Hypergraph Representation Learning with Hyperlink Random Effects

Zimeng Li; Shihao Wu; Gongjun Xu; Ji Zhu

stat.MLOct 3, 2026

Hypergraph Representation Learning with Hyperlink Random Effects

Authors: Zimeng Li, Shihao Wu, Gongjun Xu, Ji Zhu

Organizations: Department of Statistics, Harvard University Cambridge, MA 02138 · Department of Statistics and Data Science, University of California, Davis Davis, CA 95616 · Department of Statistics, University of Michigan Ann Arbor, MI 48109

Abstract

Hypergraphs record multi-way interactions among entities. Extracting information from the combinatorial structure underlying observed multi-way interactions is a central task in many real-world problems. Existing methods face several limitations. First, many deep architectures for hypergraphs do not explicitly exploit the potential low-rank structure, which can sacrifice parsimony and interpretability in the learned representations. Second, many low-rank-based methods operate on tensor representations, which typically require hyperlinks to have uniform sizes and thus limit their applicability to general hypergraphs with non-uniform hyperlink sizes. Third, many methods ignore the fact that hyperlinks often arise from heterogeneous mechanisms. For example, medical symptoms may co-occur in the profiles of patients with very different conditions, and such heterogeneity should be incorporated into the learning process. In this work, we develop a general framework for hypergraph representation learning using hyperlink random effects while exploiting the low-rank structure in hypergraphs. The proposed framework accommodates latent heterogeneity in hyperlink formation while preserving entity interaction patterns. We establish identifiability of the model parameters and theoretical guarantees of representation-level recovery under this framework. The framework allows flexible specifications for the hyperlink random effects; in this paper, we study three choices: categorical, Gaussian mixture, and score-based effects, and develop corresponding estimation algorithms. Through simulation studies, we demonstrate the effectiveness of the proposed method in recovering latent structure and capturing heterogeneous interaction patterns. Empirical studies on real-world hypergraph datasets further illustrate the practical utility of our approach.

Figures & tables

Figure 1: Illustration of hypergraph representation learning with hyperlink random effects using a patient-profile toy example. The data are represented as a symptom co-occurrence hypergraph with five nodes, corresponding to symptoms, and six observed hyperlinks, corresponding to symptom sets observed in individual patient profiles. Each symptom is assigned a latent embedding and degree parameter. To capture patient-level heterogeneity, each profile is assigned a hyperlink random-effect embedding representing latent patient characteristics that drive profile formation.

Figure 2: Recovery of node embeddings under Gaussian-mixture (left) and t-mixture (right), with

n=180

	Gaussian mixture						$t$ -mixture, df = 4
Method	1000		3000		5000		1000		3000		5000
	ARI $\uparrow$	NMI $\uparrow$	ARI $\uparrow$	NMI $\uparrow$	ARI $\uparrow$	NMI $\uparrow$	ARI $\uparrow$	NMI $\uparrow$	ARI $\uparrow$	NMI $\uparrow$	ARI $\uparrow$	NMI $\uparrow$
HRL-Cat	0.958	0.956	0.961	0.955	0.963	0.956	0.343	0.515	0.322	0.488	0.357	0.518
HRL-GM	0.962	0.959	0.962	0.956	0.964	0.956	0.343	0.517	0.344	0.510	0.358	0.519
HRL-Score	0.964	0.960	0.965	0.959	0.966	0.959	0.948	0.942	0.954	0.943	0.951	0.939
HRL-Fixed	0.962	0.960	0.962	0.956	0.964	0.956	0.338	0.501	0.314	0.471	0.351	0.504

Table 1: Clustering performance for Gaussian-mixture and heavy-tailed

t

-mixture designs. Detailed tables with standard errors are in Appendix C.1 .

	Food Recipe				MIMIC-III				MovieLens
Method	AUC $\uparrow$	AUPRC $\uparrow$	BCE $\downarrow$	Brier $\downarrow$	AUC $\uparrow$	AUPRC $\uparrow$	BCE $\downarrow$	Brier $\downarrow$	AUC $\uparrow$	AUPRC $\uparrow$	BCE $\downarrow$	Brier $\downarrow$
HRL-Cat	0.787	0.160	0.132	0.031	0.771	0.396	0.334	0.097	0.728	0.644	0.601	0.205
HRL-GM	0.827	0.194	0.121	0.030	0.795	0.437	0.315	0.093	0.783	0.712	0.545	0.184
HRL-Score	0.826	0.195	0.120	0.029	0.796	0.436	0.314	0.093	0.776	0.703	0.552	0.187
HRL-Fixed	0.769	0.130	0.206	0.041	0.745	0.337	0.506	0.111	0.767	0.682	0.617	0.199
NeuMF	0.755	0.132	0.133	0.031	0.768	0.386	0.330	0.097	0.754	0.664	0.576	0.196

Table 2: Cold-start experiment with

r_{obs}=0.2

. For each metric, the best result is highlighted in bold and the second-best result is underlined. Detailed tables with standard errors are in Appendix C.1 .

Figure 3: Performance of multi-label classification on Movie Lens datasets

Figure 4: Uniform Manifold Approximation and Projection (UMAP) of the clustering results based on ICD-9 code embeddings learned by HRL-Score with $7$ clusters. Points with the same color belong to the same group. Filled points represent core members with posterior membership probability at least $0.9$ , and hollow points represent nodes with posterior membership probability less than $0.9$ . The right panel reports the cluster names and the sizes of the core-member sets.

Appendix figures & tables31 assets

Supplementary material from the paper’s appendix.

Appendix

\widehat{\mathbf{m}}_{\mathbf{F}}=\frac{1}{B_{0}}\sum_{b=1}^{B_{0}}\mathbf{f}^{(b)},\hskip 18.49988pt\widehat{\boldsymbol{\Sigma}}_{\mathbf{F}}=\frac{1}{B_{0}}\sum_{b=1}^{B_{0}}\left(\mathbf{f}^{(b)}-\widehat{\mathbf{m}}_{\mathbf{F}}\right)\left(\mathbf{f}^{(b)}-\widehat{\mathbf{m}}_{\mathbf{F}}\right)^{\top}.

Appendix

Algorithm 4 Monte Carlo variational EM for score-based hyperlink random effects

	$\\|\hat{\mathbf{Q}}-\mathbf{Q}^{*}\\|_{F}/\sqrt{Kn}$				$\\|\hat{\mathbf{F}}-\mathbf{F}^{*}\\|_{F}/\sqrt{Kd}$				$\\|\hat{\mathbf{Z}}-\mathbf{Z}^{*}\\|_{F}/\sqrt{nd}$
	$n=30$	$n=90$	$n=270$	$n=810$	$n=30$	$n=90$	$n=270$	$n=810$	$n=30$	$n=90$	$n=270$	$n=810$
$m=500$	0.057	0.039	0.037	0.037	0.297	0.181	0.156	0.152	0.347	0.282	0.269	0.274
$m=1000$	0.032	0.026	0.025	0.025	0.148	0.097	0.093	0.091	0.204	0.175	0.169	0.171
$m=1500$	0.024	0.021	0.020	0.020	0.107	0.077	0.075	0.069	0.150	0.135	0.133	0.133
$m=2000$	0.021	0.018	0.017	0.017	0.087	0.066	0.063	0.059	0.123	0.113	0.113	0.113

Appendix

Table 3: Parameter recovery under the balanced categorical design. Values are averaged over 100 Monte Carlo repetitions.

Figure 5: Monte Carlo distribution of normalized estimation errors under the balanced categorical design. The number of hyperlinks $m$ varies along the horizontal axis, and different box colors correspond to different node sizes $n$ .

	$\\|\hat{\mathbf{Q}}-\mathbf{Q}^{*}\\|_{F}/\sqrt{Kn}$				$\\|\hat{\mathbf{F}}-\mathbf{F}^{*}\\|_{F}/\sqrt{Kd}$				$\\|\hat{\mathbf{Z}}-\mathbf{Z}^{*}\\|_{F}/\sqrt{nd}$
	$n=30$	$n=90$	$n=270$	$n=810$	$n=30$	$n=90$	$n=270$	$n=810$	$n=30$	$n=90$	$n=270$	$n=810$
$m=500$	0.083	0.054	0.050	0.049	0.452	0.267	0.239	0.234	0.446	0.351	0.341	0.341
$m=1000$	0.048	0.032	0.031	0.031	0.241	0.129	0.120	0.118	0.293	0.223	0.217	0.218
$m=1500$	0.034	0.025	0.024	0.024	0.152	0.093	0.083	0.080	0.211	0.168	0.162	0.163
$m=2000$	0.027	0.021	0.020	0.020	0.115	0.068	0.064	0.062	0.166	0.132	0.132	0.134

Appendix

Table 4: Parameter recovery under the unbalanced categorical design. The mixing probabilities satisfy

\pi_{1}^{*}:\cdots:\pi_{8}^{*}=1:1:2:2:3:3:4:4

. Values are averaged over 100 Monte Carlo repetitions.

Figure 6: Monte Carlo distribution of normalized estimation errors under the unbalanced categorical design. The number of hyperlinks $m$ varies along the horizontal axis, and different box colors correspond to different node sizes $n$ .

	$\\|\hat{\mathbf{Q}}-\mathbf{Q}^{*}\\|_{F}/\sqrt{Kn}$				$\\|\hat{\mathbf{F}}-\mathbf{F}^{*}\\|_{F}/\sqrt{Kd}$				$\\|\hat{\mathbf{Z}}-\mathbf{Z}^{*}\\|_{F}/\sqrt{nd}$
	$n=50$	$n=250$	$n=500$	$n=1000$	$n=50$	$n=250$	$n=500$	$n=1000$	$n=50$	$n=250$	$n=500$	$n=1000$
$m=1000$	0.112	0.056	0.042	0.041	0.421	0.189	0.156	0.160	0.451	0.313	0.286	0.288
$m=2000$	0.061	0.027	0.025	0.025	0.219	0.119	0.116	0.114	0.291	0.205	0.205	0.206
$m=3000$	0.037	0.020	0.020	0.020	0.144	0.100	0.096	0.094	0.209	0.174	0.170	0.168
$m=4000$	0.027	0.017	0.017	0.017	0.115	0.083	0.084	0.080	0.173	0.146	0.149	0.145

Appendix

Table 5: Parameter recovery under the larger categorical design with

K=32

latent hyperlink types and embedding dimension

d=5

. Values are averaged over 100 Monte Carlo repetitions.

Figure 7: Validation curves for selecting the latent embedding dimension $d$ under the balanced categorical design. The true embedding dimension is $d^{*}=3$ .

Figure 8: Validation curves for selecting the latent embedding dimension $d$ under the larger categorical design. The true embedding dimension is $d^{*}=5$

Figure 9: Recovery of node degree parameters $\boldsymbol{\alpha}$ under Gaussian-mixture (left) and t-mixture (right), with $n=180$ .

	Gaussian mixture
Method	1000		3000		5000
	ARI $\uparrow$	NMI $\uparrow$	ARI $\uparrow$	NMI $\uparrow$	ARI $\uparrow$	NMI $\uparrow$
HRL-Cat	0.958(0.0027)	0.956(0.0026)	0.961(0.0014)	0.955(0.0014)	0.963(0.0010)	0.956(0.0009)
HRL-GM	0.962(0.0025)	0.959(0.0025)	0.962(0.0013)	0.956(0.0013)	0.964(0.0009)	0.956(0.0009)
HRL-Score	0.964(0.0015)	0.960(0.0015)	0.965(0.0012)	0.959(0.0013)	0.966(0.0007)	0.959(0.0008)
HRL-Fixed	0.962(0.0024)	0.960(0.0024)	0.962(0.0013)	0.956(0.0013)	0.964(0.0010)	0.956(0.0009)

Appendix

Table 6: Clustering performance for Gaussian-mixture design, mean(standard error), based on 20 repetitions

	$t$ -mixture, df = 4
Method	1000		3000		5000
	ARI $\uparrow$	NMI $\uparrow$	ARI $\uparrow$	NMI $\uparrow$	ARI $\uparrow$	NMI $\uparrow$
HRL-Cat	0.343(0.0135)	0.515(0.0105)	0.322(0.0157)	0.488(0.0146)	0.357(0.0117)	0.518(0.0101)
HRL-GM	0.343(0.0135)	0.517(0.0103)	0.344(0.0195)	0.510(0.0111)	0.358(0.0118)	0.519(0.0101)
HRL-Score	0.948(0.0028)	0.942(0.0030)	0.954(0.0013)	0.943(0.0014)	0.951(0.0010)	0.939(0.0012)
HRL-Fixed	0.338(0.0136)	0.501(0.0123)	0.314(0.0171)	0.471(0.0164)	0.351(0.0130)	0.504(0.0117)

Appendix

Table 7: Clustering performance for t-mixture design, mean(standard error), based on 20 repetitions

Initialization	$\mathrm{Err}_{Z}\downarrow$	$\mathrm{Err}_{F}\downarrow$	ARI $\uparrow$	NMI $\uparrow$	Runtime (s)
USVT–GMM	0.1123 (0.0077)	0.1384 (0.0079)	0.9591 (0.0015)	0.9529 (0.0020)	164.9 (4.5)
USVT $(Z,\alpha)$ + $N(0,I_{d})$ prior	0.1124 (0.0083)	0.1288 (0.0056)	0.9560 (0.0014)	0.9508 (0.0014)	152.7 (3.9)
Random $Z$ + degree $\alpha$ + $N(0,I_{d})$ prior	0.7070 (0.1839)	0.5733 (0.1354)	0.8450 (0.0796)	0.8745 (0.0515)	157.6 (4.5)

Appendix

Table 8: HRL-Score sensitivity to initialization. Values are mean (standard error) over five repetitions.

Particles	$\mathrm{Err}_{Z}\downarrow$	$\mathrm{Err}_{F}\downarrow$	ARI $\uparrow$	NMI $\uparrow$	Runtime (s)
64	0.8559 (0.1847)	0.7584 (0.0263)	0.5607 (0.0233)	0.6788 (0.0202)	94.7 (0.9)
128	0.7264 (0.3129)	0.4787 (0.0904)	0.8539 (0.0285)	0.8803 (0.0171)	106.6 (2.0)
256	0.8919 (0.7597)	0.3196 (0.1329)	0.8440 (0.0825)	0.8708 (0.0584)	121.0 (2.1)
512	0.1123 (0.0077)	0.1384 (0.0079)	0.9591 (0.0015)	0.9529 (0.0020)	164.9 (4.5)
1024	0.1087 (0.0065)	0.1108 (0.0070)	0.9645 (0.0014)	0.9587 (0.0013)	248.0 (2.4)

Appendix

Table 9: HRL-Score sensitivity to the number of Monte Carlo particles.

Factor	Setting / evaluation window	$\mathrm{Err}_{Z}\downarrow$	$\mathrm{Err}_{F}\downarrow$	ARI $\uparrow$	NMI $\uparrow$	Runtime (s)
Pretrain	200	0.1284 (0.0206)	0.1899 (0.0597)	0.9570 (0.0015)	0.9511 (0.0017)	165.8 (1.8)
	400	0.1123 (0.0077)	0.1384 (0.0079)	0.9591 (0.0015)	0.9529 (0.0020)	164.9 (4.5)
	600	0.1124 (0.0076)	0.1358 (0.0071)	0.9562 (0.0022)	0.9516 (0.0017)	171.7 (2.0)
	800	0.1201 (0.0077)	0.1569 (0.0249)	0.9582 (0.0013)	0.9526 (0.0015)	163.3 (6.1)
Width	$H=32$ (epochs 1200–1400)	0.1178 (0.0069)	0.1397 (0.0100)	0.9596 (0.0019)	0.9546 (0.0019)	160.9 (4.2)
	$H=64$ (epochs 1000–1200)	0.1075 (0.0045)	0.1242 (0.0051)	0.9592 (0.0021)	0.9538 (0.0022)	156.8 (2.8)

Appendix

Table 10: HRL-Score sensitivity to pretraining, hidden width, and learning rate. Widths

H=64

and

H=32

were extended beyond

1000

epochs. All other settings use

1000

epochs.

Method	AUROC $\uparrow$	AUPRC $\uparrow$	BCE $\downarrow$	Brier $\downarrow$
HRL-Cat	0.771 (0.0022)	0.396 (0.0041)	0.334 (0.0020)	0.097 (0.0006)
HRL-GM	0.795 (0.0014)	0.437 (0.0038)	0.315 (0.0011)	0.093 (0.0004)
HRL-Score ( $H=26$ )	0.796 (0.0014)	0.436 (0.0035)	0.314 (0.0012)	0.093 (0.0004)
HRL-Fixed	0.745 (0.0031)	0.337 (0.0069)	0.506 (0.0115)	0.111 (0.0011)
NeuMF	0.768 (0.0014)	0.386 (0.0031)	0.330 (0.0016)	0.097 (0.0006)
AllSet	0.765 (0.0022)	0.378 (0.0021)	0.331 (0.0018)	0.098 (0.0006)

Appendix

Table 11: MIMIC-III: cold-start prediction at

r_{\rm obs}=0.2

. Values are mean (standard error) across five repetitions.

$r_{\rm obs}$	Method	AUROC $\uparrow$	AUPRC $\uparrow$	BCE $\downarrow$	Brier $\downarrow$
0.2	HRL-Cat	0.771 (0.0022)	0.396 (0.0041)	0.334 (0.0020)	0.097 (0.0006)
	HRL-GM	0.795 (0.0014)	0.437 (0.0038)	0.315 (0.0011)	0.093 (0.0004)
	HRL-Score	0.796 (0.0010)	0.435 (0.0040)	0.314 (0.0010)	0.093 (0.0004)
	HRL-Fixed	0.745 (0.0031)	0.337 (0.0069)	0.506 (0.0115)	0.111 (0.0011)
0.5	HRL-Cat	0.785 (0.0020)	0.413 (0.0034)	0.323 (0.0014)	0.094 (0.0005)
	HRL-GM	0.809 (0.0017)	0.460 (0.0048)	0.305 (0.0009)	0.090 (0.0003)

Appendix

Table 12: MIMIC-III reveal-fraction study with 70% global training rows. Values are mean (standard error) over shared repetitions. Best values are bold and second-best values are underlined.

Cluster	Core size	Manual name	Top 5 representative ICD-9 concepts
1	34	Cardiac disease	Native coronary atherosclerosis (414.01) PCI status (V45.82) Old myocardial infarction (412) CABG status (V45.81) Atrial fibrillation (427.31)
2	33	Cardiometabolic comorbidities	Essential hypertension (401.9) Morbid obesity (278.01) Chronic diastolic heart failure (428.32) Obstructive sleep apnea (327.23) Acute-on-chronic diastolic heart failure (428.33)
3	63	Postoperative complications	Implant/graft-related surgical complications (E878.2) Post-procedural hematoma (998.12) Post-procedural hemorrhage (998.11) Postoperative infection (998.59) Other procedure-related complications (E878.8)
4	12	Diabetes-related complications	Type 2 diabetes with neurologic complications (250.60) Diabetic polyneuropathy (357.2) Diabetic retinopathy (362.01) Type 2 diabetes with renal complications (250.40) Nephritis/nephropathy(583.81)
5	4	Kidney-related	Chronic kidney disease, unspecified (585.9) Hypertensive CKD, stage I–IV or unspecified (403.90) Stage III CKD (585.3) Acute kidney failure (584.9)
6	31	Liver-related complications	Portal hypertension (572.3) Hepatic encephalopathy (572.2) Liver cirrhosis (571.5) Ascites (789.59) Alcoholic cirrhosis (571.2)

Appendix

Table 13: ICD-9 code cluster learned from HRL-Score with

G=7

Cluster	Core Size	Manual name	Top 5 ICD-9 concepts
1	22	Liver-related complications	Portal hypertension (572.3) Other ascites (789.59) Hepatic encephalopathy (572.2) Cirrhosis of liver without mention of alcohol (571.5) Alcoholic cirrhosis of liver (571.2)
2	27	Cardiometabolic	Unspecified essential hypertension (401.9) Chronic diastolic heart failure (428.32) Morbid obesity (278.01) Obstructive sleep apnea (adult)(pediatric) (327.23) Iatrogenic pneumothorax (512.1)
3	35	Cardiac disease	Aortocoronary bypass status (V45.81) Bypass-graft coronary atherosclerosis (414.02) Native coronary atherosclerosis (414.01) Cardiac device/graft complications(996.72) Subendocardial infarction, initial episode of care (410.71)
4	53	Malignancy / neurologic complications	Obstructive hydrocephalus (331.4) Intracerebral hemorrhage (431) Secondary malignancy, other sites (198.89) History of lung cancer (V10.11) Brain/spinal cord metastasis (198.3)
5	22	Hematologic comorbidity	Acute posthemorrhagic anemia (285.1) Venous peripheral insufficiency, unspecified (459.81) Do not resuscitate status (V49.86) Other and unspecified hyperlipidemia (272.4) Hyperpotassemia (276.7)
6	8	Diabetes-related complications	Diabetic retinopathy (362.01) Type 2 diabetes with neurologic complications (250.60) Diabetic polyneuropathy (357.2) Type 2 diabetes with renal complications (250.40) Nephropathy in diseases classified elsewhere (583.81)

Appendix

Table 14: ICD-9 code cluster learned from HRL-Score with

G=11

Cluster	Core size	Manual name	Top 5 representative ICD-9 concepts
1	33	Cardiorenal comorbidity	Subendocardial infarction, initial episode of care (410.71) Pleural effusion (511.9) Hypertensive CKD, stage V or ESRD (403.91) Acute kidney failure with tubular necrosis (584.5) Hypovolemia (276.52)
2	60	Neurovascular / malignancy-related	Lung contusion (861.21) Thoracic aneurysm (441.2) Alcohol abuse, continuous (305.01) Subarachnoid hemorrhage (430) Cerebral edema (348.5)
3	18	Anticoagulation / infection comorbidity	Long-term anticoagulant use (V58.61) Thrombocytopenia (287.5) Aspiration pneumonitis (507.0) Acquired hypothyroidism (244.9) Clostridium difficile infection (008.45)
4	20	Liver / fluid-overload complications	Hematemesis (578.0) Urinary complications (997.5) Hepatic encephalopathy (572.2) Alcoholic cirrhosis (571.2) Fluid overload (276.6)
5	69	Liver / diabetes mixed residual	Tricuspid valve disease (397.0) Dysthymic disorder (300.4) Chronic hepatitis C (070.54) Acute/subacute liver necrosis (570) Diabetic polyneuropathy (357.2)
6	7	Acute cardiorenal comorbidity	Acute kidney failure (584.9) Congestive heart failure (428.0) Atrial fibrillation (427.31) Native coronary atherosclerosis (414.01) Essential hypertension (401.9)

Appendix

Table 15: ICD-9 code clusters learned from LightGCN with

G=7

Method	AUROC $\uparrow$	AUPRC $\uparrow$	BCE $\downarrow$	Brier $\downarrow$
HRL-Cat	0.787 (0.0055)	0.160 (0.0067)	0.132 (0.0014)	0.031 (0.0002)
HRL-GM	0.827 (0.0036)	0.194 (0.0093)	0.121 (0.0010)	0.030 (0.0002)
HRL-Score	0.826 (0.0034)	0.195 (0.0069)	0.120 (0.0007)	0.029 (0.0001)
HRL-Fixed	0.769 (0.0075)	0.130 (0.0061)	0.206 (0.0067)	0.041 (0.0006)
NeuMF	0.755 (0.0051)	0.132 (0.0048)	0.133 (0.0002)	0.031 (0.0001)
AllSet	0.758 (0.0062)	0.123 (0.0053)	0.131 (0.0005)	0.031 (0.0001)

Appendix

Table 16: Food Recipe: cold-start prediction at

r_{\rm obs}=0.2

. Values are mean (standard error) across five repetitions.

Cluster	Cluster name	Size	Representative ingredients
1	Seafood	22	oyster, salmon, cod, snapper, halibut, fish, lobster, crab, scallop, bass
2	Richer flavor profile	42	whiskey, kumquat, cottage cheese, pumpkin, tea, sherry, jam or jelly, bourbon, duck, apple
3	Lighter flavor profile	68	nut, nectarine, egg, cantaloupe, honeydew, butter, hazelnut, orange, amaretto, watermelon
4	Asian/Latin ingredients	40	paprika, lime juice, tofu, coriander, sake, plum, sesame, tomatillo, cumin, bok choy
5	Vegetable-legume savory dishes	77	legume, meat, vegetable, chard, pork, chickpea, parsley, olive, butternut squash, herb

Appendix

Table 17: Ingredient clusters learned from HRL-GM with 5 clusters.

Cluster	Cluster name	Size	Representative ingredients
1	Aromatics	39	lime juice, ginger, coriander, seed, chile, lemongrass, mango, lime, paprika, tamarind
2	Nuts	17	walnut, pistachio, pomegranate, date, tofu, chestnut, almond, dried fruit, tree nut, apricot
3	Mixed group	151	lentil, phyllo/puff pastry dough, mustard, vinegar, breadcrumbs, ricotta, herb, pork, buttermilk, bread
4	Seafood	26	fish, salmon, oyster, crab, cod, halibut, shrimp, snapper, squid, bass
5	Vegetables	16	yellow squash, eggplant, broccoli, zucchini, corn, chickpea, plantain, barley, squash, okra

Appendix

Table 18: Ingredient clusters learned from HRL-Fixed with 5 clusters

Cluster	Cluster name	Size	Representative ingredients
1	Mixed fruit, dairy & protein flavors	46	duck, pecan, raisin, pear, honey, orange, brie, swiss cheese, brussels sprout, fig
2	Mixed savory ingredients	91	root vegetable, dairy, celery, green bean, citrus, beef, potato, fish, mushroom, shellfish
3	Rich fruit and alcohol flavors	43	nutmeg, clove, cantaloupe, blackberry, nectarine, melon, blueberry, brandy, sake, asian pear
4	Dairy, grains & savory staples	62	bulgur, pumpkin, seed, lemongrass, horseradish, chard, mint, phyllo/puff pastry dough, sour cream, bok choy
5	Drinks and spirits	7	champagne, sparkling wine, gin, whiskey, anchovy, honeydew, amaretto

Appendix

Table 19: Ingredient clusters learned from EHGNN with 5 clusters.

Method	AUROC $\uparrow$	AUPRC $\uparrow$	BCE $\downarrow$	Brier $\downarrow$
HRL-Cat	0.728 (0.0017)	0.644 (0.0051)	0.601 (0.0005)	0.205 (0.0003)
HRL-GM	0.783 (0.0018)	0.712 (0.0049)	0.545 (0.0012)	0.184 (0.0005)
HRL-Score	0.776 (0.0021)	0.703 (0.0050)	0.552 (0.0012)	0.187 (0.0005)
HRL-Fixed	0.767 (0.0018)	0.682 (0.0055)	0.617 (0.0031)	0.199 (0.0007)
NeuMF	0.754 (0.0034)	0.664 (0.0083)	0.576 (0.0023)	0.196 (0.0011)
AllSet	0.659 (0.0020)	0.572 (0.0054)	0.635 (0.0010)	0.222 (0.0005)

Appendix

Table 20: MovieLens: cold-start prediction at

r_{\rm obs}=0.2

. Each entry reports mean (standard error) over five repetitions.

Figure 10: Performance of Multi-label classification on MovieLens datasets: one embedding and 100 training/testing splits

Dataset	Source and version	License / terms of use	Use in this paper
MIMIC-III	MIMIC-III Clinical Database, version 1.4, hosted on PhysioNet. We cite both the PhysioNet database record and the original Scientific Data publication.	PhysioNet Credentialed Health Data License 1.5.0 and Credentialed Health Data Use Agreement 1.5.0. Access is restricted to credentialed users who complete the required training and sign the data use agreement.	Used to construct an ICD-9 code co-occurrence hypergraph. We only use it for method evaluation, we do not redistribute the raw data.
Epicurious recipes	Kaggle dataset “Epicurious – Recipes with Rating and Nutrition” by Hugo Darwood; downloaded from https://www.kaggle.com/datasets/hugodarwood/epirecipes/data .	The Kaggle page lists the license as unknown. We therefore treat the dataset as a third-party asset with unspecified licensing terms, use it only for academic evaluation, and do not redistribute the raw dataset.	Used to construct a recipe–ingredient hypergraph. We only use it to evaluate the proposed method. We do not re-distribute the raw data.
MovieLens 1M	GroupLens MovieLens 1M Dataset, released 02/2003; downloaded from https://grouplens.org/datasets/movielens/1m/ .	GroupLens usage terms as stated in the MovieLens 1M README. The dataset may be used for research with acknowledgment, but may not be redistributed or used for commercial or revenue-bearing purposes without permission.	Used to construct the movie-preference hypergraph, used for model performance evaluation. We do not redistribute the raw data.

Appendix

Table 21: Sources, versions, and license or usage terms of the real-world datasets.

Method	Retained parameters	Dense work	Principal working arrays
HRL-Cat	$\scriptstyle n(d+1)+K(d+1)-1$	$\scriptstyle O(mnK)$	$\scriptstyle O(mK+Kn+nd)$
HRL-GM	$\scriptstyle n(d+1)+K(d+1)-1$	$\scriptstyle O(mnKd^{2}+(mK+n)d^{3})$	$\scriptstyle O(mnK+mKd^{2}+nd)$
HRL-Score	$\scriptstyle n(d+1)+P_{\boldsymbol{\theta}}$	$\scriptstyle O(mnd^{2}+mB_{\mathrm{MC}}d^{2}+(m+n)d^{3}+C_{\mathrm{score}})$	$\scriptstyle O(mn+mB_{\mathrm{MC}}+nd+(m+B_{\mathrm{MC}})d^{2}+P_{\boldsymbol{\theta}})$
HRL-Fixed	$\scriptstyle n(d+1)+md$	$\scriptstyle O(mnd)$	$\scriptstyle O(nd+md)$

Appendix

Table 22: Retained representation size and conservative dense-operation counts for one fitting iteration or epoch. GM uses a fixed common covariance. Working arrays exclude the common incidence matrix; initialization and final normalization/factorization are excluded from per-iteration work.

	$n=90$		$n=180$		$n=360$
Method	$\mathrm{Err}_{Z}\downarrow$	NMI $\uparrow$	$\mathrm{Err}_{Z}\downarrow$	NMI $\uparrow$	$\mathrm{Err}_{Z}\downarrow$	NMI $\uparrow$
HRL-Cat	0.0808	0.8824	0.0783	0.9482	0.0784	0.9744
HRL-GM	0.0612	0.8984	0.0530	0.9573	0.0518	0.9818
HRL-Score	0.0672	0.8878	0.0689	0.9476	0.0894	0.9694
HRL-Fixed	0.1566	0.8976	0.0810	0.9569	0.0606	0.9819

Appendix

Table 23: Gaussian-mixture design with

m=3000

: representation recovery and computational cost. Accuracy values are means across simulation seeds. Bold and underlined entries mark the best and second-best reported accuracy values within each node count.

	$n=90$		$n=180$		$n=360$
Method	$\mathrm{Err}_{Z}\downarrow$	NMI $\uparrow$	$\mathrm{Err}_{Z}\downarrow$	NMI $\uparrow$	$\mathrm{Err}_{Z}\downarrow$	NMI $\uparrow$
HRL-Cat	0.4124	0.4143	0.4152	0.4330	0.4042	0.4396
HRL-GM	0.3930	0.4141	0.3818	0.4398	0.3759	0.4042
HRL-Score	0.0702	0.8769	0.0737	0.9321	0.0862	0.9486
HRL-Fixed	0.3899	0.3920	0.3813	0.4523	0.3771	0.4722

Appendix

Table 24: Heavy-tailed

t_{4}

-mixture design with

m=3000

Train	Method	Params	AUPRC $\uparrow$	BCE $\downarrow$	Time (s)	RSS (MB)
0.6	HRL-Cat	1,973	0.391 (0.0033)	0.336 (0.0019)	6.7 (1.2)	388.0 (12.1)
	HRL-GM	1,973	0.434 (0.0025)	0.315 (0.0007)	366.1 (13.7)	1278.5 (70.2)
	HRL-Score	5,358	0.429 (0.0035)	0.316 (0.0018)	29.9 (1.6)	530.0 (6.8)
	HRL-Fixed	5,525	0.332 (0.0075)	0.514 (0.0068)	45.0 (1.4)	492.3 (7.3)
0.7	HRL-Cat	1,973	0.396 (0.0041)	0.334 (0.0020)	13.6 (2.7)	402.6 (13.4)
	HRL-GM	1,973	0.437 (0.0038)	0.315 (0.0011)	332.7 (32.3)	1408.2 (62.6)

Appendix

Table 25: MIMIC-III row-disjoint cold-start prediction at

r_{\mathrm{obs}}=0.2

as the global-training portion varies. Values are mean (standard error).

Method	Random-effect structure	Advantage	Disadvantages
HRL-Cat	Discrete hyperlink types	Fast fitting and a compact, interpretable representation	Restrictive with within-type variation
HRL-GM	Mixture Gaussian	Continuous/moderately multimodal heterogeneity; a strong default	High working memory
HRL-Score	Flexible learned distribution $\mathbb{P}_{\mathbf{F}}$	Admits non-Gaussian flexibility	Needs structured initialization and enough particles
HRL-Fixed	Unrestricted effects for training hyperlinks	Dense/transductive rows or no pooling benefit	Parameters grow with ; no prior for new rows

Appendix

Table 26: Practical considerations for choosing an HRL specification, informed by the reported accuracy, cost, and stability results.

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