v2009.01.01 - Convex Optimization

670 APPENDIX E. PROJECTION
As for all subspace projectors, range of the projector is the subspace on which
projection is made: {P 1 Y P 2 | Y ∈ R m×p }. Altogether, for projectors P 1 and
P 2 of any rank, this means projection P 1 XP 2 is unique minimum-distance,
orthogonal
P 1 XP 2 − X ⊥ {P 1 Y P 2 | Y ∈ R m×p } in R mp (1870)
and P 1 and P 2 must each be symmetric (confer (1852)) to attain the infimum.
E.7.2.0.1 Proof. Minimum Frobenius norm (1869).
Defining P ∆ = A 1 (A † 1 + B 1 Z T 1 ) ,
inf ‖X − A 1 (A † 1 + B 1 Z1 T )X(A †T
2 + Z 2 B2 T )A T 2 ‖ 2 F
B 1 , B 2
= inf ‖X − PX(A †T
2 + Z 2 B2 T )A T 2 ‖ 2 F
B 1 , B 2
(
)
= inf tr (X T − A 2 (A † 2 + B 2 Z2 T )X T P T )(X − PX(A †T
2 + Z 2 B2 T )A T 2 )
B 1 , B 2
(
= inf tr X T X −X T PX(A †T
2 +Z 2 B2 T )A T 2 −A 2 (A †
B 1 , B 2
2+B 2 Z2 T )X T P T X
)
+A 2 (A † 2+B 2 Z2 T )X T P T PX(A †T
2 +Z 2 B2 T )A T 2
(1871)
Necessary conditions for a global minimum are ∇ B1 =0 and ∇ B2 =0. Terms
not containing B 2 in (1871) 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 Z T 2X T P T PXA †T
2 A T 2+A 2 B 2 Z T 2X T P T PXZ 2 B T 2A 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
(1872)
= 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 (1868) is implicit.
Were instead P T = ∆ (A †T
2 + Z 2 B2 T )A T 2 and the gradient with respect to B 1
observed, then similar results are obtained. The projector is unique.
Perpendicularity (1870) establishes uniqueness [92,4.9] of projection P 1 XP 2
on a matrix subspace. The minimum-distance projector is the orthogonal
projector, and vice versa.
)