1. Multiple Column Partitioned Min Max
Dr. Keith Evan Schubert

2. Motivating Problem Signal Separation with uncertainty
Data streams sent by multiple sources
Same channel
Multiple receivers with different gains
Different uncertainty for each source

3. Geometry of Least Squares
Project b into range of A
Error in b only b
R(A)
r=b-Ax Ax

4. Geometry of Total Least Squares
Find nearest space to R(A) and b
Project b and R(A) into it
Basic System assumed consistent b
R(A) A
Ax=b

5. Geometry of Min Max Define error cone around R(A)
Project b to the far side of cone
Basic System assumed worst in bounded region (cone) b
R(A)
h||x||

6. Algebraic Min Max Solution
Assume b not in range of A
Form of Solution
Get  from Secular Equation
Solve using root finder
Not closed form

7. Similar Problem Tikhonov Regression Closed form solution
Why not use this?

8. Cost function

9. Cost function near singularity

10. Not The Same
Under Regularized
Over Regularized

11. Multiple Error Bounds

12. Column Dependence 1=2=.25 Force x(2)=0 Allow x(2)≠0
Cost is approx 1.330
Cost is approx 1.329

13. Solutions Directly use convex cost function
Ellipsoidal algorithm
Sum of Euclidian Norms solvers
Successive over-relaxation
Subgradients
Overton’s method
General nonlinear solvers
Take gradient and set equal zero
Get secular equation
Faster because solving smaller problem with more information

14. Solution
with
Find the  that makes
gi()=0 by a rootfinder

15. Performance

16. Image Separation

17. Primary Image

18. Second Image