Landa in lasso problem


(pop) #1

hello
for sparse recovery using lasso , Landa should be chosen like this or arbitrary ?
minimize norm( Ax - b ,2) + landa* norm( x,1)
lambda_max = norm( A’b, ‘inf’ );
lambda = 0.1
lambda_max;
what about this? in this mode how should I choose error in term of A and y?
minimize norm(x,1)
s.t
norm(y-Ax)<error
Thanks


(Royi) #2

Hello,

You model is given by:

$$ {\left| A x - b \right|}^{2} + \lambda {\left| x \right|}_{1} $$

To set Lambda correctly, the way I see it, you have 2 main choices:

Bayesian View

You may look at this problem as the MAP Estimator of a Laplacian Prior and Gaussian Noise model.

Namely A x - b \sim \mathcal{N} \left( 0, {{\sigma}_{n}}^{2} \right) and x \sim \mathcal{L} \left( 0, b \right) .
Now, the parameter \lambda can be derived from {\sigma}_{n} and b of your model.

Optimization View

See the above as the some kind of Lagrangian of minimization problem of the {\ell}_{1} Norm of x under constraints of the Linear System equality (I’m not accurate here as the Lagrangian should be in Linear form but the idea…).

It is related to Lasso Regression (See its Lagrangian form) and Basis Pursuit.

People here are much more knowledge in this then me and can expand.