Combinatorial Residual Network: a decay-chain-aware classifier without explicit jet assignment

John Alison; Patrick Bryant; Alejandro Gomez Espinosa; Christian Guinto-Brody; Aniket Khanal; Chuyuan Liu; Sindhu Murthy; Guanglei Zhao

hep-exOct 29, 2026

Combinatorial Residual Network: a decay-chain-aware classifier without explicit jet assignment

Authors: John Alison, Patrick Bryant, Alejandro Gomez Espinosa, Christian Guinto-Brody, Aniket Khanal, Chuyuan Liu, Sindhu Murthy, Guanglei Zhao

Organizations: Department of Physics, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA

Abstract

Final states with several jets from the decays of heavy particles pose a combinatorial problem, since the jets must be assigned, at least implicitly, to their parent particles. We describe the Combinatorial Residual Network (CoReNet), a classifier that encodes the assumed decay chain of the final state in its architecture. CoReNet builds its event representation level by level along the decay hierarchy, evaluates every jet assignment hypothesis with weights shared across objects, and combines the hypotheses with learned weights using residual networks. The exchange symmetries of the decay are built into the construction, the size of the network does not depend on the number of hypotheses, and the network is trained with the classification loss alone, without generator-level jet assignments. The architecture was developed for the $b\bar{b}b\bar{b}$ final state and has been used in several CMS searches. On fast-simulation samples of Higgs boson pair production in the $b\bar{b}b\bar{b}$ final state against QCD multijet production, CoReNet with 1508 trainable parameters performs well above kinematic baselines and trains in minutes on a single GPU. Its hypothesis weights select the truth-matched jet pairing in most of the signal events in which it is defined, although no assignment target is used in training. Since only the encoded decay chain is specific to the final state, the construction carries over to other topologies, which we test with $HH \to b\bar{b}W^+W^-$ against semileptonic $t\bar{t}$ . There, encoding the decay chain of the background as well as that of the signal further improves the classification, at a cost of about $10^3$ parameters. In both final states its performance is comparable to that of a state-of-the-art transformer-based network more than two orders of magnitude larger. The implementation is publicly available at https://github.com/cms-cmu/CoReNet.

Figures & tables

Figure 1: The CoReNet architecture for

HH\to b\bar{b}b\bar{b}

. The four candidate jets are combined into six dijets, which are grouped into the three pairing hypotheses

v_{0}

v_{1}

and

v_{2}

. At each level, the representation of an object is built from the objects below it and from its own four-vector, and is refined by two residual layers with weights shared across objects. The additional jets enter through attention blocks that update the dijet representations. The hypotheses are pooled with the weights

q_{\mathrm{score}}

into an event representation

\mathbf{e}

, which a linear layer maps to the classifier output. No assignment target is applied to

q_{\mathrm{score}}

Figure 2: The CoReNet hierarchy for

HH\to b\bar{b}W^{+}W^{-}

with one leptonically decaying

W

boson. (a) Signal hypothesis, used in both CoReNet HH and CoReNet

{}_{HH+t\bar{t}}

: the two leading

b

-tagged jets,

b_{1}

and

b_{2}

, form the

H\to b\bar{b}

candidate, the two leading other jets

q_{1}

and

q_{2}

the hadronic

W

candidate, and the lepton and missing transverse momentum the leptonic

W

candidate. (b) Background hypothesis, added in CoReNet

{}_{HH+t\bar{t}}

: the two exclusive assignments of the

b

jets to a hadronic and a leptonic top quark candidate are evaluated and combined with learned weights, as the three pairings are for

b\bar{b}b\bar{b}

. Dashed arrows indicate the pairwise merge that connects the two halves of each hypothesis. In CoReNet

{}_{HH+t\bar{t}}

the two hypotheses and a set of relation features are combined into the classifier output, and the

t\bar{t}

hypothesis also feeds an auxiliary output.

Model	Parameters	ROC AUC	corrected $p$
Kinematic GBDT	—	$0.8294\pm 0.0028$	$6\times 10^{-5}$
Kinematic MLP	4993	$0.8295\pm 0.0041$	$3\times 10^{-4}$
CoReNet ( $d=8$ )	1508	$\mathbf{0.8664\pm 0.0031}$	—
CoReNet ( $d=8$ ), no additional-jet attention	782	$0.8578\pm 0.0013$	$0.012$
SPANet, classification only ( $\lambda_{\mathrm{assign}}=0$ )	559846	$0.8536\pm 0.0032$	$0.015$
SPANet, $\lambda_{\mathrm{assign}}=1$ (best)	559846	$0.8593\pm 0.0047$	$0.14$

Table 1: Five-fold classification performance (ROC AUC weighted by the class-normalized event weights). Uncertainties are standard deviations across folds.

p

-values are two-sided paired

t

-tests against CoReNet (

d=8

) across the five folds, with the Nadeau–Bengio variance correction for the overlap of the training sets [ 32 ] . SPANet uses the architecture of Ref. [ 4 ] trained for 500 epochs, with

\lambda_{\mathrm{det}}=\lambda_{\mathrm{cls}}=1

and the assignment-loss weight

\lambda_{\mathrm{assign}}

given in the table (

\lambda_{\mathrm{det}}=0

for the classification-only row); SPANet rows use, for each fold, the epoch checkpoint with the best validation AUC.

Figure 3: Five-fold mean ROC curves with cross-fold spread for CoReNet (

d=8

) (1508 parameters), the classification-only SPANet (

559846

parameters, 500 epochs) and the kinematic MLP baseline, on (a) linear and (b) logarithmic background-efficiency axes.

Model	Parameters	ROC AUC	corrected $p$
Kinematic GBDT	—	$0.8536\pm 0.0041$	$2\times 10^{-5}$
Kinematic MLP	5633	$0.8537\pm 0.0019$	$4\times 10^{-7}$
CoReNet HH	2550	$0.8859\pm 0.0016$	$3\times 10^{-5}$
CoReNet ${}_{HH+t\bar{t}}$	3832	$\mathbf{0.9004\pm 0.0018}$	—
CoReNet ${}_{HH+t\bar{t}}$ , auxiliary $t\bar{t}$ output	3832	$0.8753\pm 0.0005$	$4\times 10^{-5}$
SPANet, classification only ( $\lambda_{\mathrm{assign}}=0$ )	594520	$0.9126\pm 0.0025$	$0.002$

Table 2: Five-fold classification performance on

HH\to b\bar{b}W^{+}W^{-}

against semileptonic

t\bar{t}

(ROC AUC weighted by the class-normalized event weights, mean

\pm

standard deviation over the five test folds). CoReNet HH encodes the decay chain of the signal only, and CoReNet

{}_{HH+t\bar{t}}

also that of the

t\bar{t}

background (Fig. 2 ); the auxiliary row uses the auxiliary

t\bar{t}

output of CoReNet

{}_{HH+t\bar{t}}

as the discriminant, with score

1-P(t\bar{t})

. SPANet uses the architecture of Ref. [ 4 ] trained for 500 epochs, with

\lambda_{\mathrm{det}}=\lambda_{\mathrm{cls}}=1

and the assignment-loss weight given in the table (

\lambda_{\mathrm{det}}=0

for the classification-only row), and the checkpoint with the best validation AUC of each fold. The

p

-values are two-sided paired

t

-tests against CoReNet

{}_{HH+t\bar{t}}

across the five folds, with the Nadeau–Bengio correction [ 32 ] .

Figure 4: Five-fold mean ROC curves for

HH\to b\bar{b}W^{+}W^{-}

with cross-fold spread for CoReNet

{}_{HH+t\bar{t}}

(3832 parameters), SPANet trained with the classification loss alone (594520 parameters, 500 epochs) and the kinematic MLP baseline, on (a) linear and (b) logarithmic background-efficiency axes. SPANet uses, per fold, the checkpoint with the best validation AUC.

Figure 5: Classification AUC (test) versus trainable parameters for (a)

HH\to b\bar{b}b\bar{b}

and (b)

HH\to b\bar{b}W^{+}W^{-}

. CoReNet is shown at

d\in\{4,8,16,32,64\}

with five-fold standard deviations, in (b) as CoReNet

{}_{HH+t\bar{t}}

and CoReNet HH ; SPANet appears at 559846 (a) and 594520 (b) parameters at the four assignment-loss weights

\lambda_{\mathrm{assign}}=0

(classification only,

\lambda_{\mathrm{det}}=0,\ \lambda_{\mathrm{cls}}=1

), 1, 5 and 20, the latter three with

\lambda_{\mathrm{det}}=\lambda_{\mathrm{cls}}=1

, horizontally offset for visibility; the shaded band is the spread of the hand-engineered GBDT and MLP baselines.

$d$	Parameters	AUC (train)	AUC (validation)	AUC (test)
4	504	$0.8587\pm 0.0025$	$0.8575\pm 0.0039$	$0.8571\pm 0.0029$
8	1508	$0.8723\pm 0.0018$	$0.8670\pm 0.0042$	$\mathbf{0.8664\pm 0.0031}$
16	5052	$0.8835\pm 0.0020$	$0.8713\pm 0.0017$	$0.8706\pm 0.0013$
32	18284	$0.8883\pm 0.0012$	$0.8726\pm 0.0018$	$0.8725\pm 0.0017$
64	69324	$0.8887\pm 0.0046$	$0.8729\pm 0.0018$	$0.8723\pm 0.0022$

Table 3: CoReNet width scan on

HH\to b\bar{b}b\bar{b}

. Only the embedding dimension

d

changes; everything else follows the main training. ROC AUC on the training folds, the validation fold and the held-out test fold, five-fold mean

\pm

standard deviation.

CoReNet HH
$d$	Parameters	AUC (train)	AUC (validation)	AUC (test)
4	780	$0.8877\pm 0.0010$	$0.8812\pm 0.0027$	$0.8818\pm 0.0035$
8	2550	$0.8980\pm 0.0012$	$0.8863\pm 0.0020$	$\mathbf{0.8859\pm 0.0016}$
16	9066	$0.9104\pm 0.0021$	$0.8878\pm 0.0015$	$0.8877\pm 0.0013$
32	34002	$0.9139\pm 0.0102$	$0.8870\pm 0.0021$	$0.8856\pm 0.0032$
64	131490	$0.9235\pm 0.0008$	$0.8880\pm 0.0027$	$0.8872\pm 0.0021$

Table 4: CoReNet HH and CoReNet

{}_{HH+t\bar{t}}

width scans on

HH\to b\bar{b}W^{+}W^{-}

. Only the embedding dimension

d

changes; everything else follows the main training. ROC AUC on the training folds, the validation fold and the held-out test fold, five-fold mean

\pm

standard deviation.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: The residual blocks of (a) the dijet level, Eq. ( 1 ), and (b) the pairing-hypothesis level, Eq. ( 4 ). (a) The jets and dijets are laid out as six slots of (jet, jet, dijet), and a single convolution with kernel size and stride three, with the same weights for every slot, maps each slot to a dijet update. Each update is added to the initial embedding

\mathbf{d}^{(0)}_{ij}

of the same slot and passed through the SiLU nonlinearity

\sigma

. The outputs replace

\mathbf{d}^{(\ell)}_{ij}

in the next layer, and two layers are applied. Between them, each jet slot is refined once by a size-1 convolution with a skip connection to

\mathbf{j}^{(0)}

(jet path), and the second layer reads the refined jets

\mathbf{j}^{(1)}

. (b) For each pairing hypothesis, the two dijets are combined through their mean

\mathbf{s}_{k}

and absolute difference

\mathbf{a}_{k}

, which carry no trainable weights and are unchanged under exchange of the two dijets. A single convolution, shared across the three hypotheses, reads

(\mathbf{s}_{k},\mathbf{a}_{k},\mathbf{q}_{k}^{(\ell)})

, and its output is added to the initial embedding

\mathbf{q}_{k}^{(0)}

of the four-jet system. The dijets

\tilde{\mathbf{d}}

entering this level are refined once between the two layers in the same way as the jets in (a) (dijet path).

Figure 7: The dijet–jet attention block, shown for the dijet

\mathbf{d}_{12}

as the query. The values are the twelve jets after the jet–jet block, with the two jets of the dijet and the empty slots masked. For each query–jet pair, the logarithm of the invariant mass and the azimuthal separation are embedded and enter the score and the value through small networks

g_{s}

and

g_{v}

. The attention weights are the softmax of the scores multiplied by their sigmoid, and the weighted sum of the values updates the dijet as a residual correction. The jet–jet block that precedes it has the same structure.

	CoReNet ( $d=8$ )	SPANet
Folds	5 (3 training, 1 validation, 1 test)
Trainable parameters	1508	559846
Width	$d=8$	hidden dimension 32, 8 attention heads
Depth	jet $\to$ dijet $\to$ quadjet hierarchy	8 central + 6 branch encoder layers
Per-jet inputs	$p_{\mathrm{T}},\ \eta,\ \phi,\ m$	$p_{\mathrm{T}},\ \eta,\ \phi,\ m$ , $b$ tag
Optimizer	Adam, learning rate $10^{-2}$	AdamW [ 41 ] , learning rate $1.5\times 10^{-3}$

Appendix

Table 5: Training configurations for

HH\to b\bar{b}b\bar{b}

. Both networks use the same folds, the same class-normalized event weights and the same input file; the

b

tag enters CoReNet only through the choice and ordering of the candidate jets, and SPANet as a per-jet feature. SPANet’s architecture and optimizer follow Ref. [ 4 ] .

$\lambda_{\mathrm{assign}}$	50 epochs	150 epochs	500 epochs	best epoch
0 (classification only)	$0.8555\pm 0.0017$	$0.8545\pm 0.0035$	$0.8536\pm 0.0032$	29
1	$0.8520\pm 0.0058$	$0.8624\pm 0.0059$	$0.8593\pm 0.0047$	150
5	$0.8338\pm 0.0064$	$0.8361\pm 0.0051$	$0.8437\pm 0.0052$	407
20	$0.8075\pm 0.0075$	$0.8188\pm 0.0106$	$0.8261\pm 0.0011$	465

Appendix

Table 6: SPANet classification AUC on

HH\to b\bar{b}b\bar{b}

as a function of the assignment-loss weight

\lambda_{\mathrm{assign}}

(

\lambda_{\mathrm{det}}=\lambda_{\mathrm{cls}}=1

, except

\lambda_{\mathrm{det}}=0

for the classification-only row), for training budgets of 50, 150 and 500 epochs. Each entry is the five-fold mean

\pm

standard deviation of the test AUC at the best-validation checkpoint; the last column gives the mean best-validation epoch of the 500-epoch runs. For comparison, CoReNet (

d=8

) reaches

0.8664\pm 0.0031

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