Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 4: Ok but not good enough - rejection
review_content: summary: This paper proposes to use variational EM algorithm to simultaneously recover forward model and latent variable in an inverse problem. The proposed method is extensively evaluated on seismic tomography.
main_review: - Novelty: the proposed method consists of using neural nets to parametrize forward models and use variational EM to solve for parameters. Both aspects are well established and does not provide much new insights.

- Experiments: the proposed method is extensively evaluated on blind seismic tomography. However this paper lacks quantitative results and comparison with baseline methods. In blind convolution there are no quantitative comparison. It seems to me that for publication purposes, this paper should be reorganized as a  technical novelty on seismic tomography instead of a general machine learning algorithm, as in its current form the evaluation is not comprehensive enough.

- Scope: this paper discusses model mismatch, however in blind deconvolution experiments, instead of using a mismatch model the authors treat the forward model as blind. To properly deal with model mismatch, the algorithm should be able to utilize the inaccurate initial model to improve performance. This aspect is not reflected in the deconvolution case study.
limitations_and_societal_impact: Have the authors adequately addressed the limitations and potential negative societal impact of their work? If not, please include constructive suggestions for improvement：Yes
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: This paper proposes a normalizing-flow-based framework to solve inverse problems with model mismatch. The main idea is to use a normalizing flow to model the posterior distribution of the unknowns x conditioned on the current model parameters \theta and the observation y, as part of the expectation-maximization algorithm. The proposed strategy yields results that seem to outperform "traditional" methods in seismic tomography and blind deconvolution.

main_review: Originality: The paper builds on existing ideas in an original way. It proposes a creative use of normalizing flows to solve inverse problems with model mismatch. On the other hand, the central idea---using flows to model conditional distributions---is not a new one (among other places, it is described in the cited DPI paper).

Quality: The paper is well written and the empirical results seem solid.

- One somewhat philosophical comment is that model mismatch and blind inverse problems (as in blind deconvolution) are not really the same thing. In blind inverse problems we are up front about the fact that some explicit parameter of the forward model is unknown and should be identified as part of the inverse problem. In vanilla model mismatch we don't always have a model for the mismatch, and we often assume that the mismatch is in some sense small. In your examples, \theta is really a bona fide unknown, like x.

- This is especially evident in that in your seismic tomography experiments (e.g. Figure 3) you make more effort to compare the wave speed models \theta than the source locations x which are ostensibly the object you're trying to solve for. There is nothing wrong with this but to me it indicates a certain arbitrariness in the split between the unknowns of interest and the model parameters. In this case for the split you leverage independence of x and \theta (Btw, is this verified in practice? I would expect that the sources are mostly located along some geological features which indicates a dependence between the geology \theta and the source locations. Or am I wrong here?). Of course, in seismic imaging the split is very natural and it corresponds to well-known inverse problems (tomography vs source localization), but one could also say that the model is perfectly known---it is the wave equation---and you're jointly solving for the parametric RHS and the wave speed.

- I am wondering about the "seismologist-run" baseline. Why limit oneself to straight rays? The main difference between the two approaches then seems to be in your use of the EikoNet rather than in the use of flows and the EM algorithm. But EikoNet could be easily used as the forward model in the alternating scheme used by the seismologist, without involving flows. Wouldn't that be a fairer comparison? Otherwise this is not just model mismatch (as in the wave speed being different) but also model class (or physics) mismatch, since regardless of the wave speed your method uses physical, bent rays, while the seismologist's method uses unphysical straight rays. I think that a solid comparison should also use EikoNet for the baseline.

- Somehow related to the above two comments: in several places in the manuscript you advertise the "generality" of the approach, but the generality seems to equate the fact that normalizing flows can model conditional distribution which was shown already in the DPI paper. Granted, your application is a bit different. To the contrary, for each inverse problem one wants to address (here tomography and deconvolution), one has to design a differentiable, parameterized forward model (here in both cases a neural net). Thus the proposed method is not an out-of-the-box general method. The bulk of the difficulty is, I expect, in the design of this differentiable forward model. I believe this is a point worth elaborating.

- Out of curiosity, how do you expect your method to scale? It seems that you need to train a normalizing flow at every iteration of the EM algorithm? How long does one iteration of that take? Can you speculate about performance on large-scale problems?

Clarity: The papers is very clear and well written.

Significance: The paper addresses an important problem of machine learning for inverse problems with model mismatch. It is surely a significant problem for the inverse problems / wave imaging community. The machine learning contribution on the other hand is minor (in view of the existing DPI and EikoNet papers), but inverse problems have been gaining traction at NeurIPS over the past couple of years.

limitations_and_societal_impact: Yes.
review_title: null

final_score: rating: 4: Ok but not good enough - rejection
review_content: summary: This paper considers the problem of inverting a signal from a given forward model $y=f_\theta(x^*)+\epsilon$, while accounting model mismatch from true model $f_{\theta^*}$. The authors propose to solve this problem by first approximating the distribution of $x$ with $q_\phi(x) \sim G_\phi (z)$ using a deep model and carrying out an expectation over $x\sim q_\phi(x)$ of the log-likelihood $p(y|\theta,x)$. Model parameters $\theta$ are approximated via maximization over this expectation (EM framework). They call this framework DeepGEM.

The authors specifically apply this mathematical model to 1) blind seismic tomography, where the task is to estimate the velocity structure of earthquake waves. The velocity computation depends on travel time $T$ (this is the variable observed) parameterized by source location $x$ and receiver location $r$.  In this paper $T(x,s) = f_\theta(x,s) $ and $y \approx T(x,s=r)$. 2) Blind deconvolution with $y=x * k_\theta +\epsilon$.
main_review: The authors propose to use deep probabilistic models [31] to estimate a-posteriori distribution of signal x, in the context of inverse problems with model mismatch. The approach of using Expectation-Maximization has been used for solving inverse problems with model mismatch, such as blind deconvolution [20]; however this is the first paper to consider deep priors under this framework. Moreover the application of such approach to the domains of blind seismic tomography and blind deconvolution are novel. However there are several concerns.

It is unclear if this kind of approach will even converge; this paper will benefit from showing training curves showing reduction of loss value. 

In Line 142, why is the prior on $\theta$ (remains close to initial) reasonable? What if the initialization is far from the true parameters $\theta^*$. This section requires a comment on how $\theta$ is initialized.

In Figures 3 and 4 what are the metrics reported in the top right corner of each figure?

For the blind deconvolution case study, there is no baseline comparison, even though baseline models have been referenced in literature survey (eg. [20,21]). Since this paper is purely experimental in nature, baseline comparisons are important. 

Some notations need to be clarified or introduced as described below.

Some line specific comments are as below:

Line 102: define |x| to be dimension/cardinality

Step 3.1: explain what value theta assumes here. Is it randomly initialized?

Line 132: simplifying KL divergence -- N has not be defined anywhere so far.

Line 135: if y includes gaussian noise with 0 mean, then eps = || y - f(x) ||^2, 

Line 142: why should theta stay close to \tilde(theta)? Why is this a reasonable assumption?

Line 185: $\bar{x}$ is not defined at this point (it is defined later at 195)

Line 187: What is $\tilde{V}(s)$ and how does it differ from $V(s)$? 

Apart from these concerns the paper is otherwise well written in terms of introduction and explanations related to the blind seismic tomography. 
limitations_and_societal_impact: Limitations: Line 216: authors rely on domain expert to decide when to terminate the optimization for the baseline, without which the algorithm diverges. It's unclear how to gauge the accuracy of baseline.
review_title: null

final_score: rating: 8: Top 50% of accepted NeurIPS papers, clear accept
review_content: summary: This paper proposes a general variational Expectation-Maximization framework aided by a normalizing flow generative network to solve inverse problems for which a forward physical model is known, but important model mismatch exists. The approach is evaluated on a blind seismic tomographic task and showed to quantitatively and qualitatively outperform a state-of-the art baseline through various experiments, that include ablation studies. The versatility of the framework is also illustrated by a small case-study on image blind deconvolution, where the authors report convincing qualitative results on two images.
main_review: This is a very good paper. The addressed topic is timely and relevant, the methodology is novel and technically sound, the paper is clearly written, experiments are well designed and the reported results are good and convincing. Some important clarifications are nevertheless required.

Some clarifications needed in section 3:
- In both eq. (5) and (6) the authors appear to be using an approximation of the expectation via a form of re-parameterization trick (as coined in the original VAE paper by Kingma and Welling), but this is not explicitly stated.
- Some derivations are missing to explain how the log-likelihood is "exactly computed" (line 131)
- More generally, it is not clear how (5) and (6) are solved in practice, some derivations should be provided.
- On lines 114-115, the authors explain that the MAP objective will be optimized using "only measurements y". It is not clear at this stage and only becomes clear later that a *single* measurement y is used. This is what allows the authors to remove the dependency on y from q_{\phi}[x) on line 127 and later (otherwise it does not make sense). Please clarify this.

Some clarifications needed in section 4:
- Lines 201-209: How many parameters do the RealNVP and EikoNet contain? How does this tie in with the number of available observations, which seem relatively small? More generally, did the authors use training/development sets (in particular to tune hyperparameters) and how were they design?
- For both Fig. 3 and 4: How do you explain that errors get lower as the number of sources increases? shouldn't it be the opposite, as the problem is less and less well-posed? Or is it because of the strong provided priors? please comment on that.
- Line 247: the authors write that the obtained posterior is non-Gaussian, but is it multimodal? (unclear from figure).

Typos:
-Line 87: can be used _to_ approximate
-Line 265: emperically -> empirically
-Line 294: by also applying the approach to the simple, but challenging, blind deconvolution problem -> to _a_ simple ...

limitations_and_societal_impact: The authors correctly address the limitations and societal impact of their work.
review_title: null

Meta-review: The problem studies inverse problems with a partially unknown forward operator. The idea is to use a general variational Expectation-Maximization framework aided by a normalizing flow generative network. The paper is focused on blind seismic tomographic and shows very good performance. The comparisons to baselines was a bit confusing but we understand that there are limited prior works on this problem and that the proposed method will easily outperform methods that do not use deep generative models. 

There was a debate on this paper solving blind inversion versus that mismatch. In any case there is good novelty in the paper which contains novel ideas, is solid in a real and useful application and is very well written. 
Decision: Accept (Poster)