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Convergence analysis of Tikhonov regularization for non-linear statistical inverse learning problems

Abhishake Rastogi, Gilles Blanchard, Peter Mathé

Abstract

We study a non-linear statistical inverse learning problem, where we observe the noisy image of a quantity through a non-linear operator at some random design points. We consider the widely used Tikhonov regularization (or method of regularization, MOR) approach to reconstruct the estimator of the quantity for the non-linear ill-posed inverse problem. The estimator is defined as the minimizer of a Tikhonov functional, which is the sum of a data misfit term and a quadratic penalty term. We develop a theoretical analysis for the minimizer of the Tikhonov regularization scheme using the ansatz of reproducing kernel Hilbert spaces. We discuss optimal rates of convergence for the proposed scheme, uniformly over classes of admissible solutions, defined through appropriate source conditions.

Convergence analysis of Tikhonov regularization for non-linear statistical inverse learning problems

Abstract

We study a non-linear statistical inverse learning problem, where we observe the noisy image of a quantity through a non-linear operator at some random design points. We consider the widely used Tikhonov regularization (or method of regularization, MOR) approach to reconstruct the estimator of the quantity for the non-linear ill-posed inverse problem. The estimator is defined as the minimizer of a Tikhonov functional, which is the sum of a data misfit term and a quadratic penalty term. We develop a theoretical analysis for the minimizer of the Tikhonov regularization scheme using the ansatz of reproducing kernel Hilbert spaces. We discuss optimal rates of convergence for the proposed scheme, uniformly over classes of admissible solutions, defined through appropriate source conditions.

Paper Structure

This paper contains 11 sections, 18 theorems, 168 equations, 1 table.

Table of Contents

  1. Introduction
  2. Setup and basic definitions
  3. Consistency
  4. Convergence rates
  5. Upper rates of convergence
  6. Lower rates of convergence
  7. Discussion
  8. Notation and probabilistic estimates
  9. Proof of the consistency results
  10. Proof of upper rates
  11. Proof of lower rates

Key Result

Theorem 3.1

Suppose that Assumptions assmpt2, assmpt1, ass:Lipschitz hold true and $\sigma_\rho^2:=\int_Z\left\|y-A(f_\rho)(x)\right\|_Y^2d\rho(x,y)<\infty$. Let $f_{\mathbf{z},\lambda}$ denote a (not necessarily unique) solution to the minimization problem (fzl) and assume that the regularization parameter $\l Then we have that

Theorems & Definitions (37)

  • Definition 2.1
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3
  • Remark 4.1
  • Example 4.2
  • Theorem 4.3
  • Theorem 4.4
  • Theorem 4.5
  • Corollary 4.6
  • ...and 27 more