$\nu$-Flows: Conditional Neutrino Regression

1) The nu-Flow method that the authors introduce yields a posterior distribution of the neutrino momenta that is better in two ways than traditional techniques: provides full likelihood information (good for multiple solutions and error estimation) and is less biased.

2) The article is well-written and pedagogical. The example used in the text is good to use cases and illustrative. The figures are well crafted.

1) The introduction and earlier parts of the article overpromise given what the article provides.

2) The authors talk about multiple neutrino reconstruction, but only deal with one neutrino.

3) The authors talk in the beginning about how this can improve e.g. top mass measurements. This is not only not demonstrated, but I believe that this method cannot improve the final physics inference. As the authors nu-Flow is learning the correlations that already exist in the Monte Carlo.

This is a nice article to read. I thought the discussion and the ideas were interesting; however, the article overpromises and underdelivers. I believe this can be solved by editing the introduction and conclusion appropriately.

My biggest concern is that I am not sure about the usefulness of this tool. The code is learning the posterior distribution of the neutrino momenta from Monte Car for a specific process/selection. As the authors show this matches the true distributions. But why not just draw the true distributions from the Monte Carlo directly and convert this to posteriors? This seems that should provide the same answers as the author's code, but in a simpler fashion.

1) Authors need to make the introduction/conclusion more appropriate to what they are actually doing in this article. Reduce slightly the things that could be done, but have not been shown to work, and state what has been done.

2) The authors need to provide a convincing argument and/or scenario where we are indeed improving some physics parameter measurement. Just being able to plot the neutrino momenta posterior distribution does not seem sufficient.

1- introduces a completely novel method for regressing missing components in collider experiments

2- demonstrates improvement in a downstream task utilising the predictions of the method

3- all figures are overall clear and well presented

1- uses only one very basic example case study of single lepton ttbar. Many claims about the ability of the method to scale up to more complex scenarios are made, but without such examples, it is not clear that the method will perform well in under-constrained systems .

2- the downstream task is an important one, but stops short of demonstrating an improvement on a final measurement, such as on the top quark mass

2- some minor grammatical issues can affect the readability of the paper, though it rarely affects the overall message

Overall I found the paper to be a very interesting read and it shows very promising results in the simple example studied. The method is completely novel, utilising a relatively new ML architecture and applying it to a problem that has not seen similar work before.

It is held back by its restriction to this simple case and some of the claims about scaling should be revised in order not to oversell what has been demonstrated so far. It would also be desirable to see the effect of the improvements on a measureable quantity, such as the top quark mass. Nonetheless, the paper is certainly of high quality and I look forward to follow up works.

Detailed comments and questions can be found below. A further detailed editorial pass to review miscellaneous grammar issues, such as stub sentences, would also be a good idea.

Introduction
1- "...momentum must sum to zero." -> This is an overly strong statement that leaves out detector resolution effects or ambiguities about incoming parton directions, please revise

2- the Higgs boson examples are motivating but not followed up in the studies presented, and therefore seem out of place, particularly in coming before the top examples.

3- "...the possible phase space..." -> I find this phrase to bit somewhat unclear, consider revising

Method
4- I am confused by the definition of the term "ill-posed" here. Why do you say the 2-nu final state is an example, instead of the case you study here?

5- the parameter theta is not explicitly defined in the text

6- "our work falls under unfolding" -> is this really unfolding? consider rewording this sentence

7- "by default agrees with direct measurements" -> Isn't the point that it is impossible to do a direct measurement?

8- "designed and applied for" -> "designed for and applied t" or just "applied to"

Case Study
9- "produced frequently...relatively high efficiency" -> these are very vague statements and I am not sure what value they add here. Either be more precise, or remove

10- It would be great to add a "Feynman diagram" type figure to show what the decay chain and final state looks like. In the text it is assumed that the reader knows what the leptonic W decay is (ie W->lnu), but this is never defined

11- "at least two bjets, two other jets" -> this is ambiguous and probably should be inverted?

12- Please cite [26] earlier in the text, when introducing the method here. It may also be helpful to cite a ttbar example, such as https://arxiv.org/abs/1806.05463

13- please define all of the terms in these equations, and state mW=80.38 in the text

14- "several drawbacks" -> only one such drawback, ie the fixed mass, is discussed

15- Is nu-Flows really learning the resolution of the lepton kinematics? Consider revising this statement

16- It is stated that nu-Flows can scale to multiple neutrinos. While this seems true in principle, it has not been demonstated that the method can provide sensible results in these cases. Consider revising this statement

17- "decaying leptonically to a bjet..." -> this is a confusing statement, it is the W boson that decays leptonically - please revise

18- is there a b-jet cut applied in the event selection? I assume not since it is not stated, but this is unusual in ttbar analyses, please be explicit and motivate this choice

19- "These include...event observables" -> remove comma after "kinematics"

20- I presume one is selecting the 10 jets with highest pT, but this is not explicitly stated - please do so. And why do you limit to 10 jets here? Couldn't the DeepSets method take all jets in the events?

21- What is the motivation for regressing px and py, but eta instead of pz? Can a performance comparison be added here, if it differs? If the regression depends strongly on the representation of the desired quantity, this is an important detail that should receive discussion.

22- "For cross-validation, 10% of the training is..." -> "For cross-validation, 10% of the training dataset is..."

23- Please add some discussion of the training time, inference speed, memory overhead etc of the network

Performance
24- What exactly is the likelihood used to select the best prediction in the "mode" variation? Can this be plotted for the 256 predictions in the example events?

25- nu-FF details seem out of place - perhaps move them to a dedicated subsection of Sec3

26- "true values of the neutrino" -> "true values of the neutrino momenta"

27- "may be due to several reasons" -> only one such reason is discussed, please elaborate

28- were studies performed to check the validity of the deepset identifying the blep hypothesis, such as performing the reconstruction first and inputting only the ttbar system to the network? This statement suggests a cyclic dependence that may motivate further studies of a combined neutrino prediction and jet-parton assignment method, if it can be confirmed

29- "it can be proposed" -> this seems like it could trivially be checked and discussed in more detail here

30- further studies on the poorly reconstructed event(s) would be good to see. Is there some common features of these events - for example, are they always high (or low) (b-) jet multiplicity?

31- Similarly, it would be good to see a study that explicitly does the suggested removal of poorly estimated events, and check the effect this would have on the jet-parton assignments

32- "the negative bias in nu-FF is..." -> this sentence should be moved to the previous paragraph, and a similar sentence for the baseline is missing

33- "too high a variance" -> "a higher variance"

34- Could the specific values used to plot Fig5 also be added to Fig3 for the "mode" variation?

35- "when looking at correlations of mW" -> It would be good to have these plots at least in backup material.

36- Some of the figures, especially 3/4, would be difficult to read in greyscale or if the reader is colorblind. It would be god to at least try to make this easier (of course I realise this is difficult to achieve without worsening the general readability - I leave this to the authors disgression).

37 - Figure 5 (+9,10,11) has no z-axis label

38- all of the citations for the chi2 method are in the all-hadronic channel. Is there an example where this has been used in leptonic decays?

39- Some additional references for jet-parton assignment with deep learning are missing eg https://arxiv.org/abs/2010.09206 and https://arxiv.org/abs/2012.03542

40- Please include the fitted values of each sigma somewhere for reproducibility

41- How is the truth assignment defined for the chi2 method? Please describe this

42- It would be nice to also see numbers for bhad and Whad in the results. Although these are second order effects, they are in principle non-zero, and improvements here would help further sell the method.

Conclusions
43- "significant improvements in downstream tasks" -> "in the downstream task of jet-parton assignment"

References
44- one minor typo in reference 7, "collaboration" -> "Collaboration"