#### Title

A theory for optimal regularization in the finite dimensional case

#### Document Type

Article

#### Publication Date

1-1-1982

#### Abstract

Consider the matrix problem Ax = y + ε = y ̃ in the case where A is known precisely, the problem is ill conditioned, and ε is a random noise vector. Compute regularized "ridge" estimates, x ̃λ = (A*A + λI)-1 A* y ̃,where * denotes matrix transpose. Of great concern is the determination of the value of λ for which x̃λ "best" approximates x0 = A + y. Let Q = {norm of matrix} x ̃ λ - x0{norm of matrix}2,and define λ0 to be the value of λ for which Q is a minimum. We look for λ0 among solutions of dQ/dλ = 0. Though Q is not computable (since ε is unknown), we can use this approach to study the behavior of λ0 as a function of y and ε. Theorems involving "noise to signal ratios" determine when λ0 exists and define the cases λ0 > 0 and λ0 = ∞. Estimates for λ0 and the minimum square error Q0 = Q(λ0) are derived. © 1982.

#### Publication Title

Linear Algebra and Its Applications

#### Recommended Citation

Hilgers, J.
(1982).
A theory for optimal regularization in the finite dimensional case.
*
Linear Algebra and Its Applications,
48*(C), 359-379.
http://doi.org/10.1016/0024-3795(82)90121-5

Retrieved from: https://digitalcommons.mtu.edu/michigantech-p/5392