SSTD_R1: Sparse Symmetric Tensor Decomposition (SSTD)

In snQTL: Spectral Network Quantitative Trait Loci (snQTL) Analysis

Sparse Symmetric Tensor Decomposition (SSTD)

Description

SSTD solves the rank-1 approximation to the a p-by-p-by-q sparse symmetric tensor \mathcal{D}:

\min_{\Lambda, v, u} ||\mathcal{D} - \Lambda v \circ v \circ u||_F^2

subject to

\Lambda > 0, v \in R^p, u \in R^q, ||v||_2 = ||u||_2 = 1, ||v||_0 <= R

The solution \Lambda is the sparse leading tensor eigenvalue (sLTE), v is the sparse leading tensor eigenvector, and u is the loading vector.

The Symmetric Penalized Matrix Decomposition symmPMD() is used in the iterative algorithm.

Usage

SSTD_R1(
  T_obs,
  u_ini,
  v_ini,
  max_iter = 20,
  sumabs = 0.5,
  niter = 20,
  rho = 1000,
  tol = 10^(-3),
  verbose = FALSE
)

Arguments

T_obs	array, a p-by-p-by-q tensor; each p-by-p layer in T_obs should be symmetric
u_ini	vector, with length q; the random initialization for loading vector
v_ini	vector, with length p; the random initialization for tensor eigenvector
max_iter	integer, the maximal iteration number
sumabs	number, the number specify the sparsity level in the matrix/tensor eigenvector; sumabs takes value between 1/sqrt(p) and 1, where p is the dimension; sumabs*sqrt(p) is the upperbound of the L1 norm of the leading matrix/tensor eigenvector (see symmPMD())
niter	integer, the number of iterations to use in the PMD algorithm (see symmPMD())
rho	number, a large positive constant adding to the diagonal elements to ensure positive definiteness in symmetric matrix spectral decomposition
tol	number, the tolerance threshold for SSTD convergence; if the error difference between two iterations is smaller than tol, then we stop the iteration and consider the algorithm converges
verbose	logic variable, whether to print the progress during permutation tests

Value

a list containing the following:

u_hat	vector, with length q; the estimated loading vector
v_hat	vector, with length p; the estimated tensor eigenvector
gamma_hat	number, the estimated sLTE \Lambda

References

Hu, J., Weber, J. N., Fuess, L. E., Steinel, N. C., Bolnick, D. I., & Wang, M. (2025). A spectral framework to map QTLs affecting joint differential networks of gene co-expression. PLOS Computational Biology, 21(4), e1012953.

Sun, W. W., Lu, J., Liu, H., & Cheng, G. (2017). "Provable sparse tensor decomposition." Journal of the Royal Statistical Society Series B: Statistical Methodology, 79(3), 899-916.

See Also

symmPMD()

snQTL documentation built on June 8, 2025, 1:53 p.m.