Tikhonov–Phillips regularizations in linear models with blurred design

Abstract

The paper deals with recovering an unknown vector β ∈ ℝ^p based on the observations Y = Xβ + єξ and Z = X + σζ, where X is an unknown n × p matrix with n ≥ p, ξ ∈ ℝ^p is a standard white Gaussian noise, ζ is an n × p matrix with i.i.d. standard Gaussian entries, and є, σ ∈ ℝ⁺ are known noise levels. It is assumed that X has a large condition number and p is large. Therefore, in order to estimate β, the simple Tikhonov–Phillips regularization (ridge regression) with a data-driven regularization parameter is used. For this estimation method, we study the effect of noise σζ on the quality of recovering Xβ using concentration inequalities for the prediction error.