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Fast Inexact Bilevel Optimization for Analytical Deep Image Priors

Mohammad Sadegh Salehi, Tatiana A. Bubba, Yury Korolev

TL;DR

The paper tackles the computational bottleneck of solving the analytic deep image prior (ADP) formulation by introducing the Method of Adaptive Inexact Descent (MAID) for inexact bilevel optimization. It extends MAID to separable Hilbert spaces and applies it to the Sobolev-regularized ADP-$\beta$ problem, providing convergence guarantees and practical guidelines (adaptive accuracies and descent conditions). Theoretical results establish bounds on the lower- and upper-level Hessian-related constants under strong convexity assumptions, while numerical experiments in 1D and 2D deconvolution demonstrate substantial speed-ups with competitive reconstruction quality. The work enables applying ADP to larger-scale image reconstruction tasks and points to future development of adaptive inexact PALM for nondifferentiable upper-level terms in infinite dimensions.

Abstract

The analytical deep image prior (ADP) introduced by Dittmer et al. (2020) establishes a link between deep image priors and classical regularization theory via bilevel optimization. While this is an elegant construction, it involves expensive computations if the lower-level problem is to be solved accurately. To overcome this issue, we propose to use adaptive inexact bilevel optimization to solve ADP problems. We discuss an extension of a recent inexact bilevel method called the method of adaptive inexact descent of Salehi et al.(2024) to an infinite-dimensional setting required by the ADP framework. In our numerical experiments we demonstrate that the computational speed-up achieved by adaptive inexact bilevel optimization allows one to use ADP on larger-scale problems than in the previous literature, e.g. in deblurring of 2D color images.

Fast Inexact Bilevel Optimization for Analytical Deep Image Priors

TL;DR

The paper tackles the computational bottleneck of solving the analytic deep image prior (ADP) formulation by introducing the Method of Adaptive Inexact Descent (MAID) for inexact bilevel optimization. It extends MAID to separable Hilbert spaces and applies it to the Sobolev-regularized ADP- problem, providing convergence guarantees and practical guidelines (adaptive accuracies and descent conditions). Theoretical results establish bounds on the lower- and upper-level Hessian-related constants under strong convexity assumptions, while numerical experiments in 1D and 2D deconvolution demonstrate substantial speed-ups with competitive reconstruction quality. The work enables applying ADP to larger-scale image reconstruction tasks and points to future development of adaptive inexact PALM for nondifferentiable upper-level terms in infinite dimensions.

Abstract

The analytical deep image prior (ADP) introduced by Dittmer et al. (2020) establishes a link between deep image priors and classical regularization theory via bilevel optimization. While this is an elegant construction, it involves expensive computations if the lower-level problem is to be solved accurately. To overcome this issue, we propose to use adaptive inexact bilevel optimization to solve ADP problems. We discuss an extension of a recent inexact bilevel method called the method of adaptive inexact descent of Salehi et al.(2024) to an infinite-dimensional setting required by the ADP framework. In our numerical experiments we demonstrate that the computational speed-up achieved by adaptive inexact bilevel optimization allows one to use ADP on larger-scale problems than in the previous literature, e.g. in deblurring of 2D color images.

Paper Structure

This paper contains 9 sections, 5 theorems, 18 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem statement and summary of existing results
  3. Inexact bilevel optimization
  4. Method of Adaptive Inexact Descent
  5. Inexact bilevel optimization applied to ADP-$\beta$
  6. Numerical Experiments
  7. 1D deconvolution
  8. 2D deconvolution and semi-blind deconvolution
  9. Conclusions

Key Result

Lemma 1

Suppose that the condition $\psi(\alpha_k) \leq 0$ is satisfied at $\alpha_k$. Then the sufficient decrease in the exact upper-level function $g(\hat{x}(b_{k+1}) + r(b_{k+1}) - g(\hat{x}(b_{k}) - r(b_{k}) \leq -\lambda \alpha_k \|z_k\|^2$ holds.

Figures (7)

  • Figure 1: Deblurring of a 1D signal using ADP with elastic-net regularizer, solved via ADP LISTA (left), ADP IFT (center), and ADP-$\beta$-MAID (right).
  • Figure 2: Forward operator in 1D deconvolution. (a) Initial operator $A=B_0$ (Gaussian kernel); $B^*$ with (b) ADP LISTA, (c) ADP IFT, and (d) ADP-$\beta$-MAID.
  • Figure 3: Comparison of ADP IFT, ADP LISTA and ADP-$\beta$-MAID in terms of upper-level loss as a function of wall-clock time and lower-level iterations.
  • Figure 4: Kernels (first channel) of the 2D motion blur forward operator. (a) Initial kernel $b_0$. (b) Optimal kernel $b^*$ recovered by ADP-$\beta$-MAID. (c) Difference $|b^*-b_0|$.
  • Figure 5: Reconstructions in deblurring with a motion blur
  • ...and 2 more figures

Theorems & Definitions (11)

  • Lemma 1: Sufficient decrease condition, salehi2024adaptivelyinexactfirstordermethod
  • Theorem 1: convergence to a stationary point, salehi2024adaptivelyinexactfirstordermethod
  • Remark 1
  • Lemma 2
  • proof
  • Remark 2
  • Corollary 1
  • proof
  • Proposition 1
  • proof
  • ...and 1 more