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Decoupled Solution for Composite Sparse-plus-Smooth Inverse Problems

Adrian Jarret, Julien Fageot

TL;DR

This work studies a variational framework formulated as an optimization problem over the pairs of components using two regularization terms, each of them acting on a different part of the solution, and reveals a decoupling between the two components.

Abstract

We consider composite linear inverse problems where the signal to recover is modeled as a sum of two functions. We study a variational framework formulated as an optimization problem over the pairs of components using two regularization terms, each of them acting on a different part of the solution. The specificity of our work is to study the case where one component is regularized with an atomic norm over a Banach space, which is known to promote sparse reconstruction, while the other is regularized with a quadratic norm over a Hilbert space, which promotes smooth solution. We show how this composite optimization problem can be reduced to an optimization problem over the Banach space component only up to a linear problem. This reveals a decoupling between the two components, allowing for a new composite representer theorem. It naturally induces a decoupled numerical procedure to solve the composite optimization problem. We exemplify our main result with a composite deconvolution problem of Dirac recovery over a smooth background. In this setting, we illustrate the relevance of a composite model and show a significant temporal gain on signal reconstruction, which results from our decoupled algorithmic approach.

Decoupled Solution for Composite Sparse-plus-Smooth Inverse Problems

TL;DR

This work studies a variational framework formulated as an optimization problem over the pairs of components using two regularization terms, each of them acting on a different part of the solution, and reveals a decoupling between the two components.

Abstract

We consider composite linear inverse problems where the signal to recover is modeled as a sum of two functions. We study a variational framework formulated as an optimization problem over the pairs of components using two regularization terms, each of them acting on a different part of the solution. The specificity of our work is to study the case where one component is regularized with an atomic norm over a Banach space, which is known to promote sparse reconstruction, while the other is regularized with a quadratic norm over a Hilbert space, which promotes smooth solution. We show how this composite optimization problem can be reduced to an optimization problem over the Banach space component only up to a linear problem. This reveals a decoupling between the two components, allowing for a new composite representer theorem. It naturally induces a decoupled numerical procedure to solve the composite optimization problem. We exemplify our main result with a composite deconvolution problem of Dirac recovery over a smooth background. In this setting, we illustrate the relevance of a composite model and show a significant temporal gain on signal reconstruction, which results from our decoupled algorithmic approach.
Paper Structure (33 sections, 7 theorems, 71 equations, 12 figures, 3 tables)

This paper contains 33 sections, 7 theorems, 71 equations, 12 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Composite linear inverse problems
  3. From sparse-versus-smooth to sparse-plus-smooth regularization
  4. Contributions and outline
  5. Representer Theorems for Single-Component Problems
  6. Smooth regularization of inverse problems
  7. Sparse regularization of inverse problems
  8. Sparse-plus-Smooth Composite Representer Theorem
  9. Main theorem
  10. Maximum regularization parameter for TV-norm
  11. Application to Super Resolution Deconvolution
  12. Definition of the composite model
  13. Function spaces
  14. Measurement operator
  15. Composite representer theorem
  16. ...and 18 more sections

Key Result

Proposition 1

Let $\bm{y}\in \mathbb{R}^L$, ${\bm{\Phi}_\mathcal{H}}= (\phi^\mathcal{H}_1 ,\ldots , \phi^\mathcal{H}_L) \in \mathcal{H}^L$, and $\lambda > 0$. Given that the measurement functionals $\phi^\mathcal{H}_1, \dots, \phi^\mathcal{H}_L$ are linearly independent, the optimization problem admits a unique solution $\widehat{s}_2 \in \mathcal{H}$ which is given by

Figures (12)

  • Figure 1: Observation of the radio sky at coordinates from the GLEAM survey accessible at https://gleamoscope.icrar.org/gleamoscope/trunk/src/, J2000 coordinates (9h37min15.21s, 50°25'03.1").
  • Figure 2: Illustration of the effect of the measurement operator $\bm{\Phi}_\mathcal{B}$ applied to a Dirac signal $s_1^\dagger = \delta_{x_0}$ with $x_0 \approx 0.695$.
  • Figure 3: Simulated source signal. Left: Sparse component. Center: Background smooth component. Right: Sum of the two components.
  • Figure 4: Simulated measurements. Left: Contribution of the sparse component. Center: Contribution of background. Right: Total noisy observations $\bm{y}$. In practice, only the information of the right-hand plot are accessible and the problem does not know the respective contribution of the components.
  • Figure 5: Recovered signals with regularization parameters $\lambda_2 = 8$ and $\alpha_1 = 0.1$. Left: Sparse foreground component. Right: Smooth background.
  • ...and 7 more figures

Theorems & Definitions (13)

  • Proposition 1: Representer Theorem on Hilbert spaces
  • Proposition 2
  • Theorem 1
  • Corollary 1
  • Remark 1: Decoupling beyond Hilbert-plus-Banach
  • Proposition 3: Maximum value of $\lambda_1$
  • proof
  • Proposition 4: Lemma 10 in Berlinet2011reproducing
  • proof
  • Remark 2: Synthesis operation within $\mathcal{H}_k$
  • ...and 3 more