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External Division of Two Bregman Proximity Operators for Poisson Inverse Problems

Kazuki Haishima, Kyohei Suzuki, Konstantinos Slavakis

TL;DR

This work tackles sparse recovery from Poisson-corrupted measurements by augmenting the NoLips proximal framework with an external-division operator built from two Bregman proximity operators. The operator, denoted $T_{\omega,\eta_1,\eta_2, a}^{h}$, uses a shifted $\ell_1$ proximal term around $a$ and is designed to emulate the identity mapping for large coefficients while reducing estimation bias induced by $\ell_1$ regularization. The authors provide two reformulations that reveal a primal-space bias reduction and dual-space sparsity promotion, and embed the operator in a plug-and-play fashion inside NoLips with $f(\bm{x}) = D_{\phi}(\bm{A}\bm{x}, \bm{b})$ (Boltzmann-Shannon entropy). Numerical tests on synthetic data and image restoration demonstrate faster, more stable convergence and superior reconstruction quality compared to standard KL-based approaches. The work also outlines a path toward convergence guarantees and deeper theoretical understanding via the dual-space interpretation of the operator.

Abstract

This paper presents a novel method for recovering sparse vectors from linear models corrupted by Poisson noise. The contribution is twofold. First, an operator defined via the external division of two Bregman proximity operators is introduced to promote sparse solutions while mitigating the estimation bias induced by classical $\ell_1$-norm regularization. This operator is then embedded into the already established NoLips algorithm, replacing the standard Bregman proximity operator in a plug-and-play manner. Second, the geometric structure of the proposed external-division operator is elucidated through two complementary reformulations, which provide clear interpretations in terms of the primal and dual spaces of the Poisson inverse problem. Numerical tests show that the proposed method exhibits more stable convergence behavior than conventional Kullback-Leibler (KL)-based approaches and achieves significantly superior performance on synthetic data and an image restoration problem.

External Division of Two Bregman Proximity Operators for Poisson Inverse Problems

TL;DR

This work tackles sparse recovery from Poisson-corrupted measurements by augmenting the NoLips proximal framework with an external-division operator built from two Bregman proximity operators. The operator, denoted , uses a shifted proximal term around and is designed to emulate the identity mapping for large coefficients while reducing estimation bias induced by regularization. The authors provide two reformulations that reveal a primal-space bias reduction and dual-space sparsity promotion, and embed the operator in a plug-and-play fashion inside NoLips with (Boltzmann-Shannon entropy). Numerical tests on synthetic data and image restoration demonstrate faster, more stable convergence and superior reconstruction quality compared to standard KL-based approaches. The work also outlines a path toward convergence guarantees and deeper theoretical understanding via the dual-space interpretation of the operator.

Abstract

This paper presents a novel method for recovering sparse vectors from linear models corrupted by Poisson noise. The contribution is twofold. First, an operator defined via the external division of two Bregman proximity operators is introduced to promote sparse solutions while mitigating the estimation bias induced by classical -norm regularization. This operator is then embedded into the already established NoLips algorithm, replacing the standard Bregman proximity operator in a plug-and-play manner. Second, the geometric structure of the proposed external-division operator is elucidated through two complementary reformulations, which provide clear interpretations in terms of the primal and dual spaces of the Poisson inverse problem. Numerical tests show that the proposed method exhibits more stable convergence behavior than conventional Kullback-Leibler (KL)-based approaches and achieves significantly superior performance on synthetic data and an image restoration problem.
Paper Structure (12 sections, 4 theorems, 15 equations, 4 figures, 1 algorithm)

This paper contains 12 sections, 4 theorems, 15 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and definitions
  4. The NoLips algorithm
  5. The external-division operator
  6. Main Results
  7. External division of two Bregman proximity operators
  8. Two interpretations of the external-division operator
  9. Numerical Tests
  10. Synthetic data
  11. Image restoration
  12. Conclusions

Key Result

Proposition 1

Let $\eta_{2} \coloneqq \log{(\, (\omega - 1) /(\omega e^{-\eta_1}-1)\, )}$. Then, for any $\bm{x} \in \mathop{\rm int}\nolimits \mathop{\rm dom}\nolimits h$, the $i$th entry of $T_{\omega, \eta_1, \eta_2, a}^{h} (\bm{x})$ in eq:Bregman_external_division_operator, $\forall i \in \{1, 2, \ldots, n \} where $\kappa \coloneqq (\omega - 1) / (\omega e^{-\eta_1}-1)$.

Figures (4)

  • Figure 1: (a) The difference between the proposed $T_{\omega, \eta_1, \eta_2, a}^{h}$ and the standard Bregman proximity operator $\mathop{\rm Prox}\nolimits^h_{|\cdot - a|}$, where $h(\cdot)$ is the Boltzmann-Shannon entropy, and $\eta_1 = 0.3$, $\omega = 2$, $a=3$. Unlike the proposed $T_{\omega, \eta_1, \eta_2, a}^{h}$, the standard Bregman proximity operator leads to estimation bias for sufficiently large inputs. (b) The standard Bregman proximity operator $\mathop{\rm Prox}\nolimits^h_{\eta |\cdot - a|}$ with Burg's entropy $h(\cdot)$ for $a = 3$ and $\eta = 0.1$.
  • Figure 2: Convergence behavior for $m=100$, $n=150$, and $\rho=0.1$.
  • Figure 3: NMSE as a function of $\rho$ for (a) the over-determined case and (b) the under-determined case.
  • Figure 4: (a) Original, (b) Blurred and noisy, (c) R-KL, (d) F-KL, (e) proposed method ($a=0$), and (f) proposed method ($a > 0$).

Theorems & Definitions (5)

  • Proposition 1
  • Remark 1
  • Proposition 2
  • Corollary 1
  • Proposition 3