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Inverse learning in Hilbert scales

Abhishake Rastogi, Peter Mathé

Abstract

We study the linear ill-posed inverse problem with noisy data in the statistical learning setting. Approximate reconstructions from random noisy data are sought with general regularization schemes in Hilbert scale. We discuss the rates of convergence for the regularized solution under the prior assumptions and a certain link condition. We express the error in terms of certain distance functions. For regression functions with smoothness given in terms of source conditions the error bound can then be explicitly established.

Inverse learning in Hilbert scales

Abstract

We study the linear ill-posed inverse problem with noisy data in the statistical learning setting. Approximate reconstructions from random noisy data are sought with general regularization schemes in Hilbert scale. We discuss the rates of convergence for the regularized solution under the prior assumptions and a certain link condition. We express the error in terms of certain distance functions. For regression functions with smoothness given in terms of source conditions the error bound can then be explicitly established.

Paper Structure

This paper contains 20 sections, 12 theorems, 106 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Notation and Assumptions
  3. Reproducing Kernel Hilbert space and related operators
  4. Link condition
  5. Effective dimension
  6. Regularization Schemes
  7. Convergence analysis
  8. The oversmoothing case
  9. The regular case
  10. Smoothness in terms of source-wise representation
  11. The oversmoothing case
  12. The regular case
  13. Taking the behavior of effective dimension into account
  14. Optimality of the error bounds
  15. Relating smoothness
  16. ...and 5 more sections

Key Result

Proposition 2.6

Under Assumption ass:link we have Moreover, we have that

Theorems & Definitions (34)

  • Definition 2.1: Vector-valued reproducing kernel Hilbert space
  • Definition 2.2: Operator-valued positive semi-definite kernel
  • Example 2.3
  • Definition 2.4: Index function
  • Example 2.5
  • Proposition 2.6
  • proof
  • Remark 2.7
  • Example 2.8: Finitely smoothing
  • Example 2.9: Infinitely smoothing
  • ...and 24 more