# ISS solver for linear model with lasso penalty

### Description

Solver for the entire solution path of coefficients for ISS.

### Usage

1 |

### Arguments

`X` |
An n-by-p matrix of predictors |

`y` |
Response Variable |

`intercept` |
if TRUE, an intercept is included in the model (and not penalized), otherwise no intercept is included. Default is TRUE. |

`normalize` |
if normalize, each variable is scaled to have L2 norm square-root n. Default is TRUE. |

`nvar` |
Maximal number of variables allowed in the model. |

### Details

The ISS solver computes the whole regularization path for lasso-penalty for linear model. It gives the piecewise constant solution path for Bregman Inverse Scale Space Differential Inclusion. It is the asymptotic limit of LB method with kaapa goes to infinity and alpha goes to zero.

### Value

An "LB" class object is returned. The list contains the call, the family, the path, the intercept term a0 and value for alpha, kappa, iter, and meanvalue, scale factor of X, meanx and normx.

### Author(s)

Feng Ruan, Jiechao Xiong and Yuan Yao

### References

Ohser, Ruan, Xiong, Yao and Yin, Sparse Recovery via Differential Inclusions, http://arxiv.org/abs/1406.7728

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 | ```
#Examples in the reference paper
library(MASS)
library(lars)
library(MASS)
library(lars)
n = 80;p = 100;k = 30;sigma = 1
Sigma = 1/(3*p)*matrix(rep(1,p^2),p,p)
diag(Sigma) = 1
A = mvrnorm(n, rep(0, p), Sigma)
u_ref = rep(0,p)
supp_ref = 1:k
u_ref[supp_ref] = rnorm(k)
u_ref[supp_ref] = u_ref[supp_ref]+sign(u_ref[supp_ref])
b = as.vector(A%*%u_ref + sigma*rnorm(n))
lasso = lars(A,b,normalize=FALSE,intercept=FALSE,max.steps=100)
par(mfrow=c(3,2))
matplot(n/lasso$lambda, lasso$beta[1:100,], xlab = bquote(n/lambda),
ylab = "Coefficients", xlim=c(0,3),ylim=c(range(lasso$beta)),type='l', main="Lasso")
object = iss(A,b,intercept=FALSE,normalize=FALSE)
plot(object,xlim=c(0,3),main=bquote("ISS"))
kappa_list = c(4,16,64,256)
alpha_list = 1/10/kappa_list
for (i in 1:4){
object <- lb(A,b,kappa_list[i],alpha_list[i],family="gaussian",group=FALSE,
trate=20,intercept=FALSE,normalize=FALSE)
plot(object,xlim=c(0,3),main=bquote(paste("LB ",kappa,"=",.(kappa_list[i]))))
}
``` |

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