v2010.10.26 - Convex Optimization

E.10. ALTERNATING PROJECTION 749bC 2∂K ⊥ C 1 ∩ C 2(Pb) + PbC 1Figure 167: First several alternating projections in von Neumann-styleprojection (2059), of point b , converging on closest point Pb in intersectionof two closed convex sets in R 2 : C 1 and C 2 are partially drawn in vicinityof their intersection. Pointed normal cone K ⊥ (449) is translated to Pb , theunique minimum-distance projection of b on intersection. For this particularexample, it is possible to start anywhere in a large neighborhood of b and stillconverge to Pb . Alternating projections are themselves robust with respectto significant noise because they belong to translated normal cone.means we can find a point in the intersection of those subspaces in a finitenumber of steps; we find, in fact, the closest point.E.10.0.1.1 Theorem. Kronecker projector. [327,2.7]Given any projection matrices P 1 and P 2 (subspace projectors), thenP 1 ⊗ P 2 and P 1 ⊗ I (2048)are projection matrices. The product preserves symmetry if present. ⋄E.10.0.2noncommutative projectorsTypically, one considers the method of alternating projection when projectorsdo not commute; id est, when P 1 P 2 ≠P 2 P 1 .The iconic example for noncommutative projectors illustrated inFigure 167 shows the iterates converging to the closest point in theintersection of two arbitrary convex sets. Yet simple examples likeFigure 168 reveal that noncommutative alternating projection does not