Solves the main optimization problem appearing in Bien, Bunea, & Xiao (2015):
min_P  Sighat  P _F^2 + lam * sum_l  (W_l * P)_{g_l} _2
where g_l are the outermost l(l+1) elements of a square matrix.
and  (W_l * P)_g_l ^2 = sum_m<=l w_lm^2 P_s_m^2. If a nonNULL delta
is provided,
then a constraint of the form $P >= delta I_p$ is included. Problem is solved by
performing blockwise coordinate descent on the dual problem. See paper
for more explanation.
1 2 
Sighat 
The sample covariance matrix 
lam 
Nonnegative penalty parameter. Controls sparsity level. 
w 

delta 
Lower bound on eigenvalues. If this is NULL (which is default), then no eigenvalue constraint is included. 
maxiter 
Number of iterations of blockwise coordinate descent to perform. 
tol 
Only used when 
Returns the convex banded estimate of covariance.
hierband.path
hierband.cv
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  set.seed(123)
p < 100
n < 50
K < 10
true < ma(p, K)
x < matrix(rnorm(n*p), n, p) %*% true$A
Sighat < cov(x)
fit < hierband(Sighat, lam=0.4)
min(eigen(fit)$values)
fit2 < hierband(Sighat, lam=0.4, delta=0.2)
min(eigen(fit2)$values)
# Use cross validation to select lambda:
path < hierband.path(Sighat)
cv < hierband.cv(path, x)
fit < hierband(Sighat, lam=cv$lam.best)

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