Graph Fused Lasso (FGSG)
Description
Given A = {a_1,…,a_n}, the response y, and a set of edges E, this function aims to solves
min 1/2Axy^2 + λ_1x_1 + λ_2 ∑_{(i,j) in E}w_(i,j)x_ir_(i,j)x_j
where w_(i,j) is the weight of the edge (i,j), and r_(i,j) is the sign of the correlation between features a_i and a_j. The weight and sign can be specified in Rwt: w=Rwt, and r=sign(Rwt).
Usage
 gflasso(A, y, tp, s1, s2, RmaxIter = 100,
RaMaxIter = 1000, Rrho = 5, Rtau = 0.15,
Rwt = rep(1, length(tp)), Rtol = 0.001,
RaTol = 0.001, x0 = rep(0, ncol(A)))

Arguments
A 
A The data matrix of size n \times p, each row corresponds to one sample.

y 
y The response vector of length n.

tp 
tp The edges vector of length 2*g (eg. (1,2,3,4) means an edge between 1 and 2, and an edge between 3 and 4, g=2 is the number of edges).

s1 
s1 The l_1 regularization parameter, s1 >=0.

s2 
s2 Tge grouping penatly parameter, s2 >=0.

RmaxIter 
RmaxIter The maximum number of iterations in DC programming (default 100).

RaMaxIter 
RaMaxIter The maximum number of iterations in ADMM (default 1000).

Rrho 
Rrho The dual update length ofor ADMM (default 5).

Rtau 
Rtau The tuning parameter for nonconvex penalty (default 0.15).

Rwt 
Rwt The weight and signs of edges (default rep(1,g)).

Rtol 
Rtol The tolerance for convergence in DC programming (default 1e3).

RaTol 
RaTol The tolerance for convergence in ADMM (default 1e3).

x0 
x0 The returned weight vector (default rep(0,p)).

Value
Returned value x0 is the solution to the optimizaiton problem.
Author(s)
Yiwen Sun
References
S.Kim and E.Xing. Statisctical estimation of correlated genome associations to a quantitative trait network. PLoS genetics, 5(8):e1000587, 2009
Examples
 A<matrix(rnorm(25),5,5)
y<rnorm(5)
tp<c(1,2,2,3,3,4,4,5)
gflasso(A,y,tp,0,0)
