Regularized Least Squares

Now, suppose there is an error in the vector b such that b = bexact + ob, where ob = Eun, and un is the left singular vector of A corresponding to the singular value on. Let Xo be the solution of the perturbed least squares problem without regularization, Xexact be the solution to the exact least squares problem without regularization, and Xreg be the solution to the perturbed regularized least squares problem, i.e., Xo = argmin ||b - Axl?, X Xexact = argmin |bexact - Axl|?, X Xreg = argmin ||b - Axl|3 + IMx113- X (a) Show that I bexact - Axoll2 = IIIexact /12 + E-, where rexact = bexact - AXexact is the residual vector for the exact problem. (b) Now, determine a rank-1 M such that the solution to the perturbed regularized least squares satisfies bexact - AXregll2 = IT exact | |2, where |IM |2 = On/ Where bn is the length of the projection of bexact onto un. Note that the M satisfying this property is not unique. The above result shows that with an appropriate regularization, we can recover the optimal solution with respect to the exact right hand side even when solving with a perturbed right hand side