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REVIEW 4 major objections 4 minor 1 cited by

Multi-scale Optimal Transport for Complete Collider Events

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper proposes a nested two-level optimal-transport distance for complete collider events—each event a distribution over jets, each jet a distribution over calorimeter cells—and reports that it classifies boosted top-antitop and BSM…

desk verdict A sound and genuinely new nested W2-on-W2 metric for complete events, with plausible gains over single-scale OT; the unvalidated LinW2 substitution and missing reference-event details are the main reasons it needs revision rather than acceptance as-is. read the letter →

arxiv 2501.10681 v3 pith:NZJYO7A4 submitted 2025-01-18 hep-ph hep-ex

Tianji Cai , Nathaniel Craig , Katy Craig , Xinyuan Lin This is my paper · ORCID
classification hep-phhep-ex PACS 13.87.-a29.85.Fj
keywords optimaltransportWassersteindistancejetsubstructureeventclassificationmulti-scalemetriccollidereventsk-nearestneighborstransversethrust
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to answer what the right distance is between two complete collider events when relevant physics lives at two very different scales. It proposes a nested optimal-transport distance, a $W_2$-on-$W_2$ construction, in which each event is a distribution over jets and each jet is a distribution over calorimeter cells. The authors show that this distance improves signal-versus-background classification relative to single-scale optimal transport and transverse thrust in two benchmark tasks, one driven by jet substructure and one driven by jet layout. The practical payoff would be a single tunable metric that can emphasize either intra-jet or inter-jet information depending on the analysis.

What carries the argument

The central object is the two-level Wasserstein distance of Eqs. (5) and (6): an event is a distribution over the jets found by anti-kT clustering, and each jet is a distribution of transverse momentum in the rapidity-azimuth plane. The ground metric $d_{jet}$ is built from an intra-jet part, the 2-Wasserstein distance between jet shapes after recentering, and an inter-jet part, the Euclidean distance between jet positions, with coefficients $c_{intra}$ and $c_{inter}$ that set the relative physical scale. The event-level distance is then another 2-Wasserstein distance under $d_{jet}$. Since exact jet-level 2-Wasserstein distances are computationally heavy, the implementation replaces the inner distance by the linearized tangent-plane approximation LinW2, whose reference-event construction is required to satisfy the convergence condition that makes it a metric.

What would settle it

Compute the same event distances with the exact 2-Wasserstein jet distance instead of LinW2 on a few thousand events and check whether the kNN AUCs or nearest-neighbor rankings change; if they do, the linearization is carrying the result. Independently, recompute $d_{jet}$ using true jet centers of mass rather than anti-kT axes and see whether the classification gain survives.

Watch

Extended reading notes

Core claim

The paper's central claim is that a complete collider event should be compared to another as a distribution over jets, not directly as a distribution over calorimeter cells. It defines the event-level distance $W_{2,event}$ by a 2-Wasserstein optimization whose ground cost $d_{jet}$ combines two terms: the 2-Wasserstein distance between the two jets after translating each to zero center of mass, scaled by $c_{intra}$, and the Euclidean distance between the jets' centers in the rapidity-azimuth plane, scaled by $c_{inter}$. With $c_{intra}$ and $c_{inter}$ positive, $d_{jet}$ is a metric, so $W_{2,event}$ is a genuine metric on complete events. The paper argues that this nested structure separates the substructure scale inside a jet from the inter-jet scale of the whole event, and that this separation is why the distance outperforms single-scale optimal transport and transverse thrust in boosted top-antitop versus QCD and BSM dijet versus QCD classification.

Load-bearing premise

The central assumption is that the linearized LinW2 jet distance and the anti-kT jet axis are faithful stand-ins for the true jet-shape distance and jet center, so that the measured classification gain comes from the two-scale metric rather than from the approximations.

Editorial extensions

If this is right

  • The multi-scale distance improves over single-scale optimal transport and transverse thrust for almost all coefficient choices and jet radii in both benchmarks, so the nested representation itself, not the specific classifier, is the source of the gain.
  • The relative weight $c_{intra} : c_{inter}$ is tunable: for boosted top-antitop versus QCD the best performance is obtained when substructure dominates, while for resolved BSM dijets the optimal balance shifts toward inter-jet geometry.
  • Weighting jets by transverse momentum and weighting them uniformly perform similarly, and both beat a version that compares only the hardest jet, except when large-radius jets collect the full decay products.
  • When $c_{intra}=c_{inter}$, the ground metric reduces to the ordinary 2-Wasserstein distance between jets, so the framework continuously interpolates between single-scale and multi-scale comparisons.
  • Because the event-level distance is a metric, any distance-based machine learning method, not only k-nearest neighbors, can consume it directly for classification or anomaly detection.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because $c_{intra}$ and $c_{inter}$ are free weights, the reported $(10,1)$ default is not special; a per-task scan over the ratio should be seen as part of the method, and the paper's own curves show the optimum moves with jet radius.
  • The anti-kT axis proxy is a testable threat: replacing it with a true center of mass would isolate whether the multi-scale geometry or the preprocessing details produce the gains.
  • The W2-on-W2 construction is really a recipe for any nested point cloud, so the same distance could be applied to trackers, calorimeter layers, or other hierarchical detector data without changing the formalism.
  • Extending the event-level distance to an unbalanced Hellinger-Kantorovich distance would restore information lost when each event is normalized to unit total transverse momentum or unit jet count, making the metric sensitive to overall event activity as well as shape and layout.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes a hierarchical optimal-transport distance for complete collider events. Section II defines a 'W2-on-W2' construction: an event is represented as a distribution over its anti-kT jets, each jet is itself a distribution on the rapidity-azimuth plane, and the ground distance djet in Eq. (6) combines a weighted intra-jet W2 term between centered jets with a weighted inter-jet center-of-mass term. The authors state that djet is a metric for exact W2,jet and hence W2,event is a metric on events. In practice, W2,jet is replaced by the linearized LinW2 of Eq. (3), the anti-kT axis is used as a proxy for the jet center of mass, and each event is normalized to unit total mass, the latter detail appearing only in the conclusions. Section III applies the distance with a kNN classifier to t-tbar vs. QCD and BSM paired-dijet vs. QCD classification at jet radii R=0.3, 0.5, 1.0, reporting AUC curves as functions of cintra:cinter and comparing with single-scale OT and transverse thrust. The authors find that multi-scale OT generally outperforms the two benchmarks and interpret this as evidence that the nested metric captures both intra-jet substructure and inter-jet spatial correlations.

Significance. If validated, the construction gives LHC analyses a principled two-scale metric for complete events, with a tunable balance between jet-substructure information and jet-location information. The metric definition is mathematically standard for exact W2,jet, and the toy example plus two classification benchmarks are appropriate demonstrations of the intended behavior. The paper also provides useful appendices relating jet-mass distributions to the classification results. However, the numerical claims currently rest on unquantified substitutions: LinW2 instead of exact W2,jet, an unspecified reference event in the linearization, and the anti-kT axis instead of the jet center of mass. These choices could affect the reported AUC differences, so the empirical contribution is not yet fully supported as written.

major comments (4)
  1. [II C, Eq. (6)] The numerical experiments replace W2,jet in Eq. (6) with the linearized LinW2 of Eq. (3), but the reference event R used for the projection is never described; the statement that the conditions of [12, Proposition 1] are 'explicitly enforced' is not verifiable without specifying R. Because LinW2 is not a metric and its deviation from W2,jet is unquantified, the AUC gains in Figs. 3 and 4 could be an artifact of the linearization rather than evidence for the multi-scale metric. Please specify the construction of R and report a comparison between LinW2 and exact W2,jet on representative jet pairs, including the error as a function of jet multiplicity and substructure.
  2. [II C, Eq. (6)] Eq. (6) defines the inter-jet term using the jet center of mass, but the implementation uses the anti-kT jet axis as a proxy. For asymmetric jets, the axis and the center of mass differ by O(R), which is comparable to the intra-jet scale O(0.1) quoted in Section II C; this can change the effective cintra:cinter balance. The paper should quantify the axis-vs-center-of-mass offset, for example by reporting its distribution or by rerunning one benchmark with true centers of mass, before the optimal ratios in Section III can be interpreted.
  3. [III, Figs. 3 and 4] The paper states that the kNN hyperparameter k is selected on the validation set, but it does not state whether the coefficient pair (cintra,cinter) is also selected on validation. The narrative 'highest AUC is achieved around (10,1)' suggests the pair may have been chosen from the test-set curves; if so, the reported best AUCs are optimistically biased. Please clarify the selection protocol and, if the pair is scanned, select it on validation and report the test AUC at that selected point.
  4. [IV, Conclusions] The statement that each event is normalized to unit total mass before computing Eq. (6) appears only in the conclusions, not in Section II C where the jet weights Js are defined. This normalization determines the balanced optimal transport problem and is essential for reproducibility; it should be stated explicitly in the method section and its effect on both weighting schemes should be discussed.
minor comments (4)
  1. [II A, Eq. (2)] The text says 'To remove the rotational symmetry on the y−phi plane, we translate each jet...'; the symmetry being removed by translation is translational symmetry in the plane, so the terminology should be clarified.
  2. [III A] The sentence 'pp → jjjj, where a jet j is originated by either a gluon or u, d, c, squarks' appears to contain a typo; 'squarks' should probably read 'quarks'.
  3. [II C] The passage 'C ∼ O(5) is the diameter of the detector and M ∼ O(5), leading to an O(1) length scale' uses M without a definition; if M denotes the number of jets or something else, please clarify.
  4. [References, [54]] Reference [54] is the present paper itself; citing it in the conclusions for the Hellinger-Kantorovich extension does not supply the claimed content and should be replaced by an appropriate external reference.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the multi-scale OT distance is defined ex ante, its coefficients are scanned rather than fitted, and the reported AUCs are held-out evaluations.

full rationale

The central claim is that the multi-scale distance in Eqs. (5) and (6) improves event classification. The distance is constructed directly from the 2-Wasserstein distance and the weights cintra, cinter; these weights are scanned over a fixed grid, k is selected on a validation split, and the final AUC is evaluated on a held-out test set (Sec. III). No parameter is fitted to test labels, so no reported number reduces to a fit. The LinW2 substitution for W2,jet in Eq. (6) is an approximation whose error is not quantified, and the anti-kT jet axis is used as a proxy for the jet center of mass; these are correctness and reproducibility concerns, not circularity, because the distance is not defined in terms of the classifier output or any fitted value. The one self-referential passage is the future-direction sentence in Sec. IV citing [54] (the present paper) alongside [34] as having used the Hellinger-Kantorovich extension; this citation is erroneous but not load-bearing, since it occurs in a forward-looking remark and the external Hellinger-Kantorovich reference [34] is co-cited. The invocation of [12, Proposition 1] to justify LinW2 convergence to a metric is a self-citation of a formal theorem with stated assumptions, not an input to the classification benchmark; the reference-event construction is not described, which is a missing-support issue rather than circular derivation. Overall, the derivation chain is self-contained: the metric is defined ex ante and tested on external simulation benchmarks.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper introduces a distance, not a new physical entity; no new particles, forces, or conserved quantities are postulated. The central load-bearing inputs are the hand-chosen coefficients, the unvalidated LinW2 reference, and the jet clustering assumptions.

free parameters (4)
  • cintra and cinter = scanned values including (1,0), (10,1), (5,1), (0,1), and others
    Relative weights for jet substructure versus jet location in Eq. (6); the optimal ratio is selected from validation AUC curves, not derived from first principles.
  • k in kNN classifier = selected from 10 to 200 in steps of 10 on the validation set
    Hyperparameter of the classifier chosen by validation performance.
  • jet radius R = 0.3, 0.5, 1.0
    Clustering radius is varied by hand to probe intra- versus inter-jet sensitivity; it influences all results but is not fitted to classification labels.
  • LinW2 reference event = not specified
    The paper does not detail how the reference event is constructed; its choice affects all linearized jet distances and the claimed metric property.
assumptions (5)
  • standard math The 2-Wasserstein distance is a metric, and the Wasserstein space over a metric space is itself a metric space.
    The proof that d_jet and W2,event are metrics relies on classical OT results cited as [45,46].
  • domain assumption Anti-kT clustering with a fixed radius partitions the event into jets that faithfully represent the relevant physical scales.
    Section II B assumes jets found by anti-kT with R = 0.5, and datasets keep only the four highest-pT jets, discarding additional radiation.
  • domain assumption Rotational alignment of jets via principal component analysis removes only symmetry and preserves classification-relevant information.
    Section II B preprocesses each jet by translating its axis to the origin and aligning the principal component; this is a modeling choice that could erase correlations, and the paper acknowledges color-flow correlations are ignored.
  • ad hoc to paper The reference event used for LinW2 satisfies the conditions in [12, Proposition 1], so LinW2 approximates a metric.
    The paper asserts this is 'explicitly enforced' but does not describe the construction or provide evidence for the claim.
  • domain assumption Monte Carlo samples generated with MadGraph and Pythia represent the relevant LHC event distributions.
    All classification results are based on simulated signal and background samples, so detector-level effects and generator modeling choices are inherited.

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Cite this review

Pith. "Pith review of Multi-scale Optimal Transport for Complete Collider Events." pith.science (2026). https://pith.science/paper/NZJYO7A4

@misc{pith2026250110681,
  author       = {Pith},
  title        = {Pith review of: Multi-scale Optimal Transport for Complete Collider Events},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NZJYO7A4}},
  note         = {Machine review of arXiv:2501.10681}
}
read the original abstract

Building upon the success of optimal transport metrics defined for single collinear jets, we develop a multi-scale framework that models entire collider events as distributions on the manifold of their constituent jets, which are themselves distributions on the ground space of the calorimeter. This hierarchical structure of optimal transport effectively captures relevant physics at different scales. We demonstrate the versatility of our method in two event classification tasks, which respectively emphasize intra-jet substructure and inter-jet spatial correlations. Our results highlight the relevance of a nested structure of manifolds in the treatment of full collider events, broadening the applicability of optimal transport methods in collider analyses.

Figures

Figures reproduced from arXiv: 2501.10681 by the authors.

Figure 1
Figure 1. FIG. 1: A sample [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: (a) A series of multi-scale OT distances between the two events under different ( [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. shows the classification performance (mea￾sured in AUC) on the test set for both the single-scale and multi-scale OT frameworks coupled to a kNN clas￾sifier, where jets are clustered with different radii R = 0.3, 0.5, 1.0. Also displayed for comparison is the perfor￾mance of transverse thrust, which stays the same for all three datasets as the underlying events are the same. When using the single-scale OT method, ev… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4: AUC scores for the BSM [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Two artificially generated events. Each event [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Mass of the first four highest [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Jet multiplicity in each event, where jets are [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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