Full Event Particle-Level Unfolding with Variable-Length Latent Variational Diffusion

1. Innovative Approach: The paper presents a novel modification of the Variational Latent Diffusion (VLD) model, already considered by the authors in the context of unfolding, extending it to handle variable-dimensional feature spaces, which is a significant advancement in generative unfolding methods.
2. Comprehensive Evaluation: The performance of the method is thoroughly evaluated in the context of semi-leptonic top quark pair production at the LHC, providing clear and detailed results.
3. Addressing High-Dimensional Data: The method effectively addresses the challenge of unfolding unbinned distributions in high-dimensional spaces, which is a substantial improvement over traditional techniques.
4. Potential Impact: The proposed method pushes the edges of the performances of generative unfolding methods setting new standards for full-event unfolding.

1. Sparse Data Regions: As acknowledged by the authors, the method shows some mis-modeling of kinematic distributions at the edges of the selection due to a lack of training samples. This highlights a limitation in handling extremely sparse regions of the data space, which is however an "irreducible" limitation.
2. Computational Complexity: The training and inference processes for the VLD model are computationally intensive, which might limit its practical applicability for extremely large datasets or for use in environments with limited computational resources.
3. Specificity of top pair production: While the method is tested on semi-leptonic top quark pair production, its performance on other processes or in different experimental setups is not explored, leaving , as expected, questions about its generalizability.
4. Iterative Prior Adjustment: The paper mentions that the dependence on the training set prior might be mitigated through an iterative method, but this approach is not demonstrated in the current work. Further exploration and validation of this iterative adjustment would strengthen the findings.

The paper presents a significant advancement in the field of experimental particle physics by introducing a modified VLD model capable of full-event unfolding with variable-dimensional feature spaces. The methodological innovation is substantial, addressing key challenges in high-dimensional data unfolding and offering a generative approach that mitigates some limitations of discriminative methods.

The comprehensive evaluation on semi-leptonic top pair production at the LHC provides convincing evidence of the effectiveness of the method, although the results feature areas of the distributions where the model struggles, particularly in regions with sparse data. The computational demands of the model and its performance on other types of data are also areas that could benefit from further exploration.

The paper certainly meets the criteria for acceptance in this journal. It presents original and significant contributions to the field, demonstrates clear and thorough methodology, and discusses the results and limitations transparently. The clarity of the writing is excellent.

1. All metrics considered by the authors (reported in Ref. [18], and that it would be useful to report here) seem to be sums of 1D distances computed on the 1D marginal distributions corresponding to the features of interest. Such metrics are therefore not naturally expected to be sensitive to correlations among features. I suggest the authors to consider adding at least one metric with some expected sensitivity on the correlation. One option, which does not require too intensive calculations and that is still based on 1D distances, could be the Sliced Wasserstein Distance.
2. To better understand and visualize the performances beyond the 1D marginal distributions, on which evaluation metrics are computed, I would suggest the authors to add corner plots.
3. The code and dataset need to be made public (for instance on GitHub and Zenodo) to allow full reproducibility of the analysis and broader usage of the method.

Ask for minor revision

- The method and architecture is well described and a lot of details are given
- It fills a gap in the literature: it is the first method to unfold full-event (variable dimensional) collider data

- one of the main results of the paper, that the unfolding performance is independent of the training data distribution, could be explained / investigated in more detail

The manuscript describes a novel machine learning architecture to unfold full particle physics events, i.e. variable dimensional collider data. Unfolding describes the inversion of the detector effects, allowing for long-term usage of experimental data without the need of expensive detector simulation. The manuscript is very well written and provides a lot of details on the architecture. In particular, I like the illustrative figure 1 and the many subsections in section 3 that describe all the individual elements of the model.

My biggest concern is about the observations in section 4.3 in combination with statement in section 6: "This lack of prior dependence strongly motivates the use of VLD for unfolding.". It is true that the model performs well on a different distribution as seen in training on the dataset considered here. But, it is not clear that this generalizes beyond this example. I understand that the authors cannot look at many processes within the scope of this manuscript, but I would at least like to see some explanation or investigation of why such a behavior would be expected. If the authors want to keep the sentence in the conclusion, they need to provide more explanantion / tests / examples to support the claim.
Apart from this, I have a few minor proposals that could enhance the manuscript, see the list below.

I therefore think that the manuscript should be published in SciPost Physics after the concerns have been addressed.

- One of the strengths of generative unfolding: event-by-event uncertainties, (since a given detector level input can be unfolded several times) could be mentioned in the introduction, since this an additional reason to use this method compared to discriminative methods.
- In section 3.1, is O_P_0 treated differently by the position-equivariant transformer with respect to the other O_P_i? Please explain.
- In section 3.3, why is y_0 needed? Please explain.
- In section 3.4, how is ensured that the ordering is constant over t? Or is that not a problem to be concerned of?
- In section 4.1, have both coordinate representations P^cart and P^Polar been used simultaneously (concatenated)?
- In sections 4.2 and 4.3, to better visualize the correlations between the observables and how well these are learned by the model, I suggest to add corner plots (for example to an appendix). Maybe this also explains the performance on down-stream observables a bit more.
- In sections 4.2 and 4.3, in addition to the metrics shown in the tables, I think it would be nice to see how well a neural classifier (see discussion in 2305.16774) would be able to distinguish unfolded from true events.
- In figures 5c and 7b (c), I suggest to zoom in on the bottom panel a bit.
- In the tables, I suggest to add a test of 'truth vs truth' to get a better feeling for the natural spread of the metrics. (i.e. is a distance of 0.04 a lot?)
- The authors refer a lot to reference 18, which is ok. However, it would be nice if the definitions of the metrics used in the tables could be replicated here as well and are not kept in Appendix C of Ref 18 only.
- In table 3, the errors of VAE and diffusion seem to add perfectly to the Unfolding error. Is that by construction or a non-trivial cross check on how the metrics are evaluated?
- Please make your code and training data available via git / zenodo / others.

Ask for minor revision