attributes. This requires prioritization of the important features in the data and ways to query the data in order to identify these features. Similar problems occur when numerically treating physical, chemical, and biological models describing complex phenomena that occur e.g. in oceanographic and atmospheric modeling. This research will identify new techniques, which can be used when confronted with such high dimensional problems.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Aug 12, 2016
- Source ID
- N000141512181
Entities
People
- Ronald A. Devore
Organizations
- Office of Naval Research
- Texas A&M University
- United States Navy