CAREER: High-dimensional Tensor Learning: The Good, the Bad, and the Pragmatic

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). Higher-order tensor datasets are rising ubiquitously in modern data science applications. A tensor provides an effective representation of a data structure that classical low-order methods fail to capture. However, empirical success has uncovered a myriad of new challenges. Unlike matrices, higher-order tensor problems are computationally hard, and the statistical properties crucially hinge on the choice of algorithms. The PI plans to investigate the fundamental computational-statistical tradeoffs for a range of tensor problems. The PI will develop a suite of statistical learning theory, efficient algorithms, and data-driven solutions for high-dimensional tensor estimation. The developed tools will allow domain scientists to examine complex tensor data, thereby providing solutions to questions that traditional analyses cannot address. Education and research will be integrated through developing new courses, providing summer research-training opportunities, and engaging students in the research. The project will focus on three major research areas: (i) parametric tensor models with statistical and computational optimality; (ii) nonparametric estimation and completion for high-rank tensors; (iii) predictive tensor neural network models with structure constraints. The PI will investigate the intrinsic low-dimensionality for a wide range of structured tensors, including, but not limited to, low-rankness, non-negativity, block-structure, and smoothness. Optimization landscape will be studied for non-convex problems involving exponential-family tensors, orthogonal decomposable tensors, methods-of-moment tensors, and deep tensor neural networks. The new framework will fill in the gap between statistical oracles and the empirical algorithms for addressing higher-order high-dimensional tensor problems. The research will be applied to a variety of data problems, such as classification of brain connectivity data, pattern detection in recommendation systems, and omics data integration. Software packages will be released with detailed documentations to facilitate reproducible research. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.