What are the solutions to Lasso and Bridge regressions?

I am taking a course about ridge, bridge and lasso regression. My teacher showed us that the solution to ridge regression can be written in the form

$$\beta=(X^TX+\lambda I)^{-1}X^T Y$$

My question is do lasso and bridge regression have a similar solution?

• At least for LASSO this has been answered already in the site (albeit it's written in the question). – Firebug Nov 1 '17 at 15:39
• @Firebug, I have already checked that question before asking my question but I didn't understand what closed form lasso is and how it is related to lasso regression – user3741635 Nov 1 '17 at 15:53
• It's written right at the beginning: $$\beta^{\text{lasso}}= \operatorname*{argmin}_\beta \| y-X\beta\|^2_2 + \alpha \| \beta\|_1$$ – Firebug Nov 1 '17 at 16:04
• At the end of the question is a demonstration why LASSO regression has no closed form solution (in fact some exist for special cases, like orthogonal dependent variables). – Firebug Nov 1 '17 at 16:05
• @Firebug That link doesn't demonstrate that lasso has no closed form solution: it just shows that the derivation attempted by OP is wrong. (I edited the title of the post last month since I saw someone else mistakenly reference it in the same way.) – user795305 Nov 1 '17 at 22:10