# ProxADMM: Solving penalized Frobenius problem. In spcov: Sparse Estimation of a Covariance Matrix

## 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

 `1` ```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.

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```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) ```