Alternating Manifold Proximal Gradient Method for Sparse PCA.
amanpg
Description
Uses an alternating manifold proximal gradient (AManPG) method to find sparse principal components from the given data or covariance matrix.
Only base R is required to be installed.
Usage
spca.amanpg(z, lambda1, lambda2, f_palm = 1e5, x0 = NULL, y0 = NULL, k = 0, type = 0, gamma = 0.5,
maxiter = 1e4, tol = 1e-5, normalize = TRUE, verbose = FALSE)
Arguments
| Name | Type | Description |
|---|---|---|
z | matrix | Either the data matrix or sample covariance matrix |
lambda1 | matrix | List of parameters of length n for L1-norm penalty |
lambda2 | double | L2-norm penalty term |
f_palm | double | Upper bound for the gradient value to reach convergence, default value is 1e5 |
x0 | matrix | Initial x-values for the gradient method, default value is the first n right singular vectors |
y0 | matrix | Initial y-values for the gradient method, default value is the first n right singular vectors |
k | integer | Number of principal components desired, default is 0 (returns min(n-1, p) principal components) |
type | integer | If 0, b is expected to be a data matrix, and otherwise b is expected to be a covariance matrix; default is 0 |
gamma | double | Parameter to control how quickly the step size changes in each iteration, default is 0.5 |
maxiter | integer | Maximum number of iterations allowed in the gradient method, default is 1e4 |
tol | double | Tolerance value required to indicate convergence (calculated as difference between iteration f-values), default is 1e-5 |
normalize | logical | Center and normalize rows to Euclidean length 1 if True, default is True |
verbose | logical | Function prints progress between iterations if True, default is False |
Value
Returns a dictionary with the following key-value pairs:
| Key | Value Type | Value |
|---|---|---|
iter | integer | Total number of iterations executed |
f_manpg | double | Final gradient value |
sparsity | float | Number of sparse loadings (loadings == 0) divided by number of all loadings |
time | double | Number of seconds for execution |
x | matrix | Corresponding ndarray in subproblem to the loadings |
loadings | matrix | Loadings of the sparse principal components |
Authors
Shixiang Chen, Justin Huang, Benjamin Jochem, Shiqian Ma, Lingzhou Xue and Hui Zou
References
Chen, S., Ma, S., Xue, L., and Zou, H. (2020) "An Alternating Manifold Proximal Gradient Method for Sparse Principal Component Analysis and Sparse Canonical Correlation Analysis" INFORMS Journal on Optimization 2:3, 192-208
Zou, H., Hastie, T., & Tibshirani, R. (2006). Sparse principal component analysis. Journal of Computational and Graphical Statistics, 15(2), 265-286.
Zou, H., & Xue, L. (2018). A selective overview of sparse principal component analysis. Proceedings of the IEEE, 106(8), 1311-1320.
Example
See SPCA.R for a more in-depth example.
library('SPCA')
#see SPCA.R for a more in-depth example
d <- 500 # dimension
m <- 1000 # sample size
set.seed(10)
a <- normalize(matrix(rnorm(m * d), m, d))
lambda1 <- 0.1 * matrix(data=1, nrow=4, ncol=1)
x0 <- svd(a, nv=4)$v
sprout <- spca.amanpg(a, lambda1, lambda2=Inf, f_palm=1e5, x0=x0, y0=x0, k=4, type=0, gamma=0.5,
maxiter=1e4, tol=1e-5, normalize = FALSE, verbose=FALSE)
print(paste(sprout$iter, "iterations,", sprout$sparsity, "sparsity,", sprout$time))
#extract loadings
#print(sprout$loadings)