An O(N) algorithm for computing expectation of N-dimensional truncated multi-variate normal distribution II: computing moments and sparse grid acceleration - Zheng, Chaowen; Tang, Zhuochao; Huang, Jingfang; Wu, Yichao (Advances in computational mathematics, 2022)
In a previous paper (Huang et al., Advances in Computational Mathematics 47(5):1–34, 2021), we presented the fundamentals of a new hierarchical algorithm for computing the expectation of a N -dimensional function H ( X ) where X satisfies the truncated multi-variate normal (TMVN) distribution. The algorithm assumes that H ( X ) is low-rank and the covariance matrix Σ and precision matrix A = Σ - 1 have low-rank blocks with low-dimensional features. Analysis and numerical results were presented when A is tridiagonal or given by the exponential model. In this paper, we first demonstrate how the hierarchical algorithm structure allows the simultaneous calculations of all the order M and less moments E ( H ( X ) = X 1 m 1 ⋯ X N m N | a i < X i < b i , i = 1 , … , N ) , ∑ i m i ≤ M using asymptotically optimal O ( N M ) operations when M ≥ 2 and O ( N log ( N ) ) operations when M = 1 . These O ( N M ) moments are often required in the Expectation Maximization (EM) algorithms. We illustrate the algorithm ideas using the case when A is tridiagonal or the exponential model where the off-diagonal matrix block has rank K = 1 and number of effective variables P ≤ 2 for each function associated with a hierarchical tree node. The smaller K and P values allow the use of existing FFT and Non-uniform FFT (NuFFT) solvers to accelerate the computation of the compressed features in the system. To handle cases with higher K and P values, we introduce the sparse grid technique aimed at problems with K + P ≈ 5 ∼ 20 . We present numerical results for computing both the moments and higher K and P values to demonstrate the accuracy and efficiency of the algorithms. Finally, we summarize our results and discuss the limitations and generalizations, in particular, our algorithm capability is limited by the availability of mathematical tools in higher dimensions. When K + P is greater than 20, as far as we know, there are no practical tools available for problems with 20 truly independent variables.