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Interpolating between Tikhonov regularization and spectral cutoff

Martin Sæbye Carøe, Mirza Karamehmedović, Pierre Maréchal

TL;DR

This work addresses the instability of ill-posed linear inversions $T f = g$ by introducing a one-parameter family of regularizations that interpolates between $T$ikhonov regularization and spectral cutoff via a spectral filter $q_\tau(\alpha,\sigma)$, with $q_\tau(0,\sigma)$ yielding Tikhonov and $q_\tau(\alpha,\sigma)\to$ a sharp cutoff as $\tau\to\infty$. The regularization is constructed through $R_\alpha=V[\sigma^{-1}q_\tau(\alpha,\sigma)]W^*$ (or equivalently $f_\alpha=(T^*T+H_\alpha^*H_\alpha)^{-1}T^*g$ with $H_\alpha=V[\sqrt{\alpha}(\sqrt{\alpha}/\sigma)^{\tau/2}]V^*$), and is shown to be well-defined and convergent to $T^\dagger$ as $\alpha\downarrow0$ under standard assumptions. Numerical experiments in 1D deconvolution and 2D inverse source problems illustrate that intermediate $\tau$ values can mitigate both smoothing-induced oscillations and spectral-cutoff Gibbs phenomena, offering a flexible regularization trade-off. This interpolating framework thus provides a tunable mechanism to adapt regularization to the problem, with potential integration into learning-based schemes for data-driven refinement.

Abstract

Regularizing a linear ill-posed operator equation can be achieved by manipulating the spectrum of the operator's pseudo-inverse. Tikhonov regularization and spectral cutoff are well-known techniques within this category. This paper introduces an interpolating formula that defines a one-parameter family of regularizations, where Tikhonov and spectral cutoff methods are represented as limiting cases. By adjusting the interpolating parameter taking into account the specific operator equation under consideration, it is possible to mitigate the limitations associated with both Tikhonov and spectral cutoff regularizations. The proposed approach is demonstrated through numerical simulations in the fields of signal and image processing.

Interpolating between Tikhonov regularization and spectral cutoff

TL;DR

This work addresses the instability of ill-posed linear inversions by introducing a one-parameter family of regularizations that interpolates between ikhonov regularization and spectral cutoff via a spectral filter , with yielding Tikhonov and a sharp cutoff as . The regularization is constructed through (or equivalently with ), and is shown to be well-defined and convergent to as under standard assumptions. Numerical experiments in 1D deconvolution and 2D inverse source problems illustrate that intermediate values can mitigate both smoothing-induced oscillations and spectral-cutoff Gibbs phenomena, offering a flexible regularization trade-off. This interpolating framework thus provides a tunable mechanism to adapt regularization to the problem, with potential integration into learning-based schemes for data-driven refinement.

Abstract

Regularizing a linear ill-posed operator equation can be achieved by manipulating the spectrum of the operator's pseudo-inverse. Tikhonov regularization and spectral cutoff are well-known techniques within this category. This paper introduces an interpolating formula that defines a one-parameter family of regularizations, where Tikhonov and spectral cutoff methods are represented as limiting cases. By adjusting the interpolating parameter taking into account the specific operator equation under consideration, it is possible to mitigate the limitations associated with both Tikhonov and spectral cutoff regularizations. The proposed approach is demonstrated through numerical simulations in the fields of signal and image processing.
Paper Structure (8 sections, 3 theorems, 35 equations, 6 figures, 3 tables)

This paper contains 8 sections, 3 theorems, 35 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. A family of regularization schemes
  3. Spectral regularization
  4. Interpolating between Tikhonov and spectral cutoff
  5. Numerical examples
  6. 1D example
  7. 2D example
  8. Appendix: Closedness of the operator $H_\alpha$

Key Result

Theorem 1

Let $T\colon F\to G$ be an injective bounded linear operator, where $F$ and $G$ are real Hilbert spaces. There then exist such that $T=W[\sigma]V^*$, in which $[\sigma]$ denotes the operator of multiplication by $\sigma$. Moreover, $\mathop{\mathrm{ran}}W=\overline{\mathop{\mathrm{ran}}T}$.

Figures (6)

  • Figure 1: A plot of $q_\tau$ with $\alpha=0.005$ (left) and $\alpha = 0.05$ (right) and four different values of $\tau$.
  • Figure 2: Reconstruction of $f^1$. Here we set $s_{\rm blur} = 0.1$, $N=1001$ and noise level $\sigma = 0.05$, and $\alpha$ is determined using the Morozov principle.
  • Figure 3: Reconstruction of $f^2$. Here we set $s_{\rm blur} = 0.1$, $N=1001$ and noise level $\sigma = 0.05$, and $\alpha$ is determined using the Morozov principle.
  • Figure 4: Reconstruction of $f^3$. Here we set $s_{\rm blur} = 0.03$, $N=1001$ and noise level $\sigma = 0.075$, and $\alpha$ is found using the Morozov principle.
  • Figure 5: The ground truth (top left) and reconstructions for different values of $\tau$. We set $D$ to be the unit disc, and $D_0$ to be the disc centered at the origin with radius $0.99$.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • proof
  • Corollary 1