v2009.01.01 - Convex Optimization

688 APPENDIX E. PROJECTION
b
C 2
∂K ⊥ C 1 ∩ C 2
(Pb) + Pb
C 1
Figure 142: First several alternating projections (1935) in
von Neumann-style projection of point b converging on closest point Pb
in intersection of two closed convex sets in R 2 ; C 1 and C 2 are partially
drawn in vicinity of their intersection. The pointed normal cone K ⊥ (1964)
is translated to Pb , the unique minimum-distance projection of b on
intersection. For this particular example, it is possible to start anywhere
in a large neighborhood of b and still converge to Pb . The alternating
projections are themselves robust with respect to some significant amount
of noise because they belong to translated normal cone.
number of steps; we find, in fact, the closest point.
E.10.0.1.1 Theorem. Kronecker projector. [284,2.7]
Given any projection matrices P 1 and P 2 (subspace projectors), then
P 1 ⊗ P 2 and P 1 ⊗ I (1925)
are projection matrices. The product preserves symmetry if present. ⋄
E.10.0.2
noncommutative projectors
Typically, one considers the method of alternating projection when projectors
do not commute; id est, when P 1 P 2 ≠P 2 P 1 .
The iconic example for noncommutative projectors illustrated in
Figure 142 shows the iterates converging to the closest point in the
intersection of two arbitrary convex sets. Yet simple examples like
Figure 143 reveal that noncommutative alternating projection does not