Solving penalized Frobenius problem.

Description

This function solves the optimization problem

Minimize_X (1/2)||X - A||_F^2 + lam||P*X||_1 s.t. X >= del * I.

This is the prox function for the generalized gradient descent of Bien & Tibshirani 2011 (see full reference below).

Usage

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ProxADMM(A, del, lam, P, rho=.1, tol=1e-6, maxiters=100, verb=FALSE)

Arguments

A

A symmetric matrix.

del

A non-negative scalar. Lower bound on eigenvalues.

lam

A non-negative scalar. L1 penalty parameter.

P

Matrix with non-negative elements and dimension of A. Allows for differing L1 penalty parameters.

rho

ADMM parameter. Can affect rate of convergence a lot.

tol

Convergence threshold.

maxiters

Maximum number of iterations.

verb

Controls whether to be verbose.

Details

This is the R implementation of the algorithm in Appendix 3 of Bien, J., and Tibshirani, R. (2011), "Sparse Estimation of a Covariance Matrix," Biometrika. 98(4). 807–820. It uses an ADMM approach to solve the problem

Minimize_X (1/2)||X - A||_F^2 + lam||P*X||_1 s.t. X >= del * I.

Here, the multiplication between P and X is elementwise. The inequality in the constraint is a lower bound on the minimum eigenvalue of the matrix X.

Note that there are two variables X and Z that are outputted. Both are estimates of the optimal X. However, Z has exact zeros whereas X has eigenvalues at least del. Running the ADMM algorithm long enough, these two are guaranteed to converge.

Value

X

Estimate of optimal X.

Z

Estimate of optimal X.

obj

Objective values.

Author(s)

Jacob Bien and Rob Tibshirani

References

Bien, J., and Tibshirani, R. (2011), "Sparse Estimation of a Covariance Matrix," Biometrika. 98(4). 807–820.

See Also

spcov

Examples

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set.seed(1)
n <- 100
p <- 200
# generate a covariance matrix:
model <- GenerateCliquesCovariance(ncliques=4, cliquesize=p / 4, 1)

# generate data matrix with x[i, ] ~ N(0, model$Sigma):
x <- matrix(rnorm(n * p), ncol=p) %*% model$A
S <- var(x)

# compute sparse, positive covariance estimator:
P <- matrix(1, p, p)
diag(P) <- 0
lam <- 0.1
aa <- ProxADMM(S, 0.01, lam, P)