Simplex Demixing: Disentangling Multiple Light-Flavor Jets at Colliders

arXiv:2607.24921 · hep-ph, cs.LG, hep-ex · Submitted 2026-07-27 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: I'm Tom, and with me are Jane, Lu, senior AI researcher at Tsinghua, Meng, lead engineer at a mysterious AI startup and Lalam, the in-house Large Language Model.

Jane: Today's paper: "Simplex Demixing: Disentangling Multiple Light-Flavor Jets at Colliders".

Tom: The gist The simplex demixing framework provides a practical, data-driven method to extract multiple jet flavors from mixed data samples by identifying maximally separable categories in phase space,

Jane: First, who's behind it and why it matters.

Paper summary: Tom: So we’re looking at this paper now called "Simplex Demixing: Disentangling Multiple Light-Flavor Jets at Colliders." Basically, they’re tackling this long-standing problem in collider physics about how to give us a solid, practical definition for multiple jet flavors.

Jane: That's right. Previous work gave us an operational definition for just quark and gluon jets, but they didn't have a way to generalize that robustly to more than two categories yet. This paper introduces a machine learning framework called "simplex demixing."

Lu: What they claim is that this framework lets you extract T jet flavors, or topics in statistics literature, from M data samples or mixtures with minimal constraints. The intuition behind it is identifying the maximally separable categories within the data itself.

Meng: So instead of building a definition from scratch using only theory, they are letting the data tell them what those categories actually are by finding where the distributions separate best.

Lalam: It’s like they take a bunch of mixed samples and try to find the most distinct groups hidden inside that mix. The goal is to get these jet flavors out with minimal fuss or constraints on how you define them initially.

Tom: Exactly, and what matters here is that this procedure translates that multi-category classifier on M mixtures into a bounded geometric object with T vertices. That's their main idea for the operational definition of jet flavor.

Jane: They explain this geometric interpretation by training a classifier on those mixtures using cross-entropy loss where the output distribution is bounded by a (T minus one)-simplex within the probability (M minus one)-simplex <ref:2607.24921#pg2>.

Meng: So, there are T vertices in this simplex, and these vertices are defined as the hadron-level definition of jet flavor. This sounds like they’re using geometry to define what a jet flavor is in a way that's directly tied to how it looks in data.

Lu: They walk through a three-stage training procedure called simplex demixing. Stage one is for the classifier to "learn" where the vertices are fixed at the corners of that probability simplex with L2 regularization.

Paper summary: Tom: Then there’s stage two, which they call "shape," where an auxiliary loss called Ledge encourages those vertices to tightly enclose the classifier distribution. That sounds like forcing them to be well-defined boundaries for the categories they found.

Jane: And stage three is "prune," where an L1 loss on vertex weights removes any unnecessary vertices, leaving only those that correspond to the actual operational topics. This is how they get down to T flavors from M samples.

Meng: That sounds very practical for engineers because it’s a finite procedure, not some infinite search space. It’s about cleaning up the results to find the meaningful structure in the data.

Lalam: And this whole thing is validated on a toy problem with M equals three mixtures of down-quark, up-quark, and gluon jets. They found that in this case, the distribution lies on a simplex spanned by T equals seven operational topics.

Tom: Seven topics! That’s more than just quarks and gluons. The paper then proposes a tag-and-probe strategy to extract multiple light-flavor categories in a more realistic collider setting involving dijet production.

Jane: They showed that the identifiability of these jet flavors depends on their relative abundance in the samples and the hadron-level information available to that classifier architecture.

Meng: That makes sense; if you don't have enough data for a certain flavor, it’s hard to tell it apart from another one, which is a real constraint we have in experiments.

Lu: To extract observable distributions for each operational topic, they perform the inversion of an equation called Eq. (three point eight) in a least-squares sense to get pt(O) equals X M m=one (F+) tm pm(O) <ref:2607.24921#pg2>.

Tom: This allows them to reconstruct operational topic distributions over any observable O, like constituent multiplicity Nconst or two-subjettiness tau beta equals one squared. They showed good agreement between the inferred topic distributions and the truth-level Pythia distributions for quark-like topics T3 and T5, while gluon-like topics T4 and T7 also matched the Pythia gluon distributions.

Jane: It’s interesting because they managed to map five operational topics onto the Pythia down, up, strange, anti-strange, and gluon flavors based on their identification of those seven vertices.

Paper summary: Meng: So they didn't just find some abstract mathematical concepts; they actually linked them back to the known physics from Monte Carlo generators like Pythia eight point three one zero seventy-three. That’s where the real validation happens for us in practice.

Lu: The tag-and-probe method successfully decomposed marginal distributions into mixtures modeled as pm equals X t Fmt pt(x). This mixture model allowed for the recovery of universal operational topics in the asymptotic limit, provided that Fmt has full column rank.

Tom: It suggests that even with finite data samples, you can get a good approximation of these underlying universal jet properties if your model assumptions hold up. The limitations they mention are related to the weak identifiability of anti-down and anti-up flavors because their truth-level Pythia fractions in Table two are quite small <ref:2607.24921#pg2>.

Jane: That low statistic for those flavors means the low anti-d and anti-u counts can only give us a semi-quantitative understanding of why topics T2 and T4 might be contaminated by gluons. The intrinsic reducibility factor kappa qg is nonzero for Pythia jets, meaning the gluon-like operational topic will always have some quark component.

Meng: From an engineering standpoint, having a way to systematically pull out these components from complex data sets like dijet events sounds like it could significantly help us in jet substructure searches at the LHC.

Lalam: The paper proves that this simplex demixing framework is compatible with any classifier architecture and robust to finite statistics, which is a pretty strong statement for a new method.

Tom: So to wrap up this segment, the simplex demixing framework provides a practical, data-driven method to extract multiple jet flavors from mixed data samples by identifying maximally separable categories in phase space. It opens the door to extracting substructure distributions at the LHC.

Jane: And we’re going to see how they move beyond just two categories into seven topics using this new geometric approach. Next up, we’ll discuss what these results actually mean for our understanding of jet physics and how they might be used in real experiments.

Conclusion: Tom: So, we’ve seen how they use this simplex demixing framework to pull multiple jet flavors out of mixed data samples at colliders like the LHC.

Jane: Yeah, and that framework is basically a machine learning procedure that finds the most distinct categories in a set of mixed jet data by looking at where the samples separate best in phase space.

Tom: It’s quite an elegant way to define what those individual jet flavors actually are, turning it into a bounded geometric object with these vertices.

Jane: So, when you look at the final results, they show that this works for seven operational topics in a test case involving down-quark, up-quark, and gluon jets.

Lu: What’s really cool is that they connect those mathematical vertices directly to the known jet flavors from Monte Carlo generators like Pythia.

Meng: It means we can use this method to systematically extract distributions of these jet flavors without having to rely solely on a single theoretical model for every single process.

Lalam: The authors prove that this works for any number of topics T, as long as T is less than or equal to the number of samples M you have.

Tom: So, what does this actually change for us down here in the lab? It opens up a new way to look at jet substructure distributions.

Jane: Exactly. We can now reconstruct how these different light-flavor jets behave in things like constituent multiplicity or two-subjettiness using this framework.

Tom: And they showed that the results align well with known truth distributions from Pythia for some of those topics, like up-quark and strange jets.

Jane: But they also found a way to get a semi-quantitative understanding of why some topics, like those associated with anti-down and anti-up flavors, might be mixed with gluons.

Lu: The fact that the gluon-like topic always has some quark component because of that intrinsic reducibility factor is an important piece of physics there.

Meng: From a practical side, this could let us analyze complex dijet events in a way that separates the signal from the background components more cleanly than before.

Lalam: This work shows how we can use data-driven techniques to extract these light-flavor properties at the LHC.

Tom: It’s an important step because it gives us a practical tool for disentangling complex jet structures, moving beyond just two basic categories.

Jane: And this is just the beginning of how we can use this geometric approach to understand the intricate substructure of jets.

Center for Theoretical Physics – Leinweber Institute, Massachusetts Institute of Technology · NSF AI Institute for Artificial Intelligence and Fundamental Interactions · Institut des Hautes Etudes Scientifiques · Institut de Physique Th´eorique, CEA Paris-Saclay

hep-ph, cs.LG, hep-ex

Submitted: 2026-07-27

Updated: 2026-10-09

Comments: 54 pages, 15 figures; our code is available at https://github.com/gregofuente/simplex_demixing. v2: includes a visualization of jet topics across model sizes and a study showing simplex demixing performance with PFN-Ex information

Code: https://github.com/gregofuente/simplex_demixing

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 80/100

The gist: The gist The simplex demixing framework provides a practical, data-driven method to extract multiple jet flavors from mixed data samples by identifying maximally separable categories in phase space,

Key concepts

Operational Definition
This is defined as the 'maximally separable categories in the data' when given two mixed jet samples. It establishes a practical, data-driven way to define jet flavors beyond simple two-category classifications, setting the basis for extracting more than just quark and gluon jets.
Simplex Demixing Framework
A three-stage machine learning procedure used to extract mixing fractions of operational topics from measured mixtures. It involves 'Learn' (fixing vertices), 'Shape' (using an Ledge loss to tightly enclose the distribution), and 'Prune' (using L1 loss to remove unnecessary vertices, leaving only the relevant jet flavors).
Geometric Interpretation
The classifier output distribution is bounded by a $(T-1)$-simplex within the probability simplex. The vertices of this simplex are defined as anchor regions in phase space that yield pure samples of each operational topic, directly corresponding to the hadron-level definition of jet flavors.

Terminology

Summary

The gist The simplex demixing framework provides a practical, data-driven method to extract multiple jet flavors from mixed data samples by identifying maximally separable categories in phase space, which opens the door to extracting substructure distributions at the LHC.

Operational Definition and Geometric Interpretation

The operational definition of quark and gluon jets is defined as the maximally separable categories in the data when given two mixed jet samples, translating this into a bounded geometric object with T vertices The geometric interpretation involves training a classifier on mixtures using cross-entropy loss, where the output distribution is bounded by a (T −1)-simplex within the probability (M −1)-simplex This simplex has T vertices, which correspond to regions of phase space known as anchor regions that yield pure samples of jets of each operational topic in a specific process The vertices of the simplex are defined as the hadron-level definition of jet flavor

Simplex Demixing Framework

The simplex demixing framework is a practical machine learning procedure designed to extract the mixing fractions of operational topics from measured mixtures by training a classifier on M samples The procedure involves a three-stage training strategy Stage one is for the classifier to Learn, where vertices are fixed at the corners of the probability (M −1)-simplex, and L2 regularization is applied Stage two is for Shape, where an auxiliary loss Ledge, defined as the sum of Euclidean distances between each pair of vertices, encourages vertices to tightly enclose the classifier distribution Stage three is for Prune, where an L1 loss on vertex weights removes unnecessary vertices, leaving only those corresponding to the operational topics

Application and Results

The procedure was validated on a toy problem with M = 3 mixtures of down-quark, up-quark, and gluon jets In a proof-of-concept physics application involving dijet events from Pythia 8.310 [73], the method found that the distribution lies on a simplex spanned by T = 7 operational topics The results showed that five operational topics are strongly associated with the Pythia down, up, strange, anti-strange, and gluon flavors The tag-and-probe method successfully decomposed the marginal distributions into mixtures modeled as pm(x) = X t Fmt pt(x) This mixture model allowed for the recovery of the universal operational topics in the asymptotic limit, provided that Fmt has full column rank

Substructure Distribution Extraction

To extract observable distributions for each operational topic, the inversion of Eq. (3.8) is performed in a least-squares sense to obtain pt(O) = X M m=1 (F+) tm pm(O) This allows the reconstruction of operational topic distributions over any observable O, such as constituent multiplicity Nconst and 2-subjettiness τ(β=1)2 The results showed good agreement between the inferred topic distributions and the truth-level Pythia distributions for quark-like topics T3 (Pythia up) and T5 (Pythia strange), while gluon-like topics T4 and T7 were also in good agreement with the Pythia gluon distributions

Future Directions

Future work plans include incorporating simplex demixing into a domain-agnostic procedure for choosing the number of topics T, potentially using information-theoretic metrics or stability metrics Increasing effective sample size could be achieved by lowering jet selection cuts or augmenting the method to include probe jets with less certain tag-label assignments Finally, incorporating heavy-flavor jets into the framework would allow for a more realistic study of mixture-dependent fractions

The simplex demixing framework provides a practical, data-driven method to extract multiple jet flavors from mixed data samples by identifying maximally separable categories in phase space, which opens the door to extracting substructure distributions at the LHC.

How it works

  1. The operational definition of quark and gluon jets is defined as the maximally separable categories in the data when given two mixed jet samples, translating this into a bounded geometric object with T vertices.

  2. The geometric interpretation involves training a classifier on mixtures using cross-entropy loss, where the output distribution is bounded by a (T −1)-simplex within the probability (M −1)-simplex.

  3. The simplex demixing framework is a practical machine learning procedure designed to extract the mixing fractions of operational topics from measured mixtures by training a classifier on M samples.

  4. The procedure involves a three-stage training strategy Stage one is for the classifier to Learn, where vertices are fixed at the corners of the probability (M −1)-simplex, and L2 regularization is applied Stage two is for Shape, where an auxiliary loss Ledge, defined as the sum of Euclidean distances between each pair of vertices, encourages vertices to tightly enclose the classifier distribution Stage three is for Prune, where an L1 loss on vertex weights removes unnecessary vertices, leaving only those corresponding to the operational topics.

Identifying Operational Topics

The final results show that each vertex is an endpoint of the classifier distribution in at least one projection, so all seven vertices contribute to the convex hull The inferred coefficients pb(tc) and pb(ct) matrices establish a robust one-to-one correspondence between the Pythia d, u, s, anti-s, and g flavors and the T1 through T7 operational topics

Probabilistic Interpretation

The weak identifiability of the anti-down and anti-up flavors is expected from the small truth-level Pythia fractions shown in Table 2 The low anti-d and anti-u statistics can also provide a semi-quantitative understanding of why T2 and T4 are contaminated by gluons The intrinsic reducibility factor κqg is nonzero for Pythia jets, meaning the gluon-like operational topic necessarily contains a nonzero quark component

Extracting Substructure Distributions

To extract observable distributions for each operational topic, the inversion of Eq. (3.8) is performed in a least-squares sense to obtain pt(O) = X M m=1 (F+) tm pm(O) This allows the reconstruction of operational topic distributions over any observable O, such as constituent multiplicity Nconst and 2-subjettiness τ(β=1)2

Conclusions

In this paper, we proposed a data-driven strategy for extracting multiple light-flavor categories from jets at the LHC The key to our approach is a novel simplex demixing framework, which extends the operational definition of quark and gluon jets to any number of jet flavors T in any number of jet samples M, provided that T ≤ M In the asymptotic limit, we proved that a classifier trained on the M jet samples outputs points within a (T − 1)-simplex in classifier space, whose T vertices correspond to the latent jet flavors The positions of these vertices allow us to solve for the mixing fractions of the operational topics in the samples, and thereby extract the distributional properties of light-flavor jets With this proof in hand, we designed a practical three-stage machine learning procedure to demix the samples, compatible with any classifier architecture and robust to finite statistics

Code Availability

Our simplex demixing implementation can be found at Ref. [130] and the three-mixture datasets are at Ref. [131] The code is available at https://github.com/gregofuente/simplex demixing

Acknowledgments

We thank Sean Benevedes for helpful discussions, and Patrick Komiske and Eric Metodiev for early brainstorming on multi-category jet topics and subsequent feedback This work was supported by the U.S. National Science Foundation (NSF) under Cooperative Agreement PHY-2019786 (The NSF AI Institute for Artificial Intelligence and Fundamental Interactions, http://iaifi.org/), by the U.S. Department of Energy (DOE) Office of High Energy Physics under grant number DE-SC0012567, and by the Simons Foundation through Investigator grant 929241 JT also thanks the Institut des Hautes Etudes Scientifiques (IHES) and the Institut de Physique Th´eorique (IPhT) for providing an inspiring sabbatical environment to carry out this research This work was performed using resources from the FASRC Cannon cluster, supported by the FAS Division of Science Research Computing Group at Harvard University, and the SubMIT system [132] Claude Opus 4.8 was used for helpful discussions that improved the presentation of several arguments, to streamline the implementation of bootstrap resampling, and for proof reading; the authors take full responsibility for the content of the manuscript

References

[1] H.P. Nilles and K.H.

Improvements for AI systems

  1. Improve jet flavor definition via Simplex Demixing to handle any number of light-flavor categories, as we extend the operational definition to any number of samples M and operational topics T, as long as T ≤ M. This allows AI systems to extract the mixing fractions of operational topics from the measured mixtures even when dealing with more than two jet flavors.

  2. Implement a three-stage training procedure for Simplex Demixing, which involves learning a representation, shaping the geometry using an auxiliary loss, and pruning unnecessary vertices using L1 regularization. This ensures that the learned classifier distribution is bounded by a (T − 1)-simplex and that the system retains only the relevant vertices corresponding to operational topics.

  3. Develop a data-driven strategy for defining multiple light-flavor jet categories from dijet events using a tag-and-probe method. This allows AI systems to recover the operational topics latent in the experimental dataset by decomposing marginal distributions into topic distributions, such that pm(x) = X T Fmt pt(x).

  4. Utilize inferred conditional quasi-probabilities, specifically pb(tc) = X M m=1 (G+)cmFmt, to quantify the alignment between latent operational topics and known truth-level categories. This enables AI systems to assess the degree to which the operational topics align with the categories rather than assuming that exact agreement should hold.

  5. Extract observable distributions for each operational topic by inverting Eq. (3.8) in a least-squares sense: pt(O) = X M m=1 (F+)tm pm(O). This allows AI systems to reconstruct the operational topic distributions over any observable O, such as constituent multiplicity, 2-subjettiness, and jet charge.

Abstract

Providing a practical and hadron-level definition of multiple jet flavors has been a long-standing challenge in collider physics. Previous work has introduced a data-driven, operational definition of quark and gluon jets, but no robust generalization beyond two jet categories presently exists. To address this, we introduce a machine-learning framework called "simplex demixing'' to extract T jet flavors (or topics in the statistics literature) from M data samples (or mixtures) with minimal constraints. Intuitively, our procedure identifies the maximally separable categories in the data, translating a multi-category classifier on the M mixtures into a bounded geometric object with T vertices. We first demonstrate our procedure on a toy problem to infer the truth-level fractions of down-quark, up-quark, and gluon jets from synthetic mixtures of the three pure samples. We then propose a tag-and-probe strategy to extract multiple light-flavor categories in a more realistic collider setting involving dijet production. As expected, the identifiability of jet flavors depends on their relative abundance in the samples and the hadron-level information available to the classifier architecture. Our work opens the door to data-driven extractions of multiple jet flavor properties at the Large Hadron Collider.

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