Summary
This paper studies an extension of the Plug-and-Play (PnP) framework for solving inverse imaging problems by considering descent schemes in metrics different from L2: Motivated by the fact that some data fidelity terms such as the Kullback-Leibler divergence allow for an efficient minimization with the Bregman Proximal Gradient (BPG) method (an extension of proximal gradient descent to arbitrary Bregman distances instead of squared L2), the authors develop a parametrization of a learnable denoiser which can be interpreted as a proximal operator (or descent step) of a cost function w.r.t. the underlying Bregman distance that combines well with a particular data fidelity term. Under some additional assumptions, this allows proving the convergence of the resulting Bregman PnP framework. Numerical experiments illustrate that the resulting scheme can successfully solve deblurring problems with Poisson noise.
Strengths
The paper is technically sound and presents the technical construction of the Bregman PnP approach very well. It is an elegant solution that closes a gap missing in the PnP (and RED) framework with learnable priors. I found it very convincing in terms of its theory and it even derives some general (smaller) missing pieces for convergence beyond the seminal works by Bolte, Bauschke, Teboulle and co-authors.
Despite this strengths section being short in comparison to the weaknesses, I think that the strong theoretical contribution along with an illustration of the implementation outweighs the weaknesses in terms of (benchmark) results, such that I am leaning towards accepting the paper.
Weaknesses
The numerical experiments/results are not very convincing from a practical point of view.
- First, the denoising performance of the dedicated network is not better than that of a plain network (might not be very important).
- Second, the main paper does not compare to the plain PnP approaches with L2 fidelity. The supplementary material reports tiny differences only (with the proposed approach being slightly worse for large noise and slightly better for small noise)
- According to line 275, the desired condition of $\phi_\gamma \circ \nabla h^*$ being convex is not enforced but seems to hold empirically when training the network. Thus, any convergence guarantee is lost.
- Figure 5 in the supplementary material shows a concerning risk of ending up with a bad result. The mitigation strategy of first running 100 iterations with a first set of parameters and then switching to a different set of parameters represents a significant amount of fine-tuning (possibly exceeding the number of hyperparameters and amount of fine-tuning used for the standard L2 PnP approach), such that even the small improvements in table 3 of the supplementary material need to be viewed with care.
Minor aspects:
- In 225 the author decide to do backtracking line search to avoid estimating the NoLip constant, but would backtracking on the (differentiable but not L-smooth) data fidelity term considered here not work in the L2 case?
- The authors mention that there is no result on the convergence of the PDHG method for nonconvex regularizers (line 81). As I was curious, I briefly searched online and found "Precompact convergence of the nonconvex Primal–Dual Hybrid Gradient algorithm" by Sun et al., Journal of Computational and Applied Mathematics, 2018. Precompactness of the primal variable (the image) would be easy to ensure if one restricts every value to [0,1]. Is their result applicable? (Honestly, I have not read the paper yet).
- The condition $\lambda L_f <1 $ in Theorem 2 seems to limit the amount of data fidelity one can use in order to still have a convergent algorithm - is this a limitation?
Questions
Considering a difficult (2-stage) optimization with different parameters to avoid bad minimizers as shown in Fig. 5 of the supplement, a lack of strict convergence guarantee as the convexity condition cannot be enforced, and negligible difference to a plain PnP or RED approach in terms of the PSNR, what is the advantage of the proposed method?
Rating
6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
Confidence
4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
Limitations
I don't think there is a potential negative societal impact of this work. In terms of general limitations, I think the authors should be more open about the fact, that the use of the Bregman distance framework did not result in improved results over prior algorithmic schemes.