v2010.10.26 - Convex Optimization

C.2. TRACE, SINGULAR AND EIGEN VALUES 655This nuclear norm is convex C.1 and dual to spectral norm. [203, p.214][61,A.1.6] Given singular value decomposition A = SΣQ T ∈ R m×n(A.6), then X ⋆ = SQ T ∈ R m×n is an optimal solution to maximization(confer2.3.2.0.5) while X ⋆ = SΣS T ∈ S m and Y ⋆ = QΣQ T ∈ S n is anoptimal solution to minimization [135]. Srebro [323] asserts∑iσ(A) i = 1 minimize ‖U‖ 2 2 F + ‖V U,V‖2 Fsubject to A = UV T(1687)= minimize ‖U‖ F ‖V ‖ FU,Vsubject to A = UV TC.2.0.0.1 Exercise. Optimal matrix factorization.Prove (1687). C.2For X ∈ S m , Y ∈ S n , A∈ C ⊆ R m×n for set C convex, and σ(A)denoting the singular values of A [135,3]minimizeA∑σ(A) isubject to A ∈ Ci≡1minimize 2A , X , Ysubject totrX + trY[ ] X AA T ≽ 0YA ∈ C(1688)C.1 discernible as envelope of the rank function (1362) or as supremum of functions linearin A (Figure 74).C.2 Hint: Write A = SΣQ T ∈ R m×n and[ ] [ X A UA T =Y V][ U T V T ] ≽ 0Show U ⋆ = S √ Σ∈ R m×min{m,n} and V ⋆ = Q √ Σ∈ R n×min{m,n} , hence ‖U ⋆ ‖ 2 F = ‖V ⋆ ‖ 2 F .