Recall ridge regression: $$\beta(\lambda)=(X^TX+\lambda I)^{-1}X^TY.$$
Is there an easy way I can find $\lambda$ so that the solution shrinks to the unit ball: $\beta^T(\lambda)\beta(\lambda)=1$?
My first thought is define $f(\lambda) = Y^TX(X^TX+\lambda I)^{-2}X^TY - 1$ then solve using Newton method, but I'd rather avoid iterative method if there's an analytic solution. If not, a little help with derivatives would be nice :)
I also thought about Sherman-Morrison-Woodbury, but I didn't get very far.