LASSO and Dimension Reduction Is there a way to use LASSO for dimension reduction by choosing only relevant attributes? Is there a defined procedure for doing it optimally? Is there a example dataset that can be used for it?
 A: Yes, LASSO can be used for reducing the number of attributes. Using cross-validation to select the optimal value of lambda to be used for the LASSO would be a good idea. The following example is from the book "An Introduction to Statistical Learning with Applications in R". It uses the glmnet library for performing LASSO and the Hitters data set from ISLR
library(glmnet)
library(ISLR)

attach(Hitters)

x = model.matrix(Salary~., Hitters)[,-1]
y = Hitters[row.names(x),]$Salary

set.seed(1)
train = sample(1:nrow(x), nrow(x)*0.75)
test = (-train)
y.test = y[test]

# values of lambdas to use
grid = 10 ^ seq(10,-2, length=100)

lasso.model = glmnet(x[train,], y[train], alpha=1, lambda=grid)
plot(lasso.model)

# Using Cross Validation to find the best lambda
lasso_cv_model =cv.glmnet(x[train,], y[train], alpha=1, lambda=grid)
plot(lasso_cv_model)

best_lambda = lasso_cv_model$lambda.min
best_lambda

pred = predict(lasso.model, s=best_lambda, newx=x[test,])
mean((pred-y.test)^2)

lasso.coef = predict(lasso.model, type="coefficient", s=best_lambda)[1:20,]
lasso.coef[lasso.coef!=0]   
# only non-zero coefficients

The above mentioned book explains LASSO in detail - page 219. Also check out the lab exercise. 
A: The are a few ways, the simplest may be AIC, BIC etc. But I would suggest you to use out-of-sample performance to select the best $\lambda$ then see which attributes' coefficients are 0s in the best model.
A: LASSO, as it is, is not a good way to screen-out noisy covariates, for the reason mentioned above (correlations among covariates), but not only. Unless you have a truly strong signal in the dataset, you will never be able to screen out only the relevant covariates unless you adjust the procedure. Also, it is well known that cross-validation with LASSO leads to overfitted solutions (although the prediction performance-wise usually works well), so be careful with it too. 
I think the best reference is: https://faculty.chicagobooth.edu/christian.hansen/research/Lasso%20paper%20draft%201_28FINAL.pdf. They provide detailed discussion and example codes (with STATA and R). 
