{"id":"3d403abc-c084-4cab-8915-977248c1d230","arxiv_id":"2501.10681","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A hierarchical optimal transport distance, built by measuring events as distributions of jets whose shapes are measured by optimal transport, improves classification of simulated LHC events.","lead":"This paper introduces a new way to measure how similar two particle collision events are, by treating each event as a collection of jets and comparing both jet shapes and jet locations. The method outperforms standard single-scale optimal transport on two simulated LHC classification tasks, but without released code or uncertainty estimates.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The reported AUC gains may be an artifact of the LinW2 approximation: Eq. (6)'s d_jet is never validated against exact W2_jet, and the reference event is unspecified.","rationale":"I read the paper as proposing a genuine two-level Wasserstein metric and demonstrating it on two classification tasks. The theory of W2-on-W2 is standard, and the toy example illustrates the intended behavior. The weakest point is not the concept but the gap between Eq. (6) and what was actually computed. Eq. (6) defines d_jet using exact W2_jet, but Section II.C states 'in practice, we use the LinW2 approximation for W2_jet'. LinW2 is a linearization around a reference; it is not a metric, and its fidelity depends entirely on the reference. The paper neither constructs the reference nor quantifies error. Since the headline improvement over single-scale OT at (1,0) is attributed to 'the ability to highlight substructure variation via the ground metric of Eq. (6)', the approximation is directly in the logical path of the central claim. If LinW2 happens to separate top jets and QCD jets more than exact W2 does, the reported AUC gain would be an artifact of the linearization, not of the multi-scale metric. This is not an accusation; it is a missing validation step. A concrete check is feasible: compute exact W2_jet on a subset and see whether the AUC ranking and numerical values survive. If they do, the concern is retired. If they do not, the paper needs either a different approximation or a clear statement that the classifier is intentionally using a linearized feature. I therefore keep the reader's CONDITIONAL verdict: the idea is promising, but the current evidence is insufficient.","tokens_in":15644,"tokens_out":7329,"duration_ms":78186,"concrete_test":"On a random subset of 1000 events from each task (R=0.5), replace LinW2 in (6) with exact W2_jet computed by network simplex (e.g., POT) for all jet pairs, and rerun the kNN classification for (cintra,cinter) = (1,0) and (10,1), using the same preprocessing and k selection. Also compute max and mean relative error |LinW2-W2|/W2 over 1000 jet pairs. If AUC moves by more than 0.05 or mean relative error exceeds 0.2, the linearization is load-bearing for the reported gains.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central numerical claim rests on the event distance (5) with ground distance d_jet in (6). In practice, W2_jet in (6) is replaced by the linearized LinW2 of (3), which depends on a reference event R whose construction is never described, despite the assertion that the convergence properties are 'explicitly enforced.' LinW2 is not a metric and only approximates W2_jet locally near the reference. Because the paper reports no approximation-error study, the kNN AUCs in Figs. 3 and 4 may reflect the geometry of the linearization rather than the multi-scale metric claimed. A second unquantified distortion is the use of the anti-kT axis as a proxy for the jet center of mass in the inter-jet term of (6). Without code, data, or a reference-jet specification, these distortions cannot be checked post hoc.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":15821,"tokens_out":5709,"duration_ms":58246,"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":[{"comment":"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.","section":"II C, Eq. (6)"},{"comment":"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.","section":"II C, Eq. (6)"},{"comment":"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.","section":"III, Figs. 3 and 4"},{"comment":"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.","section":"IV, Conclusions"}],"minor_comments":[{"comment":"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.","section":"II A, Eq. (2)"},{"comment":"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'.","section":"III A"},{"comment":"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.","section":"II C"},{"comment":"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.","section":"References, [54]"}],"recommendation":"major_revision","confidential_remarks":"To the editor: this is a reasonable methods paper for hep-ph, but the reproducibility gaps are substantial for a distance-based method: no code or data release, an unspecified reference event in the linearization, and an unclear validation protocol for cintra:cinter. I would make the reference-event specification and the LinW2-versus-exact-W2 validation conditions of the revision, and I would also ask the authors to correct the self-citation [54]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper is a real, incremental advance in OT-based event metrics, with a sound construction and plausible demonstrations. The core idea — treat an event as a distribution over jets, with a weighted W2-on-W2 ground metric — is new relative to the prior single-scale OT literature, and it holds up. The reservations the reader lists are real but mostly about unshown verification, not about the math.\n\nWhat earns credit: the metric statements are correct. d_jet in Eq. (6) is a product of the 2-Wasserstein distance on centered jets and Euclidean distance on jet centers of mass, so it is a genuine metric, and W2,event inherits that. The equal-weight reduction to c·W2,jet is a classical result, properly cited. The physics motivation is also good: the scale separation between intra-jet substructure and inter-jet separation is real, and the results behave the way the physics says they should — the optimal c_intra:c_inter shifts with jet radius, and the two tasks sensibly favor intra-jet and inter-jet information respectively.\n\nSoft spots, in proportion. The biggest is real: W2,jet is replaced by the LinW2 approximation in practice, with no error study and no description of the reference event, even though the paper asserts the convergence conditions are 'explicitly enforced.' The referee should ask for the reference definition and a small LinW2-versus-exact comparison. But the stress-test's stronger claim — that the reported gains may be an artifact of LinW2 — does not land. The single-scale baseline uses linearization too, so the ~25% AUC gain of multi-scale over single-scale at (c_intra, c_inter) = (1,0) in the ttbar task is attributable to the hierarchical structure itself, which is exactly the paper's claim. Smaller issues: the anti-kT axis as a proxy for jet center of mass is plausible but unquantified (probably minor), the reported AUCs have no error bars, no code or data is released, and the total-mass normalization (acknowledged in the conclusions) discards multiplicity and total-pT information. One citation slip worth fixing: reference [54] is the paper itself, and the conclusions attribute the Hellinger-Kantorovich extension to '[34, 54]', but this paper does not use HK; [34] alone is the right citation.\n\nThe paper is for readers working on OT-based collider-event metrics and ML classification at the LHC. It is a methods paper, not a discovery, but it does one useful thing cleanly. I would send it to peer review and ask for the missing verification: reference-jet specification, an approximation-error check, error bars on the AUCs, and ideally code.","headline":"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.","tokens_in":16330,"tokens_out":9830,"would_cite":true,"duration_ms":91804,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["13.87.-a","29.85.Fj"],"model":"deepseek-v4-flash","headline":"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…","keywords":["optimal transport","Wasserstein distance","jet substructure","event classification","multi-scale metric","collider events","k-nearest neighbors","transverse thrust"],"falsifier":"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.","tokens_in":15416,"feed_emoji":"⚛️","tokens_out":12206,"duration_ms":115336,"temperature":0.7,"pith_summary":"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.","feed_headline":"Two-level Wasserstein metric beats collider event baselines","feed_subtitle":"Two-level W2-on-W2 distance reads jet shapes and positions, beating single-scale OT and transverse thrust.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Provides the single-scale event metric used as the primary optimal-transport baseline for comparison.","marker":"[10]"},{"why":"Supplies the linearized 2-Wasserstein approximation used for jet-level distances and the convergence condition invoked for the reference event.","marker":"[12]"},{"why":"Provides the transverse-thrust benchmark and the rescaling convention used as the non-optimal-transport baseline.","marker":"[15]"},{"why":"Supplies the anti-kT clustering algorithm that partitions each event into jets, fixing the objects on which the jet manifold is built.","marker":"[44]"},{"why":"Provides the metric-space result that a Wasserstein distance over a metric ground space is itself a metric.","marker":"[45]"},{"why":"Supplies the proposition used to show that the ground metric reduces to the ordinary 2-Wasserstein distance when the intra-jet and inter-jet weights coincide.","marker":"[46]"},{"why":"Generates the matrix elements for the simulated signal and background samples used in both classification tasks.","marker":"[51]"},{"why":"Provides the parton shower and hadronization used to produce the simulated events for the classification benchmarks.","marker":"[52]"}],"fun_headline_variants":["Nested Wasserstein metrics classify collider events","Event distance: Wasserstein on jets beats single-scale OT","Multi-scale transport reads jet substructure and positions","Two-level OT outperforms baselines in event classification","Wasserstein over jets: a metric for whole collider events"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Nested Wasserstein metrics classify collider events","Event distance: Wasserstein on jets beats single-scale OT","Multi-scale transport reads jet substructure and positions","Two-level OT outperforms baselines in event classification","Wasserstein over jets: a metric for whole collider events"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000249,"raw_usage":{"total_tokens":1504,"prompt_tokens":852,"completion_tokens":652,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":468,"completion_tokens_details":{"reasoning_tokens":574}},"tokens_in":468,"tokens_out":652,"duration_ms":6097,"temperature":1.0,"reasoning_tokens":574,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T19:01:12.593028+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Cacciari, G","cited_arxiv_id":null,"evidence_quote":"Provides the metric-space result that a Wasserstein distance over a metric ground space is itself a metric."},{"cited_title":"Ambrosio and N","cited_arxiv_id":null,"evidence_quote":"Supplies the proposition used to show that the ground metric reduces to the ordinary 2-Wasserstein distance when the intra-jet and inter-jet weights coincide."},{"cited_title":"Banfi, G","cited_arxiv_id":null,"evidence_quote":"Generates the matrix elements for the simulated signal and background samples used in both classification tasks."}],"review_version":1}