v2007.09.13 - Convex Optimization

E.7. ON VECTORIZED MATRICES OF HIGHER RANK 609then, given compatible X‖X −P 1 XP 2 ‖ F = inf ‖X −A 1 (A † 1+B 1 Z TB 1 , B 2 ∈R n×k 1 )X(A †T2 +Z 2 B2 T )A T 2 ‖ F (1756)As for all subspace projectors, range of the projector is the subspace on whichprojection is made: {P 1 Y P 2 | Y ∈ R m×p }. Altogether, for projectors P 1 andP 2 of any rank, this means projection P 1 XP 2 is unique minimum-distance,orthogonalP 1 XP 2 − X ⊥ {P 1 Y P 2 | Y ∈ R m×p } in R mp (1757)and P 1 and P 2 must each be symmetric (confer (1739)) to attain the infimum.E.7.2.0.1 Proof. Minimum Frobenius norm (1756).Defining P ∆ = A 1 (A † 1 + B 1 Z T 1 ) ,inf ‖X − A 1 (A † 1 + B 1 Z1 T )X(A †T2 + Z 2 B2 T )A T 2 ‖ 2 FB 1 , B 2= inf ‖X − PX(A †T2 + Z 2 B2 T )A T 2 ‖ 2 FB 1 , B 2()= inf tr (X T − A 2 (A † 2 + B 2 Z2 T )X T P T )(X − PX(A †T2 + Z 2 B2 T )A T 2 )B 1 , B 2(= inf tr X T X −X T PX(A †T2 +Z 2 B2 T )A T 2 −A 2 (A †B 1 , B 22+B 2 Z2 T )X T P T X)+A 2 (A † 2+B 2 Z2 T )X T P T PX(A †T2 +Z 2 B2 T )A T 2(1758)Necessary conditions for a global minimum are ∇ B1 =0 and ∇ B2 =0. Termsnot containing B 2 in (1758) will vanish from gradient ∇ B2 ; (D.2.3)(∇ B2 tr −X T PXZ 2 B2A T T 2 −A 2 B 2 Z2X T T P T X+A 2 A † 2X T P T PXZ 2 B2A T T 2)+A 2 B 2 Z2X T T P T PXA †T2 A T 2+A 2 B 2 Z2X T T P T PXZ 2 B2A T T 2= −2A T 2X T PXZ 2 + 2A T 2A 2 A † 2X T P T PXZ 2 +)2A T 2A 2 B 2 Z2X T T P T PXZ 2= A T 2(−X T + A 2 A † 2X T P T + A 2 B 2 Z2X T T P T PXZ 2(1759)= 0 ⇔R(B 1 )⊆ N(A 1 ) and R(B 2 )⊆ N(A 2 )(or Z 2 = 0) because A T = A T AA † . Symmetry requirement (1755) is implicit.Were instead P T = ∆ (A †T2 + Z 2 B2 T )A T 2 and the gradient with respect to B 1observed, then similar results are obtained. The projector is unique.