An O(N) algorithm for computing expectation of N-dimensional truncated multi-variate normal distribution I: fundamentals
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ArticleKAUST Department
Applied Mathematics and Computational Science ProgramComputer, Electrical and Mathematical Science and Engineering (CEMSE) Division
Extreme Computing Research Center
Office of the President
Spatio-Temporal Statistics and Data Analysis Group
Statistics Program
Date
2021-09-01Online Publication Date
2021-09-01Print Publication Date
2021-10Embargo End Date
2022-09-01Submitted Date
2020-12-29Permanent link to this record
http://hdl.handle.net/10754/671102
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In this paper, we present the fundamentals of a hierarchical algorithm for computing the N-dimensional integral ϕ(a,b;A)=∫abH(x)f(x|A)dx representing the expectation of a function H(X) where f(x|A) is the truncated multi-variate normal (TMVN) distribution with zero mean, x is the vector of integration variables for the N-dimensional random vector X, A is the inverse of the covariance matrix Σ, and a and b are constant vectors. The algorithm assumes that H(x) is “low-rank” and is designed for properly clustered X so that the matrix A has “low-rank” blocks and “low-dimensional” features. We demonstrate the divide-and-conquer idea when A is a symmetric positive definite tridiagonal matrix and present the necessary building blocks and rigorous potential theory–based algorithm analysis when A is given by the exponential covariance model. The algorithm overall complexity is O(N) for N-dimensional problems, with a prefactor determined by the rank of the off-diagonal matrix blocks and number of effective variables. Very high accuracy results for N as large as 2048 are obtained on a desktop computer with 16G memory using the fast Fourier transform (FFT) and non-uniform FFT to validate the analysis. The current paper focuses on the ideas using the simple yet representative examples where the off-diagonal matrix blocks are rank 1 and the number of effective variables is bounded by 2, to allow concise notations and easier explanation. In a subsequent paper, we discuss the generalization of current scheme using the sparse grid technique for higher rank problems and demonstrate how all the moments of kth order or less (a total of O(Nk) integrals) can be computed using O(Nk) operations for k ≥ 2 and O(NlogN) operations for k = 1.Citation
Huang, J., Cao, J., Fang, F., Genton, M. G., Keyes, D. E., & Turkiyyah, G. (2021). An O(N) algorithm for computing expectation of N-dimensional truncated multi-variate normal distribution I: fundamentals. Advances in Computational Mathematics, 47(5). doi:10.1007/s10444-021-09888-1Sponsors
J. Huang was supported by the NSF grant DMS1821093, and the work was finished while he was a visiting professor at the King Abdullah University of Science and Technology, National Center for Theoretical Sciences (NCTS) in Taiwan, Mathematical Center for Interdisciplinary Research of Soochow University, and Institute for Mathematical Sciences of the National University of Singapore.Publisher
Springer Science and Business Media LLCAdditional Links
https://link.springer.com/10.1007/s10444-021-09888-1ae974a485f413a2113503eed53cd6c53
10.1007/s10444-021-09888-1