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Efficient Constrained Signal Reconstruction by Randomized Epigraphical Projection

Shunsuke Ono

Abstract

This paper proposes a randomized optimization framework for constrained signal reconstruction, where the word "constrained" implies that data-fidelity is imposed as a hard constraint instead of adding a data-fidelity term to an objective function to be minimized. Such formulation facilitates the selection of regularization terms and hyperparameters, but due to the non-separability of the data-fidelity constraint, it does not suit block-coordinate-wise randomization as is. To resolve this, we give another expression of the data-fidelity constraint via epigraphs, which enables to design a randomized solver based on a stochastic proximal algorithm with randomized epigraphical projection. Our method is very efficient especially when the problem involves non-structured large matrices. We apply our method to CT image reconstruction, where the advantage of our method over the deterministic counterpart is demonstrated.

Efficient Constrained Signal Reconstruction by Randomized Epigraphical Projection

Abstract

This paper proposes a randomized optimization framework for constrained signal reconstruction, where the word "constrained" implies that data-fidelity is imposed as a hard constraint instead of adding a data-fidelity term to an objective function to be minimized. Such formulation facilitates the selection of regularization terms and hyperparameters, but due to the non-separability of the data-fidelity constraint, it does not suit block-coordinate-wise randomization as is. To resolve this, we give another expression of the data-fidelity constraint via epigraphs, which enables to design a randomized solver based on a stochastic proximal algorithm with randomized epigraphical projection. Our method is very efficient especially when the problem involves non-structured large matrices. We apply our method to CT image reconstruction, where the advantage of our method over the deterministic counterpart is demonstrated.

Paper Structure

This paper contains 10 sections, 1 theorem, 19 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proximal Tools
  4. Stochastic Primal-Dual Hybrid Gradient Algorithm
  5. Proposed Method
  6. Problem Formulation
  7. Reformulation via Epigraphs
  8. Algorithm
  9. Numerical Experiments
  10. Conclusion

Key Result

Proposition 1

Let ${\mathbf z}\in{\mathbb R}^N$ and let ${\mathcal{S}} := \{({\mathbf x},\eta)\in{\mathbb R}^{N}\times{\mathbb R}|\|{\mathbf x}-{\mathbf z}\|^2\leq\eta\}$. Then, for every $({\mathbf y},\zeta)\in{\mathbb R}^{N}\times{\mathbb R}$, by letting $d := \|{\mathbf y}-{\mathbf z}\|$, the projection onto $ where

Figures (2)

  • Figure 1: Convergence profiles of Algorithm \ref{['alg:RandEpi']} ("Randomized") and its deterministic counterpart ("Deterministic") on CT image reconstruction in terms of the primal distance (left), objective function value (center), and constraint error (right). Note that the optimal value of the objective function (the black line in the center figure) was measured on ${\mathbf u}^\star$.
  • Figure 2: Resulting images on CT image reconstruction (200 iterations [epochs]).

Theorems & Definitions (3)

  • Remark 1: Note on Algorithm \ref{['alg:RandEpi']}
  • Remark 2: Projection computations in Algorithm \ref{['alg:RandEpi']}
  • Proposition 1: Epigraphical projection of squared distance