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.
SSTD_R1(
T_obs,
u_ini,
v_ini,
max_iter = 20,
sumabs = 0.5,
niter = 20,
rho = 1000,
tol = 10^(-3),
verbose = FALSE
)a list containing the following:
vector, with length q; the estimated loading vector
vector, with length p; the estimated tensor eigenvector
number, the estimated sLTE \(\Lambda\)
array, a p-by-p-by-q tensor; each p-by-p layer in T_obs should be symmetric
vector, with length q; the random initialization for loading vector
vector, with length p; the random initialization for tensor eigenvector
integer, the maximal iteration number
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())
integer, the number of iterations to use in the PMD algorithm (see symmPMD())
number, a large positive constant adding to the diagonal elements to ensure positive definiteness in symmetric matrix spectral decomposition
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
logic variable, whether to print the progress during permutation tests
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.
symmPMD()