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Sparse variational regularization with oversmoothing penalty term in the scale of sequence spaces

Robert Plato, Bernd Hofmann

TL;DR

This work addresses linear ill-posed operator equations $A u=v$ with $A:\ell_r\to V$ that satisfy a conditional stability bound in a weaker sequence space. It analyzes variational regularization using penalties $\mathcal{R}_p(u)=\|u\|_p^p$ and $\mathcal{R}_0(u)=\|u\|_0$, deriving stability and convergence rates even in oversmoothing scenarios where $u^\dagger$ may lie outside the penalty domain. A priori parameter choices yield explicit rates: $\|u_{\alpha_\delta}^\delta-u^\dagger\|_r=\mathcal{O}(\delta^{\gamma_1})$ and $\mathcal{R}_p(u_{\alpha_\delta}^\delta)=\mathcal{O}(\delta^{-\gamma_2})$ with $\gamma_1=(a/r)\,(r-q)/(a-q)$ and $\gamma_2=((q-p)a)/(a-q)$; oversmoothing is handled via hard-thresholding auxiliaries. Post-processing with hard-thresholding can enforce sparsity, yielding $v_{\beta_\delta}^\delta=H_{\beta_\delta}(u_{\alpha_\delta}^\delta)$ that preserves the same convergence rates while ensuring $\mathcal{R}_p(v_{\beta_\delta}^\delta)=\mathcal{O}(\delta^{-\gamma_2})$. Overall, the paper advances sparse regularization in sequence spaces, providing a rigorous framework for oversmoothing penalties and sparsity-promoting post-processing in Banach-space settings.

Abstract

In this work, we consider a class of linear ill-posed problems with operators that map from the sequence space $ \ell_r $ ($r \ge 1$) into a Banach space and in addition satisfy a conditional stability estimate in the scale of sequence spaces $ \ell_q, \, q \ge 0 $. For the regularization of such problems in the presence of deterministic noise, we consider variational regularization with a penalty functional either of the form $ \mathcal{R} =\Vert \cdot \Vert_p^p $ for some $ p > 0 $ or in form of the counting measure $\mathcal{R}_0 = \Vert \cdot \Vert_0 $. The latter case guarantees sparsity of the corresponding regularized solutions. In this framework, we present first stability and then convergence rates for suitable a priori parameter choices. The results cover the oversmoothing situation, where the desired solution does not belong to the domain of definition of the considered penalty functional. The analysis of the oversmoothing case utilizes auxiliary elements that are defined by means of hard thresholding. Such technique can also be used for post processing to guarantee sparsity.

Sparse variational regularization with oversmoothing penalty term in the scale of sequence spaces

TL;DR

This work addresses linear ill-posed operator equations with that satisfy a conditional stability bound in a weaker sequence space. It analyzes variational regularization using penalties and , deriving stability and convergence rates even in oversmoothing scenarios where may lie outside the penalty domain. A priori parameter choices yield explicit rates: and with and ; oversmoothing is handled via hard-thresholding auxiliaries. Post-processing with hard-thresholding can enforce sparsity, yielding that preserves the same convergence rates while ensuring . Overall, the paper advances sparse regularization in sequence spaces, providing a rigorous framework for oversmoothing penalties and sparsity-promoting post-processing in Banach-space settings.

Abstract

In this work, we consider a class of linear ill-posed problems with operators that map from the sequence space () into a Banach space and in addition satisfy a conditional stability estimate in the scale of sequence spaces . For the regularization of such problems in the presence of deterministic noise, we consider variational regularization with a penalty functional either of the form for some or in form of the counting measure . The latter case guarantees sparsity of the corresponding regularized solutions. In this framework, we present first stability and then convergence rates for suitable a priori parameter choices. The results cover the oversmoothing situation, where the desired solution does not belong to the domain of definition of the considered penalty functional. The analysis of the oversmoothing case utilizes auxiliary elements that are defined by means of hard thresholding. Such technique can also be used for post processing to guarantee sparsity.
Paper Structure (12 sections, 8 theorems, 41 equations, 1 figure)

This paper contains 12 sections, 8 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Outline
  3. Ill-posedness types of linear operator equations in Banach spaces and a specific model scheme
  4. General case distinctions
  5. A specific model scheme adopted to sequence spaces
  6. Well-posedness
  7. A priori parameter choice
  8. Hard thresholding
  9. The basic steps
  10. Proof of Theorem \ref{['th:main']}
  11. Post processing
  12. Conclusions and outlook

Key Result

Proposition 1

Consider for $0 < p < \infty$ the sequence $\{u_{n}\}_{n=1}^\infty \subset \ell_p\,$ as well as the element $u \in \ell_p$, and let the following two conditions be satisfied: (a) convergence of $u_n$ to $u$ as $n \to \infty$ holds componentwise, i.e. $u_{n,k} \to u_k$ as $n \to \infty$ for each $k \

Figures (1)

  • Figure 1: Relations between strict singularity, compactness and type of ill-posedness for equations in Banach spaces with injective bounded linear operators (cf. FHV15).

Theorems & Definitions (21)

  • Proposition 1: Radon--Riesz property
  • proof
  • Lemma 2
  • proof
  • Definition 3
  • Definition 4
  • Proposition 5
  • Remark 6
  • proof : Sketch of a proof of Proposition \ref{['pro:well']}
  • Theorem 8
  • ...and 11 more