Tensor Network Computation: Representations, Algebra, and Applications

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Modern data collection and computational simulations produce many high-dimensional data sets. Such high-dimensional data sets represent one of the key challenges in computational mathematics. As is well-known, the main difficulty lies in the curse of the dimensionality, that is, the number of degrees of freedom required to represent and analyze high-dimensional functions in the traditional way grows exponentially with the number of dimensions. Recently, tensor networks have emerged, mostly from the theoretical and computational physics community, as a promising tool for representing and manipulating high-dimensional functions and probabilities. This project undertakes a systematic computational study of the tensor network approach. The investigation serves as an initial step in providing a general framework for work with high-dimensional data sets. The project focuses on four aspects of tensor networks. (1) The investigator aims to improve on recent work in tensor network skeletonization, where the main task is to develop efficient tensor contraction algorithms for inhomogeneous systems and general tensor networks, with an emphasis on parallel implementation. (2) The investigator plans to develop novel algorithms for constructing a tensor network either via sampling or from an existing but redundant tensor representation. The approach will draw tools and ideas from randomized numerical algebra, nonlinear optimization, and multiscale methods. (3) The project will develop algorithms for basic algebraic operations for manipulating high-dimensional functions in the tensor network representation. These basic operations include addition, subtraction, entry-wise multiplication, entry-wise inversion, entry-wise function application, and application of linear operators in tensor network operator form. (4) The last objective is to investigate new applications of tensor networks outside the traditional realm of statistical and quantum mechanics, for example in numerical homogenization, uncertainty quantification in very high dimension, and high-dimensional partial differential equations in control and molecular dynamics.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.