THEOREM 4.25. The unique solution X to the program: (4.4.11) X min X|*+$1X -MIIF} , is given by X * = Du-1[M]. (4.4.12) (Proof of Theorem 4.25). We prove Theorem 4.25. The goal here is to show that the solution to min || X|* + 2 1X - MIIF (4.7.14) X is given by DI[M]. 1 Argue that Problem (4.7.14) is strongly convex, and hence has a unique optimal solution. 2 Show that a solution X, is optimal if and only if X. E M - all.Ilx (X*). 3 Using the condition from part 2, show that if M is diagonal, i.e., Mij = 0 for i # j, then SI[M] is the unique optimal solution to (4.7.14). 4 Use the SVD to argue that in general, DI[M] is the unique optimal solution to (4.7.14)