Good evening everybody,
I need help with an excercise on Regularised Regression.
What I need to do is turn an Elastic-net problem: \begin{equation} argmin_\omega \Vert y-\Phi(x)^T \omega \Vert_2^2 + \lambda(\alpha\Vert\omega\Vert_2^2 + (1-\alpha)\Vert w\Vert_1) \end{equation}
Into a LASSO: \begin{equation} argmin_\omega \Vert \bar y-\bar\Phi(x)^T \omega \Vert_2^2 + \bar\lambda\Vert w\Vert_1 \end{equation}
for suitable new $\bar\lambda$, $\bar y$ and $\bar \Phi(x)$.
What I have thought is that, if $y-\Phi(x)^T\omega$ and $\omega$ are orthogonal (this is the case if $\omega$ is the solution of the Least Squares problem without regularization), we have that: \begin{equation} \Vert y-\Phi(x)^T \omega \Vert_2^2 + \lambda\alpha\Vert\omega\Vert_2^2 = \Vert y-(\Phi(x)^T-\lambda^2\alpha^2 I)\omega\Vert_2^2 \end{equation}
And so, I have solved the problem. But this is not the case.
Any idea?
Thank you in advance