Multiple Column Partitioned Min Max Dr. Keith Evan Schubert

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

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

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

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||

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

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

Cost function

Cost function near singularity

Not The Same Under Regularized Over Regularized

Multiple Error Bounds

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

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

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

Performance

Image Separation

Primary Image

Second Image