ReLaSH: Reconstructing Joint Latent Spaces for Efficient Generation of Synthetic Hypergraphs with Hyperlink Attributes
Authors: Feiyan Ma
Shihao Wu
Gongjun Xu
Ji Zhu
Venue: ICLR 2026 Poster (2026)
Keywords: hypergraphs
latent space models
structured data generation
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

Abstract: Hypergraph network data, which capture multi-way interactions among entities, have become increasingly prevalent in the big data era, spanning fields such as social science, medical research, and biology. Generating synthetic hyperlinks with attributes from an observed hypergraph has broad applications in data augmentation, simulation, and advancing the understanding of real-world complex systems. This task, however, poses unique challenges due to special properties of hypergraphs, including discreteness, hyperlink sparsity, and the mixed data types of hyperlinks and their attributes, rendering many existing generative models unsuitable. In this paper, we introduce ReLaSH (REconstructing joint LAtent Spaces for Hypergraphs with attributes), a general generative framework for producing realistic synthetic hypergraph data with hyperlink attributes via training a likelihood-based joint embedding model and reconstructing the joint latent space. Given a hypergraph dataset, ReLaSH first embeds the hyperlinks and their attributes into a joint latent space by training a likelihood-based model, and then reconstructs this joint latent space using a distribution-free generator. The generation task is completed by first sampling embeddings from the distribution-free generator and then decoding them into hyperlinks and attributes through the trained likelihood-based model. Compared with existing generative models, ReLaSH explicitly accounts for the unique structure of hypergraphs and jointly models hyperlinks and their attributes. Moreover, the likelihood-based embedding model provides efficiency and interpretability relative to deep black-box architectures, while the distribution-free generator in the joint latent space ensures flexibility. We theoretically demonstrate consistency and generalizability of ReLaSH. Empirical results on a range of real-world datasets from diverse domains demonstrate the strong performance of ReLaSH, underscoring its broad utility and effectiveness in practical applications.

Published as a conference paper at ICLR 2026
RELASH: RECONSTRUCTING JOINT LATENT SPACES FOR EFFICIENT GENERATION OF SYNTHETIC HYPER- GRAPHS WITH HYPERLINK ATTRIBUTES
Feiyan Ma, Shihao Wu, Gongjun Xu, Ji Zhu
Department of Statisties University of Michigan Ann Arbor, MI 48 18109, USA
[e peiyy veiyam,,, wsh hao, gongjun, ji zhu ( @umi wwu.ch..iiiu edu
ABSTRACT
Hypergraph network data, which capture multi-way interactions among entities, have become increasingly prevalent in the big data era, spanning fields such so- cial science, medical research, and biology. Generating synthetic hyperlinks with attributes from an observed hypergraph has broad applications in data augmenta- tion, simulation, and advancing the understanding of real-world complex systems. This task, however, poses unique challenges due to special properties of hyper- graphs, including discreteness, hyperlink sparsity, and the mixed data types of hyperlinks and their attributes, rendering many existing generative models unsuit- able. In this paper, we introduce ReLaSH (REconstructing joint LAtent Spaces for Hypergraphs with attributes), general generative framework for producing real- istic synthetic hypergraph data with hyperlink attributes via training a likelihood- based joint embedding model and reconstructing the joint latent space. Given a hypergraph dataset, ReLaSH first embeds the hyperlinks and their attributes into a joint latent space by training a likelihood-basee model, and then reconstructs this joint latent space using distribution-free generator. The generation task is completed by first sampling embeddings from the distribution-fres generator and then decoding them into hyperlinks and attributes through the trained likelihood- based model. Compared with existing generative models, ReLaSH explicitly ac- counts for the unique structure of hypergraphs and jointly models hyperlinks and their attributes. Moreover, the fikelihood-base embedding model provides ef- ficiency and interpretability relative to deep black-box architectures, while the distribution-free generator in the joint latent space ensures flexibility. We theo- retically demonstrate the consistency and generalizability of ReLaSH. Empirical results range of real-world datasets from diverse domains demonstrate the strong performance of ReLaSH, underscoring its broad utility and effectiveness practical applications.
INTRODUCTION
Hypergraph data capture multi-way interactions among entities, such as co-occurrence, collabora- tion, and co-functioning, (Benson et al., 2016; Battiston et al., 2020), and have become ubiquitous, spanning areas including biology (Rhodes et al., 2005; Nepusz et al., 2012; Feng et al., 2021), medi- cal research (Johnson et al., 2016; 2023), and the social sciences (Zhu et al., 2019; Ji et al., 2022; Wu et al., 2024). Generating hypergraphs with hyperlink attributes has broad applications in data aug- mentation (Wei et al., 2022; Zhou al., 2025), simulation (Nguyen & Le, 2024), and understanding real-worid complex systems (Torres et al., 2021). For example, Intensive Care Unit (ICU) records can be viewed symptom co-occurrence hypergraph with hyperlink attributes: for each patient profile, the co-occurrence of symptoms and diseases forms hyperlink, while other patient infor- mation constitutes hyperlink attributes. Generating synthetic hyperlinks with attributes from the
"Use footmote for providing further information about author (webpage, altemaatve address) for knowledging funding agencies. Funding acknowledgements go the end of the paper.

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symptom co-occurrence hypergraph corresponds to generating synthetic patient profiles, enabling applications such as sevvacc-preservigg data sharing across medical centers and patient simulations. Fig. I showcases synthetic patient profile produced by ReLaSH. The widespread need across fields to generate realistic hypergraphs with hyperlink attributes calls for a general generative model ar- chitecture for this task.
Personal Information
Name: Jane Doe
Gender:
D
Male
Teemale
Religion: Catholic
Marital Status:
D X Married Divorced D Widowed Separated Life Partner
Single
Ethnicity: X White Black
Hispanic/Latiine
Asian D American Indian/Alasn Native D Other
Lifetime: 86.19 yrs
Hospital Stay Time: 14d 19h
ICU Stay Time: 8d 6h
Representative Major Diseases Other Diseases and Complications Record
Coronary Atherosolerosis
Hypertipidemia: Hyperpotussemmi Pueumoeuccus infection; Atrial fibrillation;
Primary cartiomyopethies Long-term (current) use of anticoomolants
Congestive Heart Failure
Chronic systolic heat fnilure, Abdemiaul aneuryam, ruptured
Chronic Kidney Discase
Embolisa and tluombesis of ilme artery; Chaonic obstructive asilamm,
Intracerebral Hemorrhage
Chronic airway obstruction; Noninfectious gastroenteritis and colitis;
Hemorthuge gastrointestinal traet; Acute kictney failure: Sinootrial node dysfinetion:
Dementia
Hemalome complicating procedure, Persoual history of muliguan neuplasm of breast.
Total Discases: 23
An example of synthetic ICU medical record forms generated from ReLaSH, trained on a symptom co-occurrence hypergraph from Johnson et al. (2016), which includes 3,000 ICU patient profiles and 2,230 distinct disease and symptom codes. The disease combinations in this synthetic record reflect the characteristics of an aged, medically complex ICU patient, where the co-occurrence of symptoms often leads to the development of new syndromes. For example, anti- coagulant use in the setting of atrial fibrillation increases the risk of intracerebral hemorrhage and gastrointestinal bleeding (Lopes et al., 2017; Scridon & Balan, 2023), and they co-occur the record.
Figure 1:
Generative models are trained to learn the distribution of real-world observations and to generate novel yet realistic samples (Kingma & Welling, 2013; Goodfellow al., 2014). Recent research has witnessed many powerful generative architectures, including variational autoencoders (VAEs) (Kingma & Welling, 2013), generative adversarial networks (GANs) (Goodfellow et al., 2014: Ar- jovsky et al., 2017), flow-based models (Dinh et al., 2016; 2014: Kingma & Dhariwal, 2018), score- based and diffusion models "Sohl-Dickstein et al., 2015; Ho et al., 2020; Song et al., 2020), and autoregressive models (Van Den Oord et al., 2016; 2017), with notable successes in tasks includ- ing image generation (Karras al., 2019; Dhariwal & Nichol, 2021; Rombach et al., 2022), audio generation (van den Oord et al., 2016; Kong et al., 2021), and speech synthesis (Shen et al., 2018). However, most of these popular models are designed for continuous data and do not directly apply to discrete structures such as hypergraphs. Another thread of work studies generative modeling for discrete data, including extending GANs for text and audio generation (Yu et al., 2017: Nie et al., 2018) 1018, and diffusion models for categorical variables (Austin et al., 2021; Hoogeboom al., 2021). These approaches, however, often suffer from computational and storage limitations and typically do not account for the special structure of hypergraphs.
On the other hand, existing graph generative models (You et al., 2018; Chen et al., 2023b) primarily focus on generating graphs that capture pairwise relations and do not extend directly to hypergraph generation. Prior work has considered hyperlink representation learning and generation (Jo et al., 2021: Wu al., 2025). but these models do not incorporate hyperlink attributes. Representation learning on pairwise graphs with edge attributes has also been studied (Wang et al., 2024), yet these methods do not apply to generating hypergraphs that capture multi-way way interactions with hyperlink attributes. A thread of recent work has studied attributed hypergraphs from different perspectives. Badalyan et al. (2024) study structure and inference in hypergraphs with node attributes, where the primary goal is to understand the observed hypergraph for tasks such community detection and link prediction, rather than generating synthetic data. Chun et al. (2025) focus on generating synthetic hyperlinks
fixed set of nodes and node attributes, aiming to realistically capture the interplay between structure and attributes in hyperlink generation, but do not extend to generat- ing new hyperlink attributes. Gailhard et al. (2025) propose hierarchical coarsening-expansionn
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framework jointly generate hypergraph topology and node attributes; in generation, their method produces new nodes, node attributes, and hyperlinks, whereas our goal is to generate new hyper- links and hyperlink attributes fixed set of nodes (for example, generating new combinations of symptoms as hyperlinks over a fixed symptom set and corresponding attributes to form new medi- cal records). A recent survey (Stoian et al., 2025) summarizes advances in tabular data generation, which can be applied to our tasks, as these methods handle general mixed-type data, but they do not scale well and also ignore the special structural properties of hypergraphs. Moreover, most existing generative modeling approaches for such tasks lack theoretical guarantees. A principled, theoretically grounded generative modeling framework for hypergraphs with hyperlink attributes is therefore greatly needed.
In this work, we introduce ReLaSH (REconstructing joint LAtent Spaces for Hypergraphs with attributes), generative model architecture for hypergraphs with hyperlink attributes. Given an observed hypergraph with hyperlink attributes, ReLaSH first trains likelihood-basee joint em- bedding model and embeds the observed hypergraph together with hyperlink attributes into a joint low-dimensional latent space, then reconstructs this joint latent space using a distribution-fii free gen- erator. Synthetic hyperlinks with attributes the distribution-fred generator in the joint latent space and then decoding the embeddings via the trained likelihood-based model. Below, we summarize our main contributions:
then generated by first sampling embeddings from
1. Methodologically, we introduce ReLaSH, generative model architecture for generating realistic synthetic hypergraphs with hyperlink attributes. ReLaSH consists of a likelihood- based joint embedding model and distribution-fiee generator the joint latent space. The iikelihood-basee joint embedding model provides efficiency and interpretability relative to deep architectures, while the distribution generator offers ficxibility in the latent space.
2. Theoretically. we show that the KL divergence between the distributions of true samples and generated ones can be decomposed into three parts. The first two parts correspond to errors arising from training the likelihood- based joint embedding model, whereas the third depends on the discrepancy between the distributions of true and generated embeddings in the joint latent space. Notably, this analysis shows that ReLaSH circumvents the curse of ambient dimensionality in high-dimensional hypergraphs by exploiting the special structure of hypergraphs through the likelihood model, so that the overall error rate is dominated by errors from the relatively low- -iwdiimeniooall latent space rather than the original high- dimensional problem.
3. Numerically, we evaluate ReLaSH on three tasks: generating synthetic medical records from the MIMIC-III dataset (Johnson et al., 6); generating new recipes from the ERRN dataset!; and generating reference author lists with top keywords from co-citation dataset (Ji et al., 2022; Ke et al., 2023). In all tasks, ReLaSH efficiently generates realistic hy- perlinks with reasonable hyperlink attributes and achieves superior performance compared with competing methods, demonstrating its broad utility and advantages for this task. Ad- ditional simulation results further demonstrate the effectiveness of ReLaSH.
The remainder of the paper is organized as follows. Section introduces the ReLaSH framework. Section 3 presents the theoretical results. Section 4 reports numerical experiments on three real- world hypergraph generation tasks. Section 5 concludes with discussion and outlines potential di- rections for future research. Additional materials, including additional theoretical results, algorithm details, simulation study results, experimental settings, and proofs, are provided in the appendix.
RECONSTRUCTING THE JOINT LATENT SPACE FOR HYPERGRAPHS
2.1 NOTATION
For positive real numbers and b. we define a V = max(a, b) and a A b = min(a, b). Let || be the Frobenius norm of matrix A, ||Al|| be the spectral norm, and Ai denote its element at the i-th row and j-th column. For two sequences of positive real numbers an and b,. we write a, = (b,,) or on S bn there exist constants N and C such that an S Ch, for all n > N. For random variable sequences Xn, and Yn, we write X, (Y,) if for any > 0, there exists
kaggle www.dataaeeeeeeoooo hwwgoeaat//eeaeeeeeeeaeeeeeeeeee hwwp:////woooooo/////o/////eppee recip/eseeerreeppeeeeeeeeepp
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P Ne. Se.
P 2 21
4 17 17
Reconstructing joint latent space
[[ Emp X.m
H([m), Ems Kmn
P (((()) Eag ,.....
Joint embedding to low-dimensional latent space
Bn, Ccp B,9
Generating hyperlinks and attributes
Figure 2: The general pipeline of ReLaSH.
constant Ce > such that sup,, X 2 (, < For two probability distribution P.Q defined on the same sample space X, we denote the Kl--divegeence from Q to P as (kk((Pl]Q) Exex P(x) log(P(s)/Q(()))
2.2 SETUP AND THE GENERAL RELASH
We denote an observed hypergraph with hyperlink attributes by H(V,, Em, Xmm, where Vn (U1,...,?%, ] is the set of n nodes in the hypergraph, &n = {ei.-.., Cm is the set of observed hy- perlinks, and Xm = {xI,..., . Im] collects the attributes associated with each of the m hyperlinks. For simplicity, let Vn =nn] nnj= (L..... n); each hyperlink is then subset of [m]] indexing the nodes that form it. Given H( E.: ), the goal is to generate synthetic hypergraph H[[[] E & where m denotes the number of generated hyperlinks, En {ēí ... Em is the of generated hyperlinks, and X = (........ .mp is the corresponding set of attributes.
In this section, we introduce the general ReLaSH framework for the hypergraph generation task. Fig. presents the pipeline of ReLaSH. In brief, ReLaSH first jointly embeds the hyperlinks and their attributes into a latent space by training a likelihood-based model; it then reconstructs this joint latent space via distribution-fres generator; and finally generates new hyperlinks with attributes by decoding sampled embeddings from the joint latent space using the trained likelihood-basee model. Below, we describe these three steps in sequence. Sections 2.3 and 2.4 introduce the specific likelihood-based model and the latent-space generator used in this paper, respectively.
Embedding hyperlinks with attributes to the joint latent space. To build the likelihood- based embedding model, partition the dimensions of the joint latent space into three blocks with di- mensions d1, 42, and 63. corresponding respectively to attributes only, attributes and hyperlinks jointly, and hyperlinks only. We associate each node with latent embedding node popularity. The overall rate
E Rks
and a degree parameter O E R to capture heterogeneity of ar i.e., Om,, n >n i= ai serves to control hyperlink sparsity in the hypergraph (Wu et al., 2024): the smaller dem. n, the sparser the hyperlinks. The attributes are associated with a latent loading matrix B E (px(()+kk) and an intercept vector Y E Rº, where p is the number of attributes. We also associate each hyperlink and its attributes with joint latent embedding (1) (() () Rk tkatk drawn from unknown distribution Pu; for conve- nience, write (12) (IT (T and (23) (2(2)T (3)TT).. Let Um (u....., Um denote the joint embeddings of the observed hypergraph M[[n), Em X. and let Zn, = fz1,.... 2n} and a, (a), (Oq,...,a,,)) Then U,, indeed collects realizations from Pv. Consider random hypergraph Hm., = H([m), (EI,... }.). (XI..., Xm)),, ) of which H(m)). gle realization. Let PH..... [M.. a,, denote the distribution of Hm,,. conditional on Um. Zn, On, B, and 7. This defines probability measure over the product space (P((n)) x A)m where P([n)) {e C [n]} is the power set of [n), and A is the attribute space. Given all embeddings and parameters, we consider the factorization
is sin-
Pmm.n Wm 2n,Gn,
P (E E,,,,
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which implies that dependency in node co-occurrence and attributes are characterized by the em- beddings, and different hyperlinks are conditionally independent: similar assumptions are widely adopted in the literature on latent space network models (Ma et al., 2020) and hypergraphs (Ke et al., 2019). Additionally, decomposing the latent space into (k1, k2, kis) dimensions permits a factorization of the joint likelihood, where
(E(()) 2,,,
P (21 ,P,"'
Here, (F(( (1())) 2,,a,, is the hyperlink-generaall model and p (i, B.y is the attribute gen- cration model; dependence between hyperlinks and their attributes is captured via the shared latent embedding 2 Under a specified likelihood, ReLaSH obtains embeddings (Um, 2n, Og, B, 4) via optimizing a joint loss from the likelihood model. Section 2.3 introduces specific likelihood model for this task.
Reconstructing the joint latent space. The joint embeddings Üm constitute estimated sample of latent characteristics of the observed hyperlinks and attributes in the joint embedding space. In the second step, ReLaSH trains distribution-free generator on this estimated sample. Examples of such generators include normalizing flows (Kingma & Dhariwal, 2018), kernel density estimation (Silverman, 2018), and score-based generative models (Song et al., 2020). Section 2.4 specifies the score-based generator and its implementation used in this paper. ReLaSH then produces Um from the generator and separates it by dimension to obtain (7(12) and (7(22))
Generating new hyperlinks with attributes. Finally, given the generated embeddings ū.02 and d(23) ReLaSH decodes them stochastically through the fitted likelihood models to obtain a syn- thetic hypergraph H[[m), Em, ),). with hyperlinks drawn from P [((()), )1 a., and attributes drawn from
Rm 2 ,0,5" respectively.
2.3 A JOINT EMBEDDING APPROACH AND ITS IDENTIFIABILITY CONDITIONS
The embedding approach in the first step of ReLaSH can be flexibly chosen and designed domain- adaptively. In this section, we specify a joint embedding likelihood model implemented this paper.
Specifically, for each e S [n]. let F (((())))))) ,2,,a, (E = e) = Ihe pr(u(23) IIge (1 () with pæ (2(25) af (u(22)) 2; + a.), where a--) = exp(-)/(1 + exp(-)) is the sigmoid function. The distribution I' ( ,B,, specified by x = -y) where e is a p-dimensional vector of independent, mean-zero random errors with sub-Gaussian tails (Vershynin, 2018) 8) chosen to accommodate different attribute types (e.g.. Gaussian for continuous attributes and Bernoulli for binary attributes). Our choice of embedding models is motivated by prior work with theoretical support on hyperlink generation (Wu et al., 2024) and joint attribute modeling in graphs (Zhang et al., 2022; Li et al., 2025), yet our setting differs substantially and therefore requires analysis and justification. These analyses are presented in Section and in Appendix A.
To obtain embeddings from the observed hypergraph H[m)), Em, Xm), propose to optimize joint loss based on the likelihood models. Specifically, let
En = - -llllogh H([n))...))14 Znian 3 3=1
[1 (iEe,,) 0 - logg + exp(0
24 + a, and la = EE 11,,11,1- 7 y Bu, 12. We optimize the joint loss E(U, Z, B, 7) = EH + Me where X > is a weight parameter that balances the contri- butions from each part. To ensure identifiability during the embedding and estimation proce- dure, we need to impose additional structural constraints. Define the node embedding matrix as Z = (z),... TT. E (ka+kk))), and let B- = (Br B2) E +ka) The identifiability of the joint embedding model, i.e., the distribution of TP and Z, ax, B, is defined as follows.
with 8 u (23)T
Definition 1. (ldentifiability of the joint embedding model.) The joint embedding model is identifi- able for any two sets of model parameters (Pu, Z, a, B) and (P'y, Z', a', r, BB).
(23) d a' + (23) and + BU(12) d 7 B'y(12) + (2)) imply (Pu, 7, B) 7', B').
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The following theorem ensures the identifiability of the embedding parameters under set of iden- tifiability conditions.
Theorem 1. Under the following conditions: (CI) Er, [U] 0; (C2) Ep, [U(2a) Z is a diagonal matrix with distinct positive diagonal elements; (C3) Ep ( B is
diagonal matrix with distinct positive diagonal elements; (C4) Ep (U()) = 02g xk2. the joint embedding model is identifiable according to Definition 1.
Such identifiability conditions have been widely considered in the literature; see, e.g., (Wu et al., 2025; Li et al., 2025). In general, if the true (Pu, Z, B, a, 7) does not satisfy the conditions in Theorem 1, we can apply a unique transformation to enforce the constraints while keeping the joint distribution of hyperlinks and attributes unchanged. Therefore, we jointly embed the hypergraph and attributes by minimizing E(U, Z, B,a,7) under identifiability constraints designed based on Theorem 1. Full algorithmic details are provided in Appendix B.1.
2.4 SCORE-BASED JOINT EMBEDDING SPACE RECONSTRUCTION
In this section, we specify the score-based generator in the joint latent embedding space imple- mented in this work. Our construction follows score-based generative modeling through stochastic differential equations (SDEs) (Song et al., 2020): a forward SDE gradually perturbs the data into Gaussian noise, and reverse-time SDE generates samples from noise using score functions learned via denoising score matching (Hyvärinen & Dayan, 2005; Song et al., 2020). A key difference in our setup is that we train score functions using unobserved embeddings rather than observed data.
Specifically, we define a continuous-time forward diffusion process initialized from Um:
(1) where (E.)r) is standard k-dimensional Wiener process with k: k1 l:g hig Let N T/h
dU, U2 dt + d2dW, Uo Lm.:
denote the number of discretization steps with step size h. Let se R x [0, - R be the score network, a multilayer perceptron (MLP) with parameters 6, to approximate the score function. The parameter 0 learned via the denoising score matching objective (Vincent, 201 1), constructed using the learned embeddings (Wu et al., 2025):
((0)
Σ El
Euyy (no so suy, Ih) V om log Pin (uya (uo) |[]]
m
1=0
J.N where {wn)) t==o are diffused samples in the forward process (1) at time lb. V Px is the score (gradient of the log-density of UV), pm (un (n)) is the marginal density of the forward process given uo and A 1/E{I/V uia log Pu. (ws uo)112) are nonnegative weights that balance different time steps. To construct the score- based generator, for each E [0. - -1 and t E [th, (1 + 1)h], hold the score network fixed the grid time and obtain
dU = (UT + 2 2g (UF-1m T - Ih)) dt + V2dW,,
where W, is a standard k-dimensional Wiener process independent of W,. To sample mn embed- dings, we simulate the above reverse-time SDE m times, each initialized at t 0 with U6 (0, Ix). Collecting the terminal states yields the synthetic embeddings Un = (ūI...., .., u..
THEORETICAL RESULTS
Our theoretical analysis in this section considers the following setup. The node embeddings 2, = { 2n ), the degree parameters a, = (a1..... a,,) and (B,7) are treated as fixed parameters. The hyperlink embeddings Um [ui...., Um consist of m ii.d. draws from the hyperlink-embeddee distribution Pv. Conditioned on (4m, Zn, On, 7),, ), the observed hypergraph one realization from P 1.mm.,.44...... a.. Our goal is to understand how close the distribution of synthetic hyperlinks and attributes generated by ReLaSH is their true distribution under this model. Let F(E,,,U) denote the joint distribution of hyperlink E, its associated attribute X, and the cor- responding joint latent embedding U. Let Pi denote the marginal distribution of hyperlink em- bedding Ú sampled from the latent space generator trained on lmm, and let IP (E,X.0) denote the joint

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distribution of U, and the hyperlink and attributes generated from P,((,),,,,,),XX1,1110 B... Note that the estimated embeddings comes from one realization of 1P 2 P6 and F FB.8,0) are indeed defined conditioned on Um, Zn. an. B, and 7.
Ens B,A) in ReLaSH, we Lemma 1. If Zn Ons B, and 7i are available and replace (Um, any have dka. ( (E((EX) (l = (PPu P))
Lemma 1 states that the generative error for high-dimensiooal hyperlinks and attributes be duced to the generative error of low-dimensional hyperlink embeddings, under an ideal scenario where the true latent embeddings are available use. practical scenarios where these embeddings are unobserved and need to be learned from the data, the error from the embedding algorithm needs be considered. Conditioned on Um, let IP (,.X,0) denote random measure for the joint distribu-
tion of a generated hyperlink embedding Ü and its associated hyperlink E and attribute X, using the estimated parameters 2,, B, ô, and 9. More formally, this measure is P/ (F,X, D,.,,...... B..,, but we the simpler notation for clarity. The randomness of this distribution arises from the observed hy- pergraph H[[m], Em Xm) given (14m Zn, B, a, 7)), which further induces randomness in the learned embeddings (Ul, 2,, B, 0, f) and consequently in the distribution of (É, N,,ÜU Ú). Similarly, define F'D P. Ulêm B.,,, the random measure on Ú conditioned on the learned embeddings. The next theorem shows that the generation error can be decomposed into three parts.
Theorem 2. The KL-divergence between the true distribution IP (E.X.U) and the generated distribu- tion P. (B,x.0) admits the following decomposition:
(KI((P || (EE A B.a,.,,a.,))-eee -esttnntttttt A -sstiiatttoonn Atatent reconstructions
where the exact forms of the three components are given in the proof of the theorem (Appendix C).
As their names indicate, the first error term depends on the estimation error of (Z,, B, a, 7). the second on the recovery of the joint latent embedding distribution, and the third on the reconstruction of the joint latent space. Next, we analyze these three terms under the specific embedding approach in Section 2.3 and the score-based generator in Section 2.4. We first analyze A(z.,B,a.)) Follow- ing the discussion in Wu et al. (2024) on how the sparsity parameter am., affects the hypergraph embedding procedure, we introduce the following assumption 0m,
Assumption (Hyperlink sparsity). As m, n, + 0c, exp(@m,,) 2 log(m V nn/(m An).
When x n, this sparsity scaling is consistent with the sufficient order in Proposition 2.2 and the necessary order in Proposition 2.1 of Wu et al. (2024), up to a logarithmic factor. Further, we consider the following assumption on the embedding space.
Assumption 2 (Embedding space). Define 0m,up {(m A A p) exp( The mini- mum eigenvalue of Epg [ is lower bounded by constant, and eigenvalues of Ep, [UU] are distinct with their gaps lower bounded by a constant. For the m realizations of Pu Ums holds that that ||m=
Σ
3/
j=1 Uj ujll
m,mpp and I| "E= U Er, [UU ]||r O(
U (2) j=1
[m 1==
(2) (1)T
|F =(o || E
(3)T |F O
Assumption 2 ensures that the latent embedding space is well-conditioned and identifiable by re- quiring a non-degenerate, spectrally separated covariance structure, while also guaranteeing that the empirical moments of the embeddings concentrate around their population counterparts to enable consistent estimation. Note that exp (xp()m...) < 1 as m, n + oc. This concentration requirement easily satisfied and is weaker than the concentration results available for many popular multivariate distributions (Vershynin, 2018) when m x n = p. which cover a wide range of real data scenario. With Assumptions 1 and 2, we have the following theorem (,,,))
Theorem 3. Suppose that Assumptions I and 2 hold, and X x exp(ām.n)) then as (m, the rate of estimation- related error satisfies
log(m V 1i)
nn V ( B. restimation = Op min m. n,pp
Consequently, when X P. have n - (2,, B,,, B,a,)))))aee......... -esttmattoooo (logn///))) n/n), thus the error in final generative performance from estimating Z,, B, at 7 shrinks as fast as log n. n1
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Theorem 3 implies that if m x cally negligible. We defer the analysis Ag estimation to Appendix due to page constraints, as requires defining discretization theoretical development Chen et al. (2022), noting that our score networks are trained on learned embeddings rather than observed data. Let pe denote the marginal density of the joint embeddings Ü. and let Pi denote the law of U, in the forward process (I). Under the following assumptions (Block et al., 2020; Lee et al., 2022; Chen et al., 2022), we bound the generative error in the joint embedding space.
p, the error introduced by estimating (Z, B, a, is asymptoti-
the support of Fv. To study Atatent Almemt-eccessmmctionn
follow the
Assumption 3. The learned score network sy(u,t) satisfies for any S N, EU~PE, V log Pi,, (U) sj(U, I eg. The distribution of estimated embeddings has bounded second moment, i.e., Mr = E S co. For L [0,
[0, Vlggt, is I.-Lipschitz
Assumption 3 requires that the learned score network approximates the true score function consis- tently in an La sense. The boundedness assumption of its second moment automatically satisfied due to the constraint in the embedding algorithm. Together with the L--ipphchitz condition, these requirements provide the regularity needed to control sampling and approximation errors in the dif- fusion dynamics as follows.
Proposition 1 (Theorem 2 in Chen et al. (2022)). Under Assumption 3, if 1, S and T 2 1. ---1-1 KT L.2 where K ky +2 k3. Then by
We have Dlatemm-recomstiiiii I (My+ KK) TT---+-7 +TE, +N-
choosing T log( (My + K)/ea) and N = ((KTL3/e)) we have Antenttnvoomstaattioon (
The first error term quantifies the distance between Ur and the standard Gaussian distribution in the forward diffusion process, which decays exponentially T. The third term accounts for errors aris- ing from discretizing the SDE. Regarding the score approximation error Eğ, Chen et al. (2023a) stud- ied a specific neural network construction and demonstrated that the upper bound on the sample com- plexity of score estimation is exponential in the score network dimension. This sample-complexity result for score estimation highlights the curse of dimensionality in high-dimensional generative modeling tasks. By jointly embedding the hypergraph and attribute data into a low-dimensional continuous space and reconstructing this joint latent space, we avoid training a high- nigh-dimensional score network or other high-dimensional distribution-fra generators, thereby significantly improv- ing efficiency.
4 EXPERIMENTS
In this section, empirically evaluate ReLaSH for generating hypergraphs with hyperlink attributes on three datasets. We first describe the experimental setup and then report the numerical results. Additional simulation study and implementation details are provided in Appendix B.
4.1 EXPERIMENT SETUP
Datasets. We use three datasets: (i) the co-citation hypergraph data from MADStat (Ji et al., 2022) and MADStaText (Ke et al., 2023), where authors are nodes, co-cited authors in statistical journal paper form hyperlinks, and TF- IDF values (Sparck Jones, 1972) of words with re- spect to the corresponding paper abstracts are hy- perlink attributes; (ii) the recipe hypergraph d where food ingredients are nodes, the set of ingredients in a recipe forms
ARv
Axm 1
Ax. 4
FED
a-FED
RcLaSH-(7,0,2)
3.260
2.989
1.435
0.532
1.084
27.794
2.989
1.435
0.013
0.193
(76,
2.624
3.681
1.655
11.738
2.047
Rel
6,230
3.681
1.655
9.049
1.574
Gau-Diff
4.268
3.497
1.719
39.731
4.387
Re
3,958
33.240
2.526
27.685
2.843
WGAN
3.506
10.534
2.176
21,053
2,654
VAE
48.450
11.499
4.134
9.374
1.376
Table 1: Results for patient profile generation. Scales of An.. Ax...
Ax,, FED, a-FED are 10 40 10 10 10 10-1, respectively.
Phttps:/ htt::://wwwww.. kaggle www//ataseteetec.eooaoa hhugoda rwood/epi recipes/eeepppeeeeeeeeeeee
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hyperlink, and metadata such cuisine type constitute hyperlink attributes; and (iii) the symptom co-occurrence hypergraph from MIMIC-III (Johnson et al., 2016), where medical symptoms are nodes, symptoms co-occurring in patient profile forms hyperlink, and the remaining patient notes are hyperlink attributes. Details of these hypergraph datasets are provided in Appendix B.4.
Baselines. We compare ReLaSH with 9 methods that be used to produce synthetic hyper- links with attributes: Gau-Diff (Song et al., 2020), RealNVP (Dinh et al., 2016), WGAN (Arjovsky al., 2017), VAE (Kingma & Welling, 2013), ForestDiffusion (Doiiceur-Martineau et al., 2024), TabPFGen (Ma et al., 2024), CTAB-GAN (Zhao et al., 2021), CTAB-GAN+ (Zhao et al., 2024), and CTGAN (Xu al., 2019). For the tabular data generation baselines (ForestDiffusion, TabPF- Gen, CTAB-GAN STAB-GANE CTGAN), these methods do not scale to the patient-profile and co-citation generation tasks, as summarized in Table I of the survey (Stoian et al., 2025), and are therefore discarded from those two comparisons. We refer readers to Appendix B.7 for more details and discussion of these methods.
4.2 GENERATING SYNTHETIC HYPERLINKS WITH ATTRIBUTES
Evaluation metrics. The task aims to generate realistic hyperlinks with attributes that preserve properties of the observed hypergraph from different models. Following Wu et al. (2025), we evaluate performance using the RMSE of the hypergraph node covariances (AH.). the attribute means (Ax..). and the attribute covari- ances (Ax;), and FED (Fréchet Embedding Dis- tance), generalization of the FID (Fréchet Inception Distance) used in evaluating visual generation tasks (Heusel et al., 2017), adapted to the hypergraph generation setting.
Ax, +
Ax
Ax.
FED J
a-FED
ReLaSHI-(2,2,2)
1.996
8.578
1.887
1.246
0,931
(22,,)
3,890
8.578
1.887
5,481
1.935
R2L,SH-(2,7,88)
1.914
8.817
1.886
1.060
0.791
ALLaSH.-(2,7.0
2.772
8.817
1.886
0.947
0.706
ReLaSH-(S8, (8,
1.626
8.608
1.887
1.454
0.871
ReL.SH.. -8, (8,8,8, 8)
2.816
8.608
1.887
2.042
Gau-Diff
1.672
10.016
1.824
5.060
1,503
RealNVP
1.668
12.646
1.863
3.948
1.256
WGAN
2.247
8,671
1,885
1.253
0.700
VAE
9.972
9.425
1,889
1.358
0.904
Table 2: Results for the co-citation hypergraph generation task. Scales
of An., Ax,, Ax,, FRD, a-FED are 10-3, 10-2, 10-1,10-1 10-1, respectively.
addition, we report a-FED, further variant of FED that adjusts for the potential bias of FED when training the embedding machine. Details of these metrics are provided in Appendix B. For each metric, lower value indicates better performance.
Results & discussion.
The results are sum- marized in Tables 1, 2, and 3. In these tables. ReLaSH-(Ki, k2, k3) de- notes ReLaSH with joint latent-space dimensions (ki, k2, ka), and R (ki, ka, ka) further denotes that, during generation, a calibration step applied at the end so that the node degree sequence of the generated hypergraph matches that of the ob- For each metric, the best result is highlighted in bold and the second-best is under- lined. Across the three
Ag.
Xm 4
A x, + 0.894
FED
a-FED +
Rel.aSH-(5.0.2)
1.978
2.236
0.293
0.356
ReLaSH..(5.0,2)
7.504
2.236
0.894
0.182
0.248
ReLaSH-(5.0.6)
2.129
2.174
0.820
0.003
0.048
ReLaSH.-(5.0.6)
3,583
2.174
0.820
0.191
0.258
(5),
2,355
1.533
1.112
0.766
0.847
ReLaSH,-(550,10
1.847
1.533
1.112
0.180
0.255
Gau-Diff
2.375
2.154
4.256
0.802
0.828
RealNVP
2.484
1.146
3.562
0.909
0.997
WGAN
2.208
21.428
1.351
0.907
0.928
VAE
21.587
9.883
5.180
11,553
10.285
CTGAN
2.519
28.799
4,983
0,847
0.865
ForestDiffusson
1.886
8.211
2.073
0.848
0.303
TabPFGen
1.565
1,915
1.205
0.297
0.884
CTAB-GAN
2.552
19.367
3.858
0.925
0.947
CTAB-GAN+ 2.488
8.330
3.821
0.898
0.902
served hypergraph.
Table 3: Results for recipe generation.
Scales of H. A
A Ax., 10 respectivelly
FED and a-FED are 10 10-2
2000 10-1,

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Published as a conference paper at ICLR 2026
tasks, ReLaSH exhibits robust and outstanding performance in terms of the quality of the generated hyperlinks and their attributes. Figure (in the Introduction) and Figure 3 show examples of generated samples for these three tasks. The rationale for the generated medical record is described in the Introduction; below we discuss the other two cases.
The left half of Fig. 3 presents generated reference- author list with top keywords by ReLaSH- (2,7,8) Fan, Hastie, Tibshirani, and Zou are among the main contributors to high-dimensional statisties, variable selection, and penalization methods, notably through their contributions to the development of approaches such as the LASSO (Tibshirani, 1996) and SCAD (Fan & Li, 2001). Keywords like "lasso," "penalty." "oracle," " "sparse," and "high-dimension" directly reference the domain of penalized approaches in sparse regression. The cited authors and keywords are highly congruent, reflecting current trends and leading contributors in high-dimensional regression.
The right half of Fig. 3 shows generated recipe named "Mediterranean Fisherman's 's Bean Stew," produced by ReLaSH--(5, 0, 16). By comparing against meals with similar ingredient combina- tions in the training set, we confirm that no identical recipe exists in the source data, meaning that the generated cuisine is genuinely novel rather than a memorized replication. The dish resembles Mediterranean or Iberian-style fish and bean stew, similar to Spanish or Portuguese coastal cui- sine. Its high-protein, low-fat profile is consistent with ingredients like lean fish and legumes, while saffron, fennel, and wine reflect authentic regional flavor. These results highlight ReLaSH pow- erful and efficient method for generating hypergraphs with attributes, even when trained on relatively small datasets, supporting applications such as the creation of new recipes. Additional remarks and extended examples are provided in Appendix B.4.
Pote: Bodiel
Juet Bernn
Janqing Fat
prodiae
Mediterranean Fisherman's Bean Stew
Freedman
Plotr Pryzlowicz
Van
Travor Hontii
Cliffred Hluriih
Johnatone
Cherle: Long
Ranze Li
Ero Mammen
Rating: 4.22/5
Stephen Porinay
Teny Spezd
LASSO
spone
Calories: 228 kccal
Fat: 9.82 g
Robert Tibihins
Chibbling
Hassheag
penlle
Protein: 46.9g
Qiwci Yao
Ming Yean
Sodium: mg
Figure 3: Examples of synthetic reference author list (left) and synthetic recipe (right) from ReLaSH.
CONCLUSION
We introduce ReLaSH, general generative framework for hypergraphs with hyperlink attributes by bridging a likelihood- based joint embedding model with a distribution-fres latent space generator. By embedding hyperlinks and their attributes into a shared latent space and reconstructing that space prior to decoding, ReLaSH explicitly accounts for the discrete nature of hypergraph structure, hyper- link sparsity, and mixed data types, while avoiding heavy training on the original nigh-dimensional data. Our analysis presents the consistency and generalizability for the framework, and experiments across diverse real-world datasets demonstrate its strong empirical performance, highlighting Re- LaSH as a practical tool for the hypergraph generation task.
This work opens several directions for future research. First, extending ReLaSH to dynamic and temporal hypergraphs, weighted hyperlinks, and richer attribute modalities would broaden its ap- plicability while introducing new challenges. Second, conditional generation (e.g., conditioning on subsets of nodes, attributes, or constraints) could enable targeted simulation and counterfactual analysis. Third, tighter theoretical results, such as uncertainty quantification for generated struc- tures, would further strengthen its theoretical guarantees. We view ReLaSH as step toward reliable and flexible generative modeling for hypergraph data.
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APPENDIX
We provide additional theoretical results, experimental details with extended numerical results, and proofs in the appendix. Appendix A presents additional theoretical results. Appendix B collects supplementary materials for experiments, including additional numerical results on simulated and real-world data and details of the evaluation metries and implementations. Proofs of the theoretical results are in Appendix C.
A ADDITIONAL THEORETICAL RESULTS
We start with discussion of the identifiability of the embedding model introduced in Section 2.3. First, we emphasize that the identifiability conditions in Theorem are sufficient but not necessary. For instance, (C3) may be replaced by (C3#) B1 contains a k1 x 1 unit lower-triangular matrix, (C3##), p 1 B B1 Ix., The identifiability conditions in Theorem the literature (Wu et al., 2025). Other identifiability conditions can be adopted in our framework well. Additionally, coordinates in the first hyperlink embedding vector and adjusting the estimators accordingly. In the following, we specify the constraint sets for the embedding and the parameter estimation proce- dure. Let the degree parameter be a = (an,-.. ,0n) T E IR", the hyperlink embedding matrix be
chosen according to
address identifiability up column sign flips by fixing the sign of all
k2 + k3. Furthermore, denote the block submatrices Un as U (U U and U (U U( Under the
Un (w1 (u..... um T =U U U E goxk where
parametrization conditions outlined in Theorem 1, we define the feasible region:
F(O) = {(Um Z, a, 1 = [0", 1, @ = lm + U113) ,0 1m + (((1 BT, || I M1, max max sE(p]
max{Ila am. 1n/l max xE(n]
max [nax||Btxl12 BB+t1 M2,
max ||u,|12 - (M1 M2), and Cm., S am.n -C'Om.n.. JE(m)
(2) where MI, M2 > 0, C" E (0, 1), and the boundary parameter Cm,ns which may diverge slowly as m, 00, accounts for the sparsity of hyperlinks. An ideal choice of Cm.. would satisfy Following Proposition 3.2 in Wu et al. (2024), we suggest setting Cm,, = -C log 1 |e,l/ (m))) for some constant C"' 1.
am,n X -Cm, and exp( x
In the embedding step, attain (Ū, 2, B, á, 4) by solving:
min
E(U, Z, B,n)).
(Un. B.a
In what follows, we present additional theoretical results that characterize the error rates of the embedding and estimation procedure.
Theorem 4 (F-consistent estimation of e" 0",0*), Let 0 1e Ox*] be the true parameters, and é = - 6x] be the estimated version derived by the optimizers (Ü, 2, B, 3, Under Assumption 1 and the condition that (U*,Z",. ZZ' B","",
a", ,²)) E F(O), we have
[V(m V() log(m V n) + 42² Vp) (exp(-Cm..) A A)
| 16-
Remark 1. Theorem 4 implies that if m X n X P. exp(a x exp(-Cmmmpp and with Assump- tion the optimizers achieve F-consistent, ie.. Tm(n+p)
0* == (1),, implies that for any
set um
Um from the distribution Pu, the estimation precise.
Remark 2. We conducted simulation experiments regarding the embedding procedure, detailed Appendix B.2, to validate the theoretical error rate.
Remark 3. The error bound in Theorem 4 depends jointly on the sparsity parameter of the hyper- graph mm and the regularization weight X. Consequently, X should be tuned in accordance with the observed sparsity in order to balance these sources of error. Simulation results presented in Appendix B.2 further indicate that while the taning of does not substantially affect the accuracy of estimation, it may influence the stability of the gradient descent procedure.
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Remark 4. Moreover, when m x x and X exp(am exp(-Cm, the error rate in Theorem dominated by the hypergraph embedding error rate (Wu et al., 2024), and the error from the attribute part is of smaller order.
Corollary 1. (F-consistent estimation of the embedding and parameters) Let (Ū, 2, B, 6, optimizers, and (U*, Z" B², a², ,a,")) be the true embeddings and parameters satisfying the con- straints in the feasible region F(O). Denote
R) be the
(m V n) exp((m,m) log(m V n) + (M
mm,
Vm (ineexp(-Cmm A A)
As (m,n) - 0o and with Assumption 1, 2, we have F-consistent error of the embedding and parameters that l|ā a² 1|2 Op(mm.pp): I1 10 =0p ( p)/(np)mm.mp)) || 1|= = =0((6m.n.)) 189 - 70 = Op(6m.m.p) | 100--l1=00p(mmmmm
To address identifiability. we apply the transformations in Remark 5, whose details deferred to Appendix C, and which introduce an additional source of error in the embedding procedure. The following theorem characterizes this error.
Theorem 5. Let M([[]), Em, Xm)
) be the hypergraph generated from underlying embeddings Zno.
and parameters a, 7, B, where Um m realizations from Pu. Let (Ü, 2, B, â, 9) be the optimiz- ers, then with Assumption 1, and as (m, n) - oo, we have | 0p (m ||Ü-Uml Umll l|
Op (mm(n= + p)/(mp)) Op(0m.mp)
2 Z F (p | 7 ( m. ||B B
Next, we discuss Are via a discretization strategy (Wu et al., 2025). In general, analyzing the joint seeeedding-rllttee error Prp-mttmation challenging, it requires comparing the distribution of the estimated embeddings based on m observations, 24, = {u....., ... Um J. from the continuous latent distribution Pt. To address this, adopt discretization strategy to bridge these quantities and analyze the error term step by step. We first introduce Assumption 4 and analyze Ar, Arg-ssimation under these conditions.
estimation
Assumption 4 (Support of Pu). The support of Pu satisfies (PP)( C {u E IR ||w||x S (M1 A 32)) for the constant M1, M2 in 2.
Let Udds denote discretized version of U, with distribution Pya =dPua//du(Uuss defined as Pru (u
Juu 2rmmiupp aedic
)ospptPP)) Po (u) du(u) for any udis E Ac, = {a e
R ||alloo C, a Tmoo.p Z Vi E [k]}, where C M1 A M2, and Tm,n.p is sequence diverging to Do as m, n. + oo. Since Pn and Pp are defined on different sample spaces, they not directly comparable. To address this, we introduce the random vector UPs defined on the same sample space as U, whose density is piecewise constant.
Without loss of generality, let supp(IPr) = I-C - (27m,n.p) C + (2 -1y;, which forms a hypercube centered at O with side length 2C + (Tmmn..) in each coordinate. This hypercube can be perfectly partitioned into finitely many disjoint intervals of length Ym.n,p per coordinate. Define PUB as follows: for any E supp(Prr), let (w) E AC, satisfy E (u) - (27m. -1, wdil (u) + (2%....)) and define the probability density of UPe dP.,Ie (u) mm.pp ([uth(u) (200))02mmm))))))))))))) -2 w (u)
as PUR (u) = du(Umn
Pu Pu(u) du(u).
Lemma 2. UPe is a well-defieed random variable on (upp(PFv) in the sense that its probability density Pu integrated to 1.
Next, we show that the distance between pur (u) and pu (u))) is uniformly bounded, which quantifies the distance between Pur and Pu.
Theorem 6. Suppose Po (u) is L-Lipschitz continuous on its support. Then, for any u E supp(P there is 1puw (u) pu (u)l L Ymmpp'
In our setting, the hypergraph with m hyperedges is constructed on the m observations Um Let Pus denote the empirical distribution of m realizations 2dds {udis uddis
disss defined by
... "..
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Puga(u) #uu /m, Yu AC..m...." We can then show that the distance between Prias and the empirical distribution Pug. based on realizations bounded in Lemma 3.
Lemma Let Um be collection of m realizations from Pysa. Let Pinax maxa P (u) and Pmin = miny (u). For any sequence of Em,.p satisfying Pmax > Em,mp. > Pmax V log(2(mm,pp)) ( (mpmin) and >> klog Alog(2CTmm. pom.n.p//Punn Pmin m, n.p. 0o, we have P(v' E Acrrming (pu (1) Pug [u)] S Em,np. + as m, - 00.
When Um is replaced by u the corresponding estimators must be adjusted. We project the es- timators {ú,,-.. 1m onto Ac..i.., by defining ui arg M C.mmm..p || üjl: Then d1 ndis im have a probability mass function defined as Púia: (u) = #(u E Um /m. To bet- ter understand the distance between the distributions of Lidh and Um. we require tighter entry-wise consistency results beyond the average error rate of the estimators, which left as future work.
B SUPPLEMENTARY FOR NUMERICAL RESULTS
B. ALGORITHMM
To jointly embed the hypergraph and its corresponding attributes by minimizing the loss function E(U, Z, B, a, 7), we consider using a projected gradient descent algorithm similar to (Ma et al., 2020), which is commonly employed for solving constrained optimization problems. The algorithm summarized in Algorithm below.
Algorithm Projected Gradient Descent for Joint Embedding
Require: Initial embeddings U(o). Z(o). a(o), initial parameters ((0) observed hypergraph connection matrix H and attribute matrix X, learning rate 17, likelihood weight parameter X, maximum number of iterations T
for 1 to T do
2:
Compute O
1m ( + (12) BT, and 0 (11) (m + (23) (1-1)
(1-1)) 2T
0:(1) (1)
(t-1) Nua (u) U1)), + An (X ()
3:
(t)
4:
U(] = -1(21
(t-1) -Tua Vow = U(21)
(t-1) + t (A(X - ( (1-1) -1) + (H
(((--)))a
5:
U(3) = (13)
(t-1) TT V (1-1) = (33
(1-1) + nu (H - o
6:
Z(e) Z(t-1) 122 V ((((111) (1t-1) 1) + m2
o 000 (2,(t-1))
7:
((()) = a((-1) 1a V Tol(e-1 (t-1) -((-1) + n1 (I o (1-1) 1mm
8:
Bi B(--1) 7/ V (( BB + \ [xx (((1)-))) (--)
9:
7()) 7(--1) 1/, V ( - 7((1)) An,, (X (X 1'x (1-1))) 1m
Project the parameters and embeddings (U()). (()), a(t), (()) to the constraint set, with the transformation in Remark 5.
end for
12:
return (U(T), Z(T), O(T), B(T), I(T)) as (Ū, 2, a, B,7)
For the initial values (U)). Z (()), ((o))((0)) and choices of step sizes, we adapt the initializa- tion method based on universal singular value thresholding (Chatterjee, 2015) and step size choice proposed by (Ma et al., 2020). The initialization algorithm shown in Algorithm 2. Also, Moreover, the step sizes Algorithm are set as:
17 Na fy 2m
1 1p
7
12
1Z
B
7B
[1o
To0
|
( (0) lo
for i = 1,2,3.
Moreover, to select the dimensions (k1, k2, ks) of the joint latent space, we propose the "hunt then trim" " (HTT) algorithm. HTT consists of three steps. In the first and second steps, it hunts for the latent space dimensions that generate the hypergraph, and then the dimensions that orthogonal
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Algorithm 2 Initialization by Singular Value Thresholding
Require: Hypergraph matrix H, hyperlink attribute matrix X, latent embedding dimensions (ki, k2; ka).
1: Let E, o,u,y be the singular value decomposition of H, and denote P = Eo, o as the low-rank approximation of H. Flementwisely project 1 into 1-e-MM 1 to obtain P, and
take @ logit(P), P)).
2:
Take ((o)
_² im. and let U'A'V'
as the singular value decomposition of 2 a()) k2 V
ao 1 =
Jm Om then take u!22)
(0) v U 42+ky A 11/2 t Z(0) V m
11/2 t
3: Treat X as noisy version of Ox denoted as OX
4: Take Y()) = m 10*1 and regress e on (0) to obtain B2, Then take the residual
R On (7]) B (0) denote its singular value decomposition ÜÄVT, and denote
B10 = V V Vk, A ki u( V A1/2 to satisfy the identifiability conditions.
5: return (U(o). (o)) a(u), B0o T()))
to those but drive attribute generation. In the third step, it then trims the dimensions that drive hypergraph formation but (almost) the attributes. The algorithm is summarized in Algorithm below.
B.2 ADDITIONAL SIMULATION RESULTS OF THE EMBEDDING PROCEDURE
In this section, we present additional simulation results of the embedding procedure.
Specifically, for a (A2 + k3) -dimensional latent space, we randomly divide the n nodes into (A2 + k3) nearly equal-sized groups, where the group sizes may differ by at most 1. The node embeddings Z,, are then independently generated as follows. For node i belonging to the 1-th group, its embedding 2, drawn from the truncated Gaussian distribution N[-1.1j kytkk ct, Ez), where e is the unit vector with the t-th element being and the others and [-1, 1] indicates truncation on each coordinate.
The hyperlink embeddings Um are generated from Gaussian distribution (1, Ev). ensuring that the identifiability condition ΣØ= U Og is approximately preserved. For the regression parameters (), )) we generate 7/s i..d. from Uniform ([-, 1l). and each column B; independently sampled from (O, ΣB). The variance-covarianc matrices Σg. Σg. Ev defined such that the (i,j)-th entry is 0. 0.2p 1-5l with p {0,0.3, 0.5). The degree heterogeneity parameters os are generated from the uniform distribution [ām.. - 1, m, + 1] for the sparsity parameter ām,, specified each experimental setting.
We first examine how the sample size and latent dimension jointly affect estimation accuracy. We set kn = ha ky with k E 9,,9, 12), a=,,, -3, and {100, 200,... .... 1000}. The weight parameter À is chosen such that À 0.2, matching the order of cxp(a"".n) Figure 4, shows the relative Frobenius error of # e e as a function of for dif- ferent latent dimensions A: The error decreases approximately at the rate (mAn/Ap agreement with the bound in Theorem 4. Increasing leads to higher estimation errors: intuitively, larger latent dimension increases model complexity, which amplifies variance in estimation. This dependence on i does not explicitly appear in the bound of Theorem 4 (which assumes fixed k), but follows from its proof in Appendix C. When the coordinates of the embeddings are more correlated (when we set p = 0.5), the error remains of similar magnitude, with mild increase as k increases.
10n 10p with
-1/2 in
To further investigate the scaling behavior, Im A A p) in Figures and 7. In both cases, the relationship appears approximately linear, which aligns with the n -1/2 rate predicted by Theorem 4. As shown in Table 4, the observed linear trend, with a slope of approximately -0.5, consistent with the theoretical bounds, which suggest that the estimation enrors for O θ and θ are of order (n122).
plot the logarithm of the estimation error against
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Algorithm Humt-Then- Trim: Selection of Joint Latent Space Dimensions (k1, k2, K3).
Require: Hypergraph matrix H. hyperlink attribute matrix X.
1: Step I: Selection of l/2 + lt3 via Cross Validation
2: for each candidate dimension h23 22 K3 do
3: Split population hypergraph into training (80%) and testing (20%) sets.
4: Perform embedding procedure training set to obtain 2 Uran
5: Perform embedding procedure on testing set to obtain U
6: Compute Fréchet Distance: FDk2s
FD U8an
7: end for
8: Select hi2 + ht = arg mink FDizs
[je(m] per embedding procedure with selected k23- 10: Step 2: Regression and Sequential Hypothesis Testing for h1
9: Obtain embedded vectors {ūz (23)
11: Let X = (EI....,Tm)) (7(23)
(3) =.23))T.
12: Regress X over (28)):
13:
X -=1,,² + (2(3) BT +
14: Obtain fitted values X.U.P. B and residuals = X.
15: for 1= to kmax do
16:
Test Ho hn = 1 H1 + S k1 S kimax-
17:
Compute r(1) = ( Øk (0kk(xkmxx+2)) where oi is the 1th largest eigenvalue of Cov(E).
If the observed r()) exceeds the threshold from the joint Tracy-Widom Widom distribution, reject Ho. and continue testing Hn kı = +++1 1.
Else accept Ho and set the estimated k I.
20: end for
21: Step 3: Simultaneous Estimation of l2 and K3
22: for I l22 + k3 do
23:
Let B.,, denote the 1-th column of the loading matrix B.
24: Test Ho B, Opxi versus H, B,, 1 + Opx1-
25:
Compute s(l)
2 (2
1/2
Ej= (mB3, Vyl). where Vir
( (1
26:
Compare s(l) to the standard normal distribution.
27:
If s(I) significant, infer E. + 0 (belonging to ((3)), else ((2).
28:
end for
29: Simultancously estimate (2)), (3)) k2g K3 according to the hypothesis testing results.
return Dimensions (ki, k2, ka) of the joint latent space.
Estimation enror of Thela'1 versus
Estimation eneer Thela "X versus n
Estimation enor of Thela versus
1.
imipen)
9.10
330-
0.-
Figure 4: Estimation error under p = versus sample size based on 30 Monte Carlo repetitions.
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emor Theta versus
emror versus n.
versus
16-
1.2
2.75
0.00
250
333
750
1000
222
300
150
1000
255
332
730
3250
Figure 5: Estimation error under p = 0.5 versus sample size based on Monte Carlo repetitions.
log-log error for e
log-log error plat for e*
log-log plot for
116" -01
5m
100
0.5
-13-
-30
45
1.5
6.5
70
45
55
6.0
6.5
7.0
53
6.3
ko(mmimm
lco(minim, pt)
Figure 6: Log-log plot of estimation sample size for p 0, based on 30 independent Monte Carlo repetitions.
log-tog error for e
log-log plot for e
log-log emor plot for
-0
0.5-
100 16-01
-30
45
2.5
70
45
6.0
20
222
lofminim, n, p))
loq(minim, n. pt)
logfmintm.co n.p
Figure 7: Log-log plot of estimation error versus sample size for 0.5, based on 30 independent Monte Carlo repetitions.
Error Type
k
p=
p= 0.5
Slope
95% CI of Slope
Slope
95% of Slope
-0,4902
0.49551, -0.8854
-0.4697
-0.4744 -0.4650
H
-0.5158
0.5365,
-0.4951
-0,5028
0.5262,
-0.4795
-0.5470 12
0.5852,
-0.0087
-0.5546
0.5948.
-0.5145
-0.5117
0.5150,
-0.5085
-0,5212
0.5270,
-0.5153
x
-0.5125
0.5163,
-0.5087
-0.5300
0.5387.
-0.5213
12
-0.5128
0.5165,
-0.5990
-0.5321
0.5418.
-0.5223
-0.4907
0.4955.
-0.4860
-0.4711
0.4758,
0.4663
θ
-0.5157
0.5360,
-0.4954
-0,5035
0.5265.
-0.4806
12
-0,5462
-0.5877,
-0.5088
-0,5541
0.5935,
-0.5147
Table 4: Slope and confidence intervals for p 0 (left) and == 0.5 (right).
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From the perspective of the error bound derived in Theorem 4, regress the estimation error of Of on log (mAn) using log-log plot, as illustrated in Figure noteworthy that the regression based on this term performs better than directly regressing on (m A n), with the slope of nealy 0.5.
(mvv)
k
0.5-
k
6 -Oill
0.0
12
6 -Oillr
0.0
m/m
m/m
log
-0.5
-0.5-
-1.0-
-10-
-4.5
-4.0
-3.0
-4.5
-.00
-3.0
Jogmax(m, nl)
logmax(m, nl) log min(m,
log
min(m,
Figure 8: Log-log plot of estimation enror of O versus (right), based on 30 independent Monte Carlo repetitions.
log (mVv) for 0 (left) and 0.5
(mAn)
k:
Slope (p 0)
95% (p = 0)
Slope (p 0.5)
95% CI (p 0.5)
6
0.5596
0.5522, 0.5669
0.5361
0.5285, 0.5438
9
0.5886
0.5621, 0.6150
0.5738
0.5443, 0.6032
12
0.6240
0.5773, 0.6707
0.6327
0.5838, 0.6817
Table 5: Slope and confidence intervals for p 0 (left) and p 0.5 (right).
To be more precise, analysis focuses two aspects: (i) the effect of varying p on the estimation error of ox, and (ii) the effect varying n on the estimation error of " Preliminary experiments suggest that the correlation parameter p in hyperlink embeddings has negligible effect the results. Therefore, we fix 0.5 for all subsequent experiments, as this correlation introduced is commonly encountered in various scenarios. Additionally, scale k k: {6, 12, 24) to demonstrate the robustness of our algorithm in handling more complex data.
0.2, p 0.5, am.x -3, and vary p {100, 200, ... 1000) estimate the errors of @ and O (Figure 9). The results confirm our theoretical expectation: the estimation error of @ remains generally unchanged as varies, because the associated error term does not depend on p and the attribute dimension has minimal impact on hypergraph structure estimation. In contrast, the estimation еттоr of ( decreases with increasing in an approximately power-law pattern. This is intuttve-hhigher-eee diigeer-iimeniionll attribute information facilitates more ac- curate recovery of the underlying saribute-basedd structure.
We fix 5000, n 500,
m6
==12
2 =4
Eree
Catmutoon Emor
2.-
0.56
Figure 9: Estimation error attribute dimension p, averaged over 30 Monte Carlo repetitions.
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To further examine this scaling relationship, we plot the estimation error of @ against p on a log- log scale (Figure 10). The observed linear trend confirms the power-law behavior. The figure also shows the 95% confidence interval for the slope and for the expected value of the error.
6
k 1212
k-24
Sispe
-1.0
ton CJ (-2.000 -3.422)
+1.0
91% +1-0 -24:71
Skopo 0.42
42-
-2.23
45
1.
2.6
14
25
50
55
43
19
10
55
60
Figure 10: Log-log plot of estimation error versus p, based on 30 independent Monte Carlo repeti- tions.
Similarly, we exanmine the estimation error of both O and O-X as varies in the set {100, 200,.... 10003. while fixing m = 5000, p = 500, Omm, = -3, p = 0.5, and À = 0.2. As shown in Figure 11, the error associated with the attribute- part, O remains relatively un- changed the node size increases. In contrast, the error associated with hypergraph estimation, OH exhibits a clear power-law trend.
k =6
k=12
2
21-
Eneraye
Estmukentrro
Catmeadon ErTor
-2-
180
Figure 11: Estimation error versus node size n, averaged over 30 Monte Carlo repetitions.
To gain further insight, we plot the estimation error on a log-log scale, as shown in Figure 12. This plot illustrates that the error associated with hypergraph modeling (0d) follows a pawer-law trend with respect to n. The figure also shows the 95% confidence interval for the slope and for the expected value of the error.
k6
k-12
224
-0
04-
4-
Skepe --
11 Is
15% Ci
25% CI -1 -1493
<3-
10g.n
Igg
10g
Figure 12: Log-log plot of estimation error versus ni, based on 30 independent Monte Carlo repeti- tions.
We next explore the influence of the sparsity parameter ām.n under varying latent dimensions. We 10n 5000, X 0.2, and -ōm, {0.5, 3,.5,,,...33..444 4). To focus
set k E (6, 9, 12), on sparse hypergraph settings, we narrow the possible interval of degree parameters by setting ag i.i.d. ~ Uniform
0.5, ām,, + 0.5] for any i E [x). Figure 13 shows that the estima- tion errors of H and θ increase approximately proportionally exp(- which aligns with
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the findings of (Wu et al., 2024), while the error of @* remains stable. This is reasonable because changes in the hypergraph sparsity will not affect the attribute component @ The slower growth rate for e compared to è also expected, because of the stabilizing contribution from O
Erartype
Er type
Emeraee
Exstraaenn Emer
Enteration
<Bon
Figure 13: Estimation error under p versus sparsity parameter Omm, based on Monte Carlo repetitions.
To further confirm the exponential scaling, we plot log-error of O O versus -ūm,n in Figure 14. The fitted slopes confirm the exponential dependence predicted by Theorem 4.
k-12
Entrappe
55% cl (0220,0.303
Skgpe
Enorype
95% CI C1-20 6.270)
Slepe 0.25
Eroo typ
60% (8217, (22217 8288)
Slape -oee--b233
-35-
Slape
Figure 14: Log-log plot of estimation error versus sparsity parameter Öm,m for 0, based on 30 independent Monte Carlo repetitions.
Finally, we investigate the influence of the weight parameter À on estimation performance. We fix k E [6, 9, 12). n 500, 0m., 22 and vary X {0.05, 0.1 1.25). To better visualize potential differences, we consider two sample sizes, m E (500, 20003. The results are presented in Figures 15 and 16. Across both settings, the estimation errors remain largely unchanged as À varies, suggesting that the choice of À has minimal impact on accuracy.
From practical standpoint, include À primarily Specifically, a smaller allows the algorithm to converge with larger step size 1), leading to faster exccution without degrading estimation accuracy.
mubbility-enhancin tuning parameter.
k2
3.4
33-
25
Figure 15: Estimation error versus varying values for 500, averaged over 30 Monte Carlo repetitions.
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K-
1-
65
Figure 16: Estimation error versus varying À values for 2000, averaged over 30 Monte Carlo
repetitions.
B.3 ADDITIONAL SIMULATION RESULTS OF THE SYNTHETIC DATA ANALYSIS
B.3.1 SIMULATION RESULTS FOR GENERATION
We conduct simulation studies on synthetic datasets to assess the performance of the whole pipeline of the proposed method ReLaSH. To demonstrate the ability of our algorithm to recover the underly- ing structure of given hypergraph and its associated attributes, models, i.c. Gaussian Diffusion Models (Gau-Difh Generative Adversarial Networks (GANs), Re- aINVP, and Variational Autoencoders (VAE). Regarding these methods, we treat each hyperedge as a binary vector and concatenate it with the attribute vector. To ensure fairness in the comparison, we calibrate these algorithms align with the 0-1 valued hyperlinks. Details of these methods are provided in the Appendix B.7.
compare it with other generative
We assess the generative performance of different methods using the Root Mean Squared Error (RMSE) of the means and variance-covariance of node co-occurrence and attributes vector. Addi- tionally, we defined task-specific error metric called the Fréchet Embedding Distance to evaluate the performance of each method. Details of error metrics can be found in Appendix B.6, B.5. The settings simulation experiments and results of error metrics are shown in details in Appendix B.3.
Regarding the settings of synthetic data analysis, specifically, for (A2 + Aa -diimensiooa latent space, we randomly divide the n nodes into (k2 + kg) nearly equal-sized groups, where the group sizes may differ by most 1. The node embeddings 2,, are then independently generated as follows. For node i belonging to the 1-th group, its embedding distribution Mi-1,1 (1 kg et, E.g), where e, is the unit vector with the I-th element being 1 and the others 0, and [-1, 1] indicates truncation on each coordinate.
is drawn from the truncated Gaussian
The hyperlink embeddings Un are generated from a Gaussian distribution (1, Ev), ensuring that the identifiability condition Σy= Uy Og is approximately preserved. For the regression pa- rameters (B, 7). we generate i.i.d. from Uniform([-1, Il), and each column B; independently sampled from (0, EB). The pariance-covariance matrices ER, Σz, Eg are defined such that their diagonal entry is 0.2, with other entries being all 0. The degree heterogeneity parameters a, are gen- erated from the uniform distribution [ām.n 1, ām.n + 1] for the sparsity parameter ōn,u specified in each experimental setting. Once all parameters and embeddings are generated, we can construct the hyperlinks according to the hyperlink generation model. The hyperlink related attributes are
generated X5t N ( (12)) B B),, ) where the generation process is independent for each [m]] and / E [p]. That is, Gaussian noise is introduced during the generation, which is commonly observed in real data structures.
We assess the generative performance of different methods using the Root Mean Squared Error (RMSE) of the means and variance-covariincos of node co-occurrence and attributes vector. Addi- tionally, defined task-specific error metric called the Fréchet Embedding Distance to evaluate the performance of each method. Details error metrics can be found in Appendix B.6, B.5. Each time generate m. 32m new hyperlinks together with attributes by each generative model.
We take the settings of E (6, 12, 24, 48) and m, n E {200, 400, 800, 1600), and limit the eval- uation to ReLaSH (and the calibrated version ReLaSH.), Gau-Diff, RealNVP, and WGAN due to
25

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Published as a conference paper at ICLR 2026
the poor performance of VAE and constraints on computational resources and time. The runtime required by the VAE is approximately four times longer than that of other competing methods. In the real data analysis section, we will demonstrate its relatively poor performance, particularly in terms of both the first and second-order moment errors for the hypergraph and attributes estimation. The RMSE results are provided in Table 6, 7, 8, 9 respectively, and the FED results are provided in Table 10, 11. The FED for each method is calculated using Algorithm 4, where the same maximum iteration steps are imposed for all methods, and identical early stopping conditions are applied.
Generally, we observe that as m increases, the error metrics generally decrease due to the increasing data size, which allows for better utilization of available information, resulting in improved perfor- mance.
Embedding Dimension k 6
(A1
kg
R3
2)
W1
Method
Objects
Tt
200
400
p
800
=
p
1600
Mean
Cov
Mean
Cov
Mcan
Cov
Mean
Cov
ReLaSH
Hypergraph
0.0242
0.0069
0.0242
0.0053
0.0243
0.0042
0.0242
0.0041
Attributes
0.0740
0.0428
0.0703
0.0409
0.0772
0.0351
0.0732
0.0351
ReLaSH,
Hypergraph
0.0243
0.0374
0.0240
0.0385
0.0241
0.0378
0.0239
0.0405
Attributes
0.0740
0.0428
0.0703
0.0409
0.0772
0.0351
0.0732
0.0351
Gau Diff
Hypergraph
0.0243
0.0054
0.0240
0.0053
0.024I
0.0053
0.0239
0.0052
200
Attributes
0.0873
0.0683
0,1071
0.0567
0.1025
0.0564
0.0965
0.0548
RealNVP
Hypergraph
0.0243
0.0054
0.0240
0.0053
0.0241
0.0052
0.0239
0.0052
Attributes
0.0731
0.0575
0.0707
0.0545
0.0769
0.0563
0.0734
0.0549
WGAN
Hypergraph
0.0243
0.0143
0.0240
0.0108
0.0241
0.0165
0.0239
0.0221
Attributes
0,3485
0.1697
0.1425
0. 0.15
0.0892
0.0919
0.0813
0.0864
ReLaSH
Hypergraph
0.0178
0.0047
0,0201
0.0039
0.0171
0.0032
0.0180
0.0030
Attributes
0.0472
0.0292
0.0511
0.0265
0.0535
0.0241
0.0519
0.0246
ReLaSH,
Hypergraph
0.0176
0.0414
0.0200
0.0436
0.0169
0.0432
0.0178
0.0434
Attributes
0.0472
0.0292
0.0511
0.0265
0.0535
0.0241
0.0519
0.0246
Gau Diff
Hypergraph
0.0176
0.0050
0.0200
0.0051
0.0169
0.0051
0.0178
0.0051
400
Attributes
0.0712
0.0676
0.0911
0.0559
0.0929
0.0547
0.0878
0.0541
RealNVP
Hypergraph
0.0176
0.0051
0.0200
0.0051
0.0169
0.0051
0.0178
0.0051
Attributes
0.0476
0.0547
0.0512
0.0526
0.0539
0.0542
0.0520
0.0540
WGAN
Hypergraph
0.0176
0.0131
0.0200
0,0133
0.0169
0.0138
0.0178
0.0175
Attributes
0.2332
0.1714
0.0790
0,1279
0.0634
0.0639
0.0587
0.0588
ReLaSH
Hypergraph
0.0138
0.0037
0.0130
0.0028
0.0125
0.0023
0.0127
0.0021
Attributes
0.0359
0.0238
0.0411
0.0207
0.0363
0.0178
0.0381
0.0184
ReLaSH,
Hypergraph
0.0137
0.0427
0.0129
0.0441
0.0122
0.0461
0.0125
0.0470
Attributes
0.0359
0.0238
0.0411
0.0207
0.0363
0.0178
0.0381
0.0184
Gau Diff
Hypergraph
0.0137
0.0047
0.0129
0.0050
0.0122
0.0051
0.0125
0.0051
800
Attributes
0.0544
0.0788
0.0944
0.0579
0.0960
0.0529
0.0896
0.0556
RealNVP
Hypergraph
0.0137
0.0050
0.0129
0.0050
0.0122
0.0051
0.0125
0.0051
Attributes
0.0352
0.0648
0.0414
0.0545
0.0367
0.0521
0.0384
0.0555
WGAN
Hypergraph
0.0137
0.0085
0.0129
0.0088
0.0122
0.0111
0.0125
0.0195
Attrbuutes
0.0789
0.1812
0.0614
0.0958
0.0537
0.0506
0.0491
0.0406
ReLaSH
Hypergraph
0.0089
0.0033
0.0091
0.0022
0.0091
0.0017
0.0088
0.0016
Attributes
0.0278
0.0185
0.0286
0.0160
0.0262
0.0140
0.0274
0.0138
ReLaSH,
Hypergraph
0.0084
0.0439
0,0088
0.0449
0.0090
0.0458
0.0087
0,0469
Auributes
0.0278
0.0185
0.0286
0.0160
0.0262
0.0140
0.0274
0.0138
Gau Diff
Hypergraph
0.0084
0.0046
0.0088
0.0049
0.0090
0.0050
0.0087
0.0050
1600
Attributes
0.0516
0.0741
0.0879
0.0581
0.0820
0.0552
0.0831
0.0547
RealNVP
Hypergraph
0.0084
0.0048
0.0088
0.0049
0.0090
0.0050
0.0087
0,0050
Attributes
0.0277
0.0578
0.0287
0.0537
0.0263
0.0543
0.0275
0.0545
WGAN
Hypergraph 0.0084
0.0084
0.0088
0.0095
0.0090
0.0136
0.0087
0.0205
Attributes
0.0587
0.1883
0.0437
0.0590
0.0458
0.0452
0.0426
0.0358
Table 6: RMSE results for means (columns with "Mean") & covariances (columns with "Cov") of ReLaSH, ReLaSH,, Gau-Diff, RealNVP and WGAN when latent dimension is A = 6. Each value comes from the mean of 20 repetitions.
We use bold to highlight the best result in each experimental setting and underline the second-best result. For the RMSE of the hyperlink vector mean, note that all calibrated methods yield the same result, we do not emphasize these results. It is important note that the RMSE for the calibrated
26

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Published as a conference paper at ICLR 2026
Embedding Dimension & (kg Kg
= 4)
Method
Objects
Tt =
200
n
= 400
= p
800
T
1600
Mcan
Cov
Mean
Cov
Mcan
Cov
Mean
Cov
ReLaSH
Hypergraph
0.0351
0.0137
0.0252
0.0081
0.0298
0.0097
0.0302
0.0099
Attributes
0.0782
0.0583
0.0748
0.0510
0.0782
0.0502
0.0736
0.0461
ReLaSH
Hypergraph
0.0310
0.0367
0.0251
0,0306
0.0263
0.0336
0.0285
0.0280
Attributes
0.0782
0.0583
0.0748
0.0510
0.0782
0.0502
0.0736
0.0461
Gau Diff
Hypergraph
0.0310
0.0130
0.0251
0.0130
0.0263
0.0132
0.0285
0.0130
200
Attributes
0.0907
0.0852
0.1151
0.0800
0.1074
0.0753
0.0970
0.0797
RealNVP
Hypergraph
0.0310
0.0133
0.0251
0,0130
0.0263
0.0132
0.0285
0.0130
Auributes
0.0797
0.0765
0.0772
0.0786
0.0780
0.0754
0.0734
0.0799
WGAN
Hypergraph
0.0310
0.0212
0.0251
0.0126
0.0263
0.0158
0.0285
0.0188
Attributes
0.3304
0.2025
0.2052
1018
0.1106
0.0942
0.0816
0.0719
ReLaSH
Hypergraph
0.0190
0.0081
0.0175
0.0057
0.0176
0.0047
0.0179
0.0044
Attributes
0.0534
0.0427
0,0514
0.0339
0.0553
0.0339
0.0554
0.0335
ReLaSH,
Hypergraph
0.0184
0.0357
0.0182
0.0356
0.0179
0.0397
0.0178
0.0389
Attributes
0.0534
0.0427
0.0514
0.0339
0.0553
0.0339
0.0554
0.0335
Gau Diff
Hypergraph
0.0184
0.0128
0.0182
0.0129
0.0179
0.0129
0.0178
0.0129
400
Attributes
0.0792
0.0863
0.0960
0.0741
0.1055
0.0783
0.0905
0.0803
ReaINVP
Hypergraph
0.0184
0.0131
0.0182
0.0130
0.0179
0.0129
0.0178
0.0129
Attributes
0.0555
0.0780
0.0525
0.0720
0.0557
0.0782
0.0554
0.0804
WGAN
Hypergraph
0.0184
0.0152
0.0182
0.0131
0.0179
0.0184
0.0178
0.0296
Attributes
0.1838
0.1588
0.0822
0.1146
0.0640
0.0834
0.0610
0.0628
ReLaSH
Hypergraph
0.0121
0.0053
0,0123
0.0039
0.0126
0,0035
0.0129
0,0030
Attributes
0.0402
0.0301
0.0380
0.0262
0.0366
0.0244
0.0402
0.0236
ReLaSH,
Hypergraph
0.0126
0.0364
0.0122
0.0408
0.0126
0.0449
0.0128
0.0471
Auributes
0.0402
0.0301
0.0380
0.0262
0.0366
0.0244
0.0402
0.0236
Gau Diff
Hypergraph
0.0126
0.0125
0.0122
0,0127
0.0126
0.0129
0.0128
0.0131
Attributes
0.0618
0.0862
0.0950
0.0785
0.0994
0.0758
0.0865
0.0767
RealNVP
Hypergraph
0.0126
0.0129
0.0122
0.0128
0.0126
0.0129
0.0128
0.0131
Attributes
0.0394
0.0731
0.0385
0.0760
0.0368
0.0755
0.0399
0.0766
WGAN
Hypergraph
0.0126
0.0103
0.0122
0,0118
0.0126
0.0171
0.0128
0.0385
Attributes
0.0779
0.1672
0.0505
0.0615
0.0502
0.0519
0.0486
0.0469
ReLaSH
Hypergraph
0.0098
0.0046
0.0099
0.0029
0.0T02
0.0026
0.0T05
0.0020
Attributes
0.0284
0.0221
0.0259
0.0192
0.0247
0.0172
0.0236
0.0168
RcLaSH,
Hypergraph
0.0095
0.0405
0.0096
0,0419
0.0099
0.0401
0.0101
0.0399
Attrbbutes
0.0284
0.0221
0.0259
0.0192
0.0247
0.0172
0.0236
0.0168
Gau Diff
Hypergraph
0.0095
0.0123
0.0096
0.0128
0.0099
0.0130
0.0101
0.0872
1600
Attrbbutes
0.0547
0.0877
0.0850
0.0779
0.0836
0.0789
0.0128
0.0794
RealNVP
Hypergraph 0.0095
0.0128
0.0096
0.0129
0.0099
0.0130
0.0101
0.0128
Attributes
0.0286
0.0735
0.0262
0.0753
0.0277
0.0785
0.0298
0.0793
WGAN
Hypergraph 0.0095
0.0082
0.0096
0.0089
0.0099
0.0095
0.0101
0.0098
Attributes
0.0570
0.0885
0.0450
0.0620
0.0467
0.0597
0.0402
0.0610
Table 7: RMSE results for means (columns with "Mean") & covariances (columns with "Cov") of
ReLaSH, ReLaSH,, Gau-Diff, RealNVP and WGAN when latent dimension is k = 12. Each value comes from the mean of 20 repetitions.
27

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Published as a conference paper at ICLR 2026
Embedding Dimension &
(k)
Kg
8)
Method
Objects
Tt =
200
n ==
400
800
T
1600
Mcan
Cov
Mean
Cov
Mcan
Cov
Mean
Cov
ReLaSH
Hypergraph
0.0329
0.0140
0.0389
0,0133
0.0345
0.0149
0.0371
0.0128
Attributes
0.0808
0.0709
0.0817
0.0703
0.0790
0.0653
0.0834
0.0685
ReLaSH
Hypergraph
0.0245
0.0179
0.0264
0.0191
0.0319
0.0187
0.0246
0.0240
Attributes
0.0808
0.0709
0.0817
0.0703
0.0790
0.0653
0.0834
0.0685
Gau Diff
Hypergraph
0.0245
0.0284
0.0264
0.0289
0.0319
0.0296
0.0246
0.0296
200
Attributes
0.0971
0.1089
0.1198
0.1104
0.1071
0.1131
0.0978
0.1117
RealNVP
Hypergraph
0.0245
0.0288
0.0264
0.0289
0.0319
0.0296
0.0246
0.0296
Auributes
0.0818
0.1012
0.0815
0.1102
0.0789
0.1134
0.0839
0.1119
WGAN
Hypergraph
0.0245
0.0257
0.0264
0.0212
0.0319
0,0257
0.0246
0.0309
Attributes
0.2503
0.2018
0.1372
0.1167
0.2365
0.1291
0.3456
0.1334
ReLaSH
Hypergraph
0.0185
0.0093
0.0180
0.0077
0.0177
0.0071
0.0208
0.0063
Attributes
0.0563
0.0541
0,0554
0.0492
0.0584
0.0470
0.0580
0.0481
ReLaSH,
Hypergraph
0.0186
0.0204
0.0195
0.0271
0.0182
0,0315
0.0191
0.0402
Attributes
0.0563
0.0541
0.0554
0.0492
0.0584
0.0470
0.0580
0.0481
Gau Diff
Hypergraph
0.0186
0.0285
0.0195
0.0287
0.0182
0.0295
0.0191
0.0288
400
Attributes
0.0799
0.1146
0.0974
O. 1080
0.0990
0.1083
0.0789
0. 1092
RealNVP
Hypergraph
0.0186
0.0289
0.0195
0.0288
0.0182
0.0296
0.0191
0.0288
Attributes
0.0576
0.1073
0.0560
0.7022
0.0585
0.1084
0.0581
O. 0.1094
WGAN
Hypergraph
0.0186
0.0193
0.0195
0.0183
0.0182
0.0263
0.0191
0.0334
Attributes
0.1951
0.1184
0.0774
0.0948
0.0717
0.0840
0,2848
0.1141
ReLaSH
Hypergraph
0.0147
0.0073
0.0130
0.0058
0.0142
0,0050
0.0127
0,0044
Attributes
0.0418
0.0402
0.0404
0.0364
0.0400
0.0339
0.0431
0.0334
ReLaSH,
Hypergraph
0.0145
0.0128
0.0128
0.0344
0.0140
0.0430
0.0128
0.0448
Autrbutee
0.0418
0.0402
0.0404
0.0364
0.0400
0.0339
0.0431
0.0334
Gau Diff
Hypergraph
0.0145
0.0281
0.0128
0.0282
0.0140
0.0289
0.0128
0.0300
800
Attributes
0.0725
0.1139
0.1022
0.1123
0.1035
0.1110
0.0757
0.1069
RealNVP
Hypergraph
0.0145
0.0287
0.0128
0.0283
0.0140
0.0289
0.0128
0.0300
Attributes
0.0407
0.1044
0.0402
0.1116
0.0405
0.1111
0.0431
0.1070
WGAN
Hypergraph
0.0145
0.0128
0.0128
0.0269
0.0140
0.0523
0.0128
0.0459
Attributes
0.0645
0.0865
0.0572
0.0744
0.0523
0.0617
0.0645
0.0757
ReLaSH
Hypergraph
0.0098
0.0058
0.0115
0.0056
0.0097
0.0057
0.0095
0.0053
Attributes
0.0286
0.0314
0.0281
0.0309
0.0275
0.0332
0.0272
0.0298
RcLaSH,
Hypergraph
0.0095
0.0328
0.0112
0.0342
0.0091
0.0309
0.0088
0.0301
Attrbbutes
0.0286
0.0314
0,0281
0.0309
0.0275
0.0332
0.0272
0.0298
Gau Diff
Hypergraph
0.0095
0.0280
0.0112
0.0272
0.0091
0.0279
0.0088
0.0283
1600
Attributes
0.0637
0.1276
0.0618
0.1173
0.0628
0.1121
0.0597
0.1083
RealNVP
Hypergraph
0.0095
0.0287
0.0112
0.0282
0.0091
0.0263
0.0088
0.0251
Attributes
0.0281
0.1172
0.0280
0.1132
0.0285
0.1104
0.0276
0.36
WGAN
Hypergraph 0.0095
0.0105
0.0112
0.0224
0.0091
0,0300
0.0088
0.0666
Attributes
0.0496
1.53
0.0458
0.0648
0.0441
0.0579
0.0457
0.0466
Table 8: RMSE results for means (columns with "Mean") & covariances (columns with "Cov") of
ReLaSH, ReLaSH,, Gau-Diff, RealNVP and WGAN when latent dimension is k = 24. Each value comes from the mean of 20 repetitions.
28

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Published as a conference paper at ICLR 2026
Embedding Dimension
kg
kg
16)
Method
Objects
T =
200
n
400
72
p
800
TI
1600
Mcan
Cov
Mean
Cov
Mcan
Cov
Mean
Cov
ReLaSH
Hypergraph
0.0810
0.0260
0.0815
0.0260
0.0990
0,0299
0.0840
0.0368
Attributes
0.0863
0.1137
0.0807
0.1043
0.0938
0.1034
0.0872
0.1000
ReLaSH
Hypergraph
0.0281
0.0289
0.0263
0.0282
0.0268
0.0314
0.0291
0.0382
Attributes
0.0863
0.1137
0.0807
0.1043
0.0938
0.1034
0.0872
0.1000
Gau Diff
Hypergraph
0.028I
0.0533
0.0263
0.0526
0.0268
0,0559
0.0291
0.0551
200
Attributes
0.0992
0.1575
0.1199
0. 0.1581
0.1241
0.1543
0.1047
0.1550
RealNVP
Hypergraph
0.0281
0.0538
0.0263
0.0526
0.0268
0.0559
0.0291
0.0551
Auributes
0.0850
0.1523
0.0808
0.1590
0.0938
0.1551
0.0874
O. 0.1553
WGAN
Hypergraph
0.0281
0.0290
0.0263
0.0427
0.0268
0.0393
0.0291
0.0288
Attributes
0.1421
0.2809
0.1414
0.5243
0.1201
0,3009
0.0936
0.1969
ReLaSH
Hypergraph
0.0418
0.0180
0.0305
0.0139
0.0198
0.0136
0.0214
0.0104
Attributes
0.0590
0.0819
0.0640
0.0717
0.0682
0.0697
0.0593
0.0678
ReLaSH,
Hypergraph
0.0179
0.0176
0.0195
0.0136
0.0194
0.0159
0.0174
0.0220
Attributes
0.0590
0.0819
0.0640
0.0717
0.0682
0.0697
0.0593
0.0678
Gau Diff
Hypergraph
0.0179
0.0529
0.0195
0.0530
0.0194
0.0556
0.0174
0.0549
400
Attributes
0.0944
0.1667
0.1114
0.1557
0.1099
0.1501
0.0834
O. 0.1999
RealNVP
Hypergraph
0.0179
0.0535
0.0195
0.0531
0.0194
0.0556
0.0174
0.0549
Attributes
0.0601
0.1623
0.0628
0.1567
0.0686
0.1510
0.0590
0.1542
WGAN
Hypergraph
0.0179
0.0247
0.0195
0.0370
0.0194
0.0255
0.0174
0.0226
Attributes
0.0939
0.2347
0.1011
0,2433
0.0767
0.1589
0.0691
0.1597
ReLaSH
Hypergraph
0.0209
0.0108
0.0139
0.0079
0.0136
0,0085
0.0130
0.0062
Attributes
0.0460
0.0579
0.0452
0.0511
0.0481
0.0505
0.0472
0.0500
ReLaSH,
Hypergraph
0.0137
0.0127
0.0122
0.0190
0.0138
0.0252
0.0129
0.0273
Aurrbutes
0.0460
0.0579
0.0452
0.0511
0.0481
0.0505
0.0472
0.0500
Gau Diff
Hypergraph
0.0137
0.0530
0.0122
0.0523
0.0138
0,0554
0.0129
0,0553
Attributes
0.0816
0.1655
0.1116
0.1514
0.1022
0.1536
0.0727
0.1548
RealNVP
Hypergraph
0.0137
0.0539
0.0122
0.0524
0.0138
0.0554
0.0129
0.0554
Attributes
0.0459
0.1583
0.0457
0.1526
0.0485
0.1544
0.0428
0.1550
WGAN
Hypergraph
0.0137
0.0208
0.0122
0.0251
0.0138
0.0107
0.0129
0.0183
Attributes
0.0811
0.1395
0.0663
O. 16663
0.0671
0.0933
0.0551
0.0754
ReLaSH
Hypergraph
0.0173
0.0097
0.0T02
0.0086
0.0097
0.0080
0.0090
0.0054
Attributes
0.0328
0.0525
0.0322
0.0491
0.0326
0.0495
0.0318
0.0472
RcLaSH,
Hypergraph
0.0105
0.0127
0.0085
0.0169
0.0095
0.0184
0.0087
0.0204
Attrbbutes
0.0328
0.0525
0.0322
0.0491
0.0326
0.0495
0.0318
0.0472
Gau Diff
Hypergraph
0.0105
0.0526
0.0085
0.0521
0.0095
0,0555
0.0087
0.0559
1600
Attributes
0.0653
0.1609
0.1061
0.1555
0.0978
0.1520
0.0811
0.1525
RealNVP
Hypergraph
0.0105
0.0535
0.0085
0.0522
0.0095
0.0555
0.0087
0.0559
Attributes
0.0319
0.1510
0.0385
0.1570
0.0333
0.1528
0.0301
0,1527
WGAN
Hypergraph 0.0105
0.0121
0.0085
0.0185
0.0095
0.0098
0.0087
0.0367
Attributes
0.0856
0.0912
0.0824
0.1415
0.0582
0.0710
0.0479
0.0547
Table 9: RMSE results for means (columns with "Mean") & covariances (columns with "Cov") of ReLaSH, ReLaSH,, Gau-Diff, RealNVP and WGAN when latent dimension is k = 48. Each value comes from the mean of 20 repetitions.
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77
Method
n2 ==
200
n =
400
8=
800
71 =
1600
Embedding Dimension & = 6
h2
kg
ReLaSH
0.3540
0.2598
0.1561
0.1452
ReLaSH:
0.3624
0.1843
0.1884
0.2111
200
Gau Diff
0.3608
0.3245
0,4331
0,5995
RealNVP
0,3817
0,3280
0,3791
0.5638
WGAN
0.4278
0,2789
0.1673
0.1933
ReLaSH
0.1642
0.1756
0.1268
0.1341
ReLaSH.
0.2977
0.2965
0,3499
0,2740
400
Gau Diff
0.1764
0,1496
0,2627
0,3947
RealNVP
0.1775
0.1467
0.2819
0.4273
WGAN
0.2202
0.2426
0.2867
0.2766
ReLaSH
0.2199
0.1862
0.0792
0.0720
ReLaSH,
0.3112
0.3379
0.3688
0,3982
800
Gau Diff
0.1725
0,0892
0.1561
0.2866
RealNVP
0.1778
0.1143
0.1682
0.2949
WGAN
0.2215
0.1901
0,2582
0,2293
ReLaSH
0.1670
0,0766
0,1381
0.1225
ReLaSH
0.1666
0.1721
0.2752
0.5741
1600
Gau Diff
0.1634
0.0793
0.1344
0.2172
RealNVP
0.1334
0.1842
0.0995
0.1994
WGAN
0.1618
0.2282
0.1962
0.1737
V/1
Method
n p 200
1 = p 400
==
p
n p 1600
Embedding Dimension k:
12
[k]
h/
kg
4)
RcLaSH
0.9794
0.6914
0.6056
0.6034
RcLaSH,
1.0132
0.6824
0.5676
0.5067
200
Gau Diff
1.1343
0,9088
1.3026
1.4973
RealNVP
1.3139
1.0591
1.2460
1.5199
WGAN
1.6482
1.4013
0.7556
1.3379
ReLaSH
0.7072
0.4392
0.2884
0.2246
ReLaSH.
0.6191
0.4158
0,3129
0,2952
400
Gau Diff
0.6172
0.4947
0.6941
0,9002
RealNVP
0.7296
0.5701
0.8043
0.9341
WGAN
0.8856
0.6497
0.3500
0.2412
ReLaSH
0.4775
0,3718
0.2037
0,2398
ReLaSH,
0.4546
0,3552
0.3431
0,3174
800
Gau Diff
0.4071
0.3575
0.5120
0.7380
RealNVP
0.4533
0,4206
0.4720
0.7130
WGAN
0.4418
0.5541
0.3687
0.3399
ReLaSH
0.3595
0.3217
0.2843
0.2204
ReLaSH.
0.3745
0,2854
0.2679
0.2589
1600
Gau Diff
0.3529
0.3321
0.3795
0.5534
RealNVP
0.4275
0.3232
0.3423
0.5177
WGAN 0.3845
0.3974
0,4582
0.4952
Table 10: FED results for ReLaSH, ReLaSH,, Gau-Diff, RealNVP and WGAN when latent dimen- sion is l: 6, 12. Each value comes from the mean of 20 repetitions.
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77
Method
n p= 200
n n= 400
n=
= 800
7 = p
1600
Embedding Dimension k
24
[k1
kg
kg
8)
ReLaSH
3.7876
2.3795
1.7776
1.4888
ReLaSH:
4.0077
2.5164
1.8760
1.3280
200
Gau Diff
3,9593
3.4421
3.2590
3,2990
RealNVP
4.1992
3.7454
3,2969
3,3342
WGAN
3.0621
1,9154
1.5121
1,1528
ReLaSH
2.3870
1,3333
0.8480
0.5496
ReLaSH
2.3198
1,3818
0.8407
0.6347
400
Gau Diff
2.0714
1.7287
1.8749
1,7820
RealNVP
2.3455
1.7374
1.8147
1,8179
WGAN
2.5758
1.7842
1.5667
0.8110
ReLaSH
1.6944
1.0641
0.6340
0.4274
ReLaSH,
1.5742
0,9923
0,7313
0,6584
800
Gau Diff
1,5422
1.1280
1.2945
1.5734
RealNVP
1.7499
1,2599
1.3955
1,5954
WGAN
1.8389
0.9427
0,9342
0.5240
ReLaSH
1.2875
0,9731
0,5482
0.3740
ReLaSH
1.1986
1.0372
0.8542
0.6594
1600
Gau Diff
1.4199
1.0037
1.4529
T.5630
RealNVP
1.4597
1.6384
1.3294
1.2934
WGAN
0.9762
0,9205
0.7708
0.7188
V/1
Method
n p 200
= p 400
11
p
p 1600
Embodding Dimension = 48
(k)
l0
16)
RcLaSH
15.3227
11.4013
9.8524
9.1217
RcLaSH,
16.9249
11.9277
9.179I
9,2782
200
Gau Diff
17.5939
14.7371
13.4529
14.4237
RealNVP
18.0527
15,0858
13.5804
14,4640
WGAN
15.9366
12.1945
10.5906
9,4059
ReLaSH
9.2837
4.2397
2.3041
I,7503
ReLaSH.
9.7989
4,4413
2,4005
I.6694
400
Gau Diff
9.1772
5,3424
4,3290
4,3520
RealNVP
9.8230
5.6104
4,4242
4,3550
WGAN
8.4115
3.8713
2.2187
1.6203
ReLaSH
6.1817
2.981I
I.7799
T.0864
ReLaSH,
6.0745
2.9594
1.7735
0.9799
800
Gau Diff
5,9200
3.3151
3.1729
3,0181
RealNVP
6.2765
3,6356
3.2323
3,0597
WGAN
5.1394
2.3107
1.4140
0.9230
ReLaSH
3,5762
1.7204
0.9726
0,5382
ReLaSH.
4.2942
1.6382
1,0038
0.6389
1600
Gau Diff
4.5040
2.2186
2.3887
2.587I
RealNVP
4.6260
2.6337
2,3902
2.6417
WGAN 3.7429
1.6919
1.0535
0.7648
Table 11: FED results for ReLaSH, ReLaSHe, Gau-Dift RealNVP and WGAN when latent dimen- sion is = 24, 48. Each value comes from the mean of 20 repetitions.
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methods reflects only the enror between the training data and the benchmarks, while for ReLaSH, the RMSE accounts for two sources of error: the error between the training data and the benchmarks, well as the error between the generated data and the training data. As result, the RMSE for ReLaSH generally larger than that of the calibrated methods. In terms of the RMSE of the mean attribute vector, ReLaSH consistently performs the best, while RealNVP performs well occasionally, and WGAN shows slight improvement as the data scale increases.
In terms of RMSE of covariances and FED, ReLaSH generally outperforms the other three gen- eration approaches. Generally, Gau-Diff requires the most memory, while RealNVP is the most time-consuming method among the four. WGAN performs comparably ReLaSH under limited computational resources, but it yields worse results in terms of еπтог metrics. As the latent dimension k, or the values of m and n, increase (i.e., as the synthetic data structure becomes more complex), the number of epochs required for achieving similar results to ReLaSH by the other methods increases substantially, leading corresponding increase running time. However, ReLaSH constructs the diffusion model in low-dimensional continuous embedding space, thereby avoiding the need to train a high-dimensional score network. This significantly reduces sample complexity and enhances both efficiency and accuracy.
B.3.2 SIMULATION RESULTS FOR DIMENSION SELECTION
To demonstrate the effectiveness of our proposed latent dimension selection algorithm HTT (see Algorithm for details), we conducted simulation study with data size n = 500, m 5000 and true latent dimensions K1 = li2 K3 4. All embeddings and parameters were generated according to the simulation settings described in Section B.3 of the Appendix. We applied the HTT
pipeline to the simulation data across 30 repetitions, and the results presented in Table 12.
h
h:
h3
Frequency
4
4
4
27
1
4
5
4
2
Table 12: Result of latent dimension selection across 30 repetitions.
The results indicate that, in most cases, the HTT algorithm successfully recovers the true latent dimensions. We further apply this algorithm in our real data analyses.
B.3.3 SENSITIVITY ANALYSIS
In this section, we conduct some sensitivity analysis to better understand the robustness and tuning requirements ReLaSH.
For the latent dimensions k1, k2, and ki., we conduct a sensitivity analysis with m = n and k 12 h3 = In this setting, all true embeddings and parameters are generated with the same simulation settings as described in Section B.3 of the Appendix. We vary the latent dimensions and present the resulting embedding and generation error metrics across 20 repetitions in Table 13,
400
14, 15, 16.
ki
k2
k3
Dmm J
AN. J
Ax
Ax. 4
FED |
ReLaSH, Ax. + ReLaSH, FED J
2
2
2
0.0186
0.0074
0.0683
0.0522
0.4984
0.0472
0.5527
4
4
4
0.0175
0.0057
0.0514
0.0339
0.4392
0.0356
0.4158
6
6
6
0.0181
0.0064
0.0581
0.0449
0.5239
0.0405
0.5204
Table 13: Result of setting different kn repetitions.
K2
kg. Each value comes from the mean of 20
The results indicate that, in general, generation performance is optimal when the true latent space dimensions are used. Although mis-specifying the latent dimension can introduce additional error, the resulting performance remains comparable the observed fluctuations in error metrics are consistent with changes in the latent dimensions. For instance, when kn and l fixed and k3 is varied, error metrics associated with hyperlinks tend to
baseline methods such RealNVP. Furthermore,
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k1
K2
K3
Hm. +
AR. 4
Xx +
Ax, 4
FED
ReLaSH, Ax. + ReLaSH, FED
4
2
4
0.0182
0.0072
0.0639
0.0553
0.4672
0.0304
0.4821
4
4
4
0.0175
0.0057
0.0514
0.0339
0.4392
0.0356
0.4158
4
8
4
0.0179
0.0053
0.0576
0.0451
0.4404
0.0398
0.4927
Table 14: Result of setting different l22 Each value comes from the mean of 20 repetitions.
k1
22
k3
A 1
An. J
Ax..
Dx. 4
FED L
ReLaSH, 4x. + ReLaSH, FED І
2
0.0182
0.0069
0.0563
0.0432
0.4782
0.0428
0.5476
4
4
4
0.0175
0.0057
0.0514
0.0339
0.4392
0.0356
0.4158
4
4
8
0.0180
0.0048
0.0497
0.0392
0.4625
0.0388
0.4624
Table 15: Result of setting different k3. Each value comes from the mean of 20 repetitions.
increase, while those related to attributes fluctuate only slightly. Conversely, varying k while keep- ing k2 and ks fixed causes greater changes in attribute-related errors. Notably, incorrect selection of ky leads to elevated errors in both hyperlink- and vuribute-related metrics, which is expected since (2) is shared by both components.
Additionally, the FED metric appears more sensitive to these changes than the moment-based error metrics, likely due its more involved computational process. Nonetheless, simulation results demonstrate that moderate deviations in latent dimension selection do not substantially degrade performance, and the results remain on par with those of baseline methods.
Furthermore, we conduct ablations to demonstrate the effectiveness of the proposed tri-partite par- titioning strategy compared to a simpler. unified latent space. A unified k-dimensional latent space is specially case of the tri-partite partition with the triple being (0, k 0). 'The tri-partite parti- tion (k1, k2, ks) is motivated by the factorization of the joint likelihood: ()) (attribute-only), (3) (hyperlink-onlyy and (2)) (shared), allowing to separate but still couple the hypergraph and its attributes for generative modeling. Using unified k-dimensional latent space would mix attribute- only and hyperlink-only variation and requires stronger assumptions to recover interpretable com- ponents; our framework, the block structure gives clean orthogonality conditions to distinguish latent effects driving the generation of hypergraph and attributes, separately and jointly.
In ablation studies shown Table 17, 18, 19, where unified latent space of total dimension k 41 + 12 + K3 is employed, the tri-partite model demonstrates superior performance, particularly on attribute-related metrics and FED, which are notably sensitive to latent dimension selection. Additionally, the tri-partite model exhibits more stable training dynamics.
B.4 ADDITIONAL RESULTS FOR REAL DATA ANALYSIS
In this section, we provide additional results real data analysis to further assess the quality of the synthetic outcomes.
B.4.1 CO-CITATION HYPERGRAPH WITH ABSTRACT ATTRIBUTES
The MADStat dataset we utilized contains citation information for over 83,000 papers published across 36 journals between 1975 and 2015, while MADStaText includes the abstracts of all these papers. We focus on the top n 1, 000 authors most frequently cited during this period, resulting in dataset of 35,143 papers that cite at least two of these top authors. Given the large size of this dataset, among all the methods discussed previously, only ReLaSH can effectively handle such
kn
kz
K3
AH.. 1
AR. 4
Ax.,
Ax.
FED L
ReLaSH, An. + ReLaSH, FED
2
4
4
0.0179
0.0072
0.0648
0.0582
0.4872
0.0397
0.5623
4
4
4
0.0175
0.0057
0.0514
0.0339
0.4392
0.0356
0.4158
8
4
4
0.0176
0.0063
0.0587
0.0322
0.4613
0.0384
0.4729
Table 16: Result of setting different kj Each value comes from the mean of 20 repetitions.
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K1
K2
K3
Ahm Hm +
AR. 4
Xxn, 4
Ax, +
ReLaSH, Az. + ReLaSH, FED
2
2
2
0.0201
0.0039
0.0511
0.0265
0.1756
0.0436
0.2965
0
6
0
0.0209
0.0041
0.0562
0.0384
0.6432
0.0412
0.5261
Table 17: Result of setting a unified latent space of total dimension h = -1 + +12 + h3. Each value comes from the mean of 20 repetitions.
k1
kg
K3
Atm.
AH.
Ax..
Ax.
FED +
ReLaSH, AH. 1 ReLaSH, FED J
4
4
0.0175
0.0057
0.0514
0.0339
0.4392
0.0356
0.4158
0
12
0
0.0181
0.0050
0.0566
0.0698
1.0164
0.0383
1.2767
Table 18: Result of setting unified latent space of total dimension k = ki + 1:2 + K3. Each value
comes from the mean of 20 repetitions.
large volume of data, owing to the dimension reduction procedure it offers. Consequently, further narrow the scope by selecting the top m 2, 000 papers that cite the greatest number of these authors, and use the top 5,000 such papers as the population data. This approach ensures that the resulting hypergraph remains both dense and informative. To construct the co-citation hypergraph, we treat each top author as a node, with each paper forming a hyperlink. Specifically, if paper cites a top author, the corresponding node appears in the hyperlink representing that paper. As noted in (Ji et al., 2022), co-citing two authors in an article suggests that they likely share common research interests. In our application, this co-citation setting provides hypergraph with sufficient density, which justifies the construction of the co-citation hypergraph as described above.
To construct the attributes corresponding each hyperlink, we use the abstract data of each paper. We consider the pre-processed corpus of 2,106 words from the dictionary generated by Ke et al. 2, 000 (2023), and further refine this list by removing words that do not appear in the top attribute papers. This leaves us with p = 2, 039 words for analysis. To construct the p-dimensional vector for each hyperlink, the i-th entry of the vector is set the TF-IDF value (Sparck Jones, 1972) of the 1-th word in relation to the abstract of the corresponding paper (1 S S p).
We use each approach generate m 32m hyperlinks and corresponding attributes. The perfor- mance of each method is evaluated using the RMSE of the sample means and covariances of the generated hyperlinks, compared against those of the overall population consisting of the top 5,000 most-cited papers from the selected authors. We also assess the performance by comparing ReLaSH with other calibrated methods (i.c., ReLaSHe, Gau-Diff, RealNVP, WGAN, VAE) in terms of the RMSE of the hypergraph node means, denoted as (AH.). ). as summarized in Table 20.
Additionally, the Fréchet Embedding Distance (FED) is computed with respect the chosen latent dimension, with the dimension selection procedure outlined in Appendix B.6. Based on the cross- validation results presented in Table 21, we select h₁ = k2 = kg and generate benchmarks for the embedding machine Ed using the population data.
To evaluate the coherence and plausibility of the synthesized statistical research articles, we ran- domly sampled several papers from the result generated by ReLaSH-(2, 7, 8), as shown in Table 22. We then analyzed the relationship between their cited authors and their top 10 TF-IDF keywords. The analysis examined whether the referenced scholars are thematically aligned with the identi- fied keywords and what research areas such combinations are likely to represent in contemporary statistical literature.
k-
kg
Anm +
An, 4
Ax. 4
Ax. 4
FED 1
ReLaSH, An.
ReLaSH, FED
8
8
8
0.0180
0.0077
0.0554
0.0492
1.3333
0.0271
1.3818 2.9824
0
24
0
0.0184
0.0072
0.0625
0.0781
2.7461
0.0304
Table 19: Result of setting a unified latent space of total dimension comes from the mean of 20 repetitions.
ki + l/2 + K3. Each value
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(eL.aSH-(2,2.2)
ReL.aSH((2.788) ReL,8S--(8,888)
Caliberated methods
Hin +
3.20
2.59
3.68
3.57
Table 20: An,, for different generation methods for the co-citation hypergraph generation task. The scale of An. is 10
Latent dimension
FD by cross validation
(k1, h2, ks) = (2
0.257
(ki, , 22, Kia) (4, 4)
0.329
(ki, R2, kis) = (8,8,8
0.627
Table 21:
results for the co-citationship hypergraph.
Table 22: Citation authors and top keywords in abstracts of sample papers. alphabetically by surname.
Authors sorted
Paper ID
Cited Authors
Top Words
56801
Bradley Efron; Frank Critchley; Alan Gelfand: Robert
posterior, Bayesian, prior, Markov chain, density, MCMC, mixture, compute, Bayes, approximation hazard, proportion, survival. sor, failure, cox, event, maximum, time, cure
Kohn; Samuel Kotz; Nan Laird; Faming Liang; Gareth
Roberts; G. A. Young
53347
Kani Chen; Anthony C. Davison; Peter Diggle: Theo Gasser; Harry Kesten; Charles Kooperberg; Jerry Lawless: Youngjo Lee; Danyu Y. Lin; Ian W. Mckeague; Jing Qin; Bernard W. Silverman; Liuquan Sun; Yanqing Sun; Lee- jen Wei; Liugen Xue; Grace Y. Yi; Guosheng Yin; Zhiliang Ying
55762 Michael Akritas; Anestis Antoniadis; Raymond Carroli; Marie Davidian: Bert Van Es; Jianqing Fan; Peter Hall; Wolfgang Härdle; M. C. Jones; Takeaki Kariya; Tze Le- ung Lai; Doug Nychka; Byeong U. Park; John Rice; Joseph Romano; Naisyin Wang
asymptotic, non- per--arrattic,, confi- dent, construct, smooth, test, empir- ical, derivative, kernel, rale
The cited authors article 56801 are widely acknowledged for their seminal contributions to Bayesian statistics, posterior inference, and computational algorithms. In particular, many have played pivotal roles advancing Markov chain Monte Carlo (MCMC) methods, mixture modeling, and the efficient computation of Bayesian posterior distributions. This constellation of expertise strongly suggests that the article represents a significant methodological advancement in Bayesian computational techniques, with emphasis novel sampling algorithms and strategies for posterior approximation. Moreover, the close alignment between the cited authors and the article's keywords further underscores its relevance cutting-edge research in statistical computation and contempo- rary Bayesian analysis.
Regarding article 53347, the cited authors are leading figures in survival analysis and event history modeling. Their collective expertise spans proportional hazards modeling, analysis of censored and failure-time data, and the application of maximum likelihood estimation within survival contexts. The principal keywords centralize themes and methods that are foundational to survival data analy- sis, particularly in biostatistics. This synergy of authorship and thematic focus strongly indicates that the article introduces methodological innovations in survival modeling potienially centering on the Cox model, cure rate models, or advanced approaches for the analysis of censored and time-to-event data.
Finally, the referenced authors article 55762 have made foundational contributions to nonpara- metric statistics, asymptotic theory, and smoothing techniques. The associated keywords point to research on nonparametric inference, construction confidence sets, and the theoretical analysis of estimator properties. The intersection of these elements compellingly suggests that the article is devoted to advancing nonparametric estimation methods, with attention to empirical and asymptotic results, and the development of new confidence regions for smooth estimators.
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Overall, the pairing of cited authors and prominent keywords in these synthesized articles is highly consistent with the structure and topical alignment found in authentic statistical methodology papers. Each article demonstrates an internally coherent thematic structure, where the cited scholars authoritative within the research area denoted by the keywords. Therefore, our generative method, ReLaSH, support the prediction of emerging trends in the statistical community and facilitate the identification of co-citation relationships between specialists and their research interests.
B.4.2 RECIPE HYPERGRAPH WITH NUTRITION ATTRIBUTES
The "Epicurious Recipes with Rating and Nutrition dataset contains 17,736 recipes information lifted from http:/ /tt::///wwwr /www. //wp:./piicu.. mouioous.. com/ coon/recipes-eeeee - each accompanied by user ratings, nutritional information, and ingredient lists. Naturally, we represent each ingredient as a node and each recipe as a hyperedge consisting of the corresponding ingredients. Ratings and nutritional variables ("calories", "sodium", "protein", and "fat") used hyperedge attributes. If an ingredient is part of a particular recipe, the hyperedge representing that recipe is connected to the node representing the ingredient.
Note that the attributes in the recipe dataset vary in both scale and interpretation, so we apply z-score standardization the attribute before training the generative models ensure comparability across features. The summary of attributes are listed in Table 23.
Ratings
Calories
Protein
Fat
Sodium
Mean
4.192
430.04
17.46
22.90
455.90
Std. Error
0.598
257.24
16.41
17.37
429.33
Table 23: Means and standard errors for recipe attributes.
For pre-processing, we consolidated ingredient names with the same or highly similar meanings (e.g., "green onion" and "spring onion", "chili pepper" and "hot pepper"), and removed ingredients appearing in only a single recipe. To mitigate issues caused by missing information, we retained only recipes containing at least seven ingredients, resulting in population dataset of 789 recipes with 200 distinct ingredients. From this population, we randomly selected 300 recipes as the training dataset, where each recipe is associated with five hyperedge attributes. We use each approach to generate m = 32m hyperlinks and corresponding attributes.
Specifically, in the recipe dataset, the attribute dimension is relatively small (p = 5) compared to the size of the hypergraph node set (n = 200), and all attributes are continuous variables. Therefore, in the embedding procedure of ReLaSH, duces to an identity mapping. Consequently, after embedding the hypergraph into a ky-dimensional latent space, we concatenate the latent embedding of each hyperedge with its corresponding at- tributes, and then train diffusion models on the concatenated data. Intuitively, the latent variables obtained through embedding represent underlying structural features of each hyperedge, whereas the attributes provide interpretable features for each recipe. Thus, concatenating the two and performing generation constitutes natural modeling strategy, which further demonstrates the flexibility of our proposed method.
set k 5. h2 0, while the attribute embedding
We also assess the performance by comparing ReLaSH with other calibrated methods (i.e., ReLaSH,, Gau-Diff, RealNVP, WGAN, VAF) in terms of the RMSE of the hypergraph node means, denoted as (AH,), 1. as summarized in Table 24.
ReLaSHH(5.0.2)
ReLaSH-(5.0.6)
ReLaSH-(5.0,16)
Caliberated methods
AH I
9.38
13.01
8.33
8.27
Table 24: An., for different generation methods for the recipe hypergraph generation task. The scale of AH, is 10
Similarly, the Fréchet Embedding Distance (FED) is computed with respect to the chosen latent dimension. Based on the cross-validation results presented in Table 25, we select latent space di-
htttps://wwwkkatt wtt.:/....o.ateeeeeeeeee
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mensions (5, 0, and generate benchmarks for the embedding machine E data.
using the population
Latent dimension
FD by cross validation
(ki, K2,, Kis)
2,)
0.194
(ki, R2, kis) = (5.0,4)
0.427
(ki, K2, ki) 15,0,8
13.630
(k1, K2, ka) -,,
14.064
Table 25: Cross- Validation results for the recipe hypergraph.
We examine individual recipes synthesized by ReLaSHe-(5. O, 16). which generally performs the best among all the generation models, to emphasize the quality of generation. Table 26 presents sev- eral randomly selected samples. For interpretability, we assign descriptive names corresponding to real-world cuisines, and all generated recipes resemble authentic dishes. Each sample demonstrates ingredient combinations and nutritional profiles that are both plausible and balanced, particularly for seafood-foree forward, legume-based. or mixed salads and stews from the Mediterranean and North- ern Europe. The coherent ingredient pairings and reasonable macronutrient compositions highlight the applicability the generated results to realistic culinary settings.
Table 26: Generated recipe samples from ReLaSH:-(5,0, 16). All nutritional values reported
per entire recipe portion.
Rating (out of 5)
Calories (kcal)
Fat (g)
Protein (g)
Sodium (mg)
Recipe Name
Ingredients
bean, chickpea, chile pepper, clam, cod, fennel, halibut, lentil, potato, saffron, tomato, onion, wine carrot, celery, cinnamon, legume, lemon, parsley, pasta, potato
Mediterranean Fisherman's Bean Stew
4.22
228
9.82
46.9
264
Hearty
Mediterrancan
4.21
430
16.36
32.96
482
Legume Soup
Scandinavian
cilantro, citrus, clam, cod, dill, milk, cream, onion
Creamy Scafood
3.74
237
10.98
17.75
1117
Chowder
Sesame Tuna
sesame, spice, tomato, tuna, pepper
Mediterranean
4.40
526
36.24
22.4
518
Salad
basil, fruit, grape, mussel, parsley, pecan, tomato, spirits
Mediterranean
4.33
778
49.77
63
623
Mussel
The "Hearty Mediterranean Legume Soup" can be characterized Mediterranean vegetable and legume soup, recalling Italian minestrone or Middle Eastern harira. With pasta, root vegetables, beans, and aromatic herbs, the nutrition profile is realistic for hearty soup, rich in protein and complex carbohydrates yet moderate in fat.
The "Scandinavian Creamy Seafood Chowder" is Scandinavian-inspiee creamy seafood chowder, featuring cod, clam, dill, and citrus. Its nutritional composition is plausible for chowder: adequate protein, moderate fat, and elevated sodium due seafood and seasoning.
The "Sesame Tuna Mediterranean Salad" represents fusion-style salad that combines Mediter- ranean and Asian influences, pairing tuna with sesame, tomato, and peppers. The higher fat content is attributable to sesame, while protein derives from tuna, reflecting the nutritional profile of con- temporary main-course salads.
Finally, the "Mediterranean Mussel" is a classical Mediterranean mussel cuisine, integrating mus- sels, pecans, tomato, and basil, with spirits to enhance flavor. Its high protein and fat levels are consistent with shellfish and nut components, while the addition of fruit and herbs signals modern, health-oriented culinary trends.
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To provide comprehensive illustration of the cuisines, construct plot that aggregates the ingredients of each recipe, as shown in Figure 17.
Mediterranean Fisherman's Bean Stew
Rating: 4.22
Calories: 228 kcal
Fat: g
Protein: 46.9 g
Sodium: 264 mg
Hearty Mediterranean Legume Soup
Rating: 4.21
Calories: 430 kcal
Fat: 16.36
Protein: 32.96 g
Sodium: 482 mg
Scandinavian Creamy Seafood Chowder
Rating: 3.74
Calories: 237
Fat: 10.98 g
Protein: 17.75 g
Sodium: 1117 mg
Sesame Tuna Mediterranean Salad
Rating: 4.40
Calories: 526 kcal
Fat: 36.24 g
Prolsin: 22.4
Sodium: 518 mg
Modern Mediterranean Mussel Fruit Salad
Rating: 4.33
Calories: 778 kcal
Fat: 49.77 g
Protein: 63 g
Sodium: 623 mg
Figure 17: Illustration of generated cuisines.
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B.4.3 MEDICAL HYPERGRAPH WITH PATIENT ATTRIBUTES
We also apply ReLaSH and other competitive approaches symptom co-occurrence hypergraph constructed from electronic medical records of ICU patients. Specifically, we use the Medical Infor- mation Mart for Intensive Care (MIMIC-III; (Johnson et al., 2016)) dataset, which contains clinical data from over 45,000 patients at Beth Israel Deaconess Medical Center in Boston between 2001 and 2012.
In MIMIC-III, we focus on more than 10,000 patients who experienced an ICU stay and for whom a death record is available. To construct the population hypergraph, we consider 4,951 informative patient records with more than 15 co-occurring diseases-an indicator of severe health conditions, covering 2,230 distinct diseases. From this population, we randomly sample 2,000 records as the training set, and retain all 4, 951 patient profiles as the population set for evaluation.
In this hypergraph, each node represents symptom, and each hyperlink corresponds to a patient profile, with the incident nodes being the symptoms recorded for that patient. The attribute set of each hyperlink is derived from patient metadata, including lifetime, ethnicity, marital status, religion, gender, ICU length of stay, and overall hospital length of stay. Among these, ethnicity, marital status, religion, and gender are categorical variables, which we encode as integers. We use each approach generate 32m hyperlinks and corresponding attributes. Since the attribute dimension is substantially smaller than the hyperlink dimension, we adopt the same identity mapping strategy as in the recipe example for generation in ReLaSH and ReLaSH
We also assess the performance by comparing ReLaSH with other calibrated methods (i.e., ReLaSHe, Gau-Diff, RealNVP, WGAN, VAE) in terms of the RMSE of the hypergraph node means, denoted as (AHa), as summarized in Table 27.
Amp ReLaSS-(7.0.22)
ReLaSH-(7.0,16 5.75
Caliberated methods
2.52
1.14
Table 27: Ax, for different generation methods for the patient profile generation task. The scale of A H. is 10-3.
Similarly, the Fréchet Embedding Distance (FED) is computed with respect to the chosen latent dimension. Based on the cross-validation results presented in Table 28, we select latent space di- mensions (7, 0, 2) and generate benchmarks for the embedding machine E using the population data.
Latent dimension
FD by cross validation
(7.0, 2)
0.030
(7, 0, 4)
0.223
(7, 070,88
0.243
(7, 0, 16)
0.365
Table 28: Cross- Vols-Validation results for the symptom co-oceurrence hypergraph.
Furthermore, we examine the patient profiles generated by ReLaSH- 0, using the MIMIC-II dataset. For categorical variables such as marital status, ethnicity, religion, and gender, calibrated the generated outputs against the original training data to preserve the marginal proportions of each category. Examination of continuous variables including ICU stay duration, hospital stay duration, and patients' lifetime data revealed no outliers or abnormal values, indicating that the generated data clinically reasonable. We randomly sampled 10 patient profiles from the generated set, as shown in Table 29.
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Table 29: Sampled Patients: Personal Information and Major Diseases
Number of Diseases
ID
Numeric Attributes
Demographics
Representative Major Discases
Coronary Atherosclerosis;
Catholic
ICU stay: 8.25 days
Congestive Heart Failure:
White
Lifetime: 86. years
23
Chronic Kidney Disease:
Married
Hospital Stay: 354.94 hrs
Intracerebral Hemorrhage;
Female
Dementia
Atrial Fibrillation:
Protestant Quaker
ICU stay: 7.88 days
Congestive Heart Failure:
White
Lifetime: 74.42 years
20
Hypertension:
Widowed
Hospital Stay: 413.36 hrs
Acute Kidney Failure;
Male
Diabetes Mcllitus
Not Specified
Malignant Ncoplasm;
ICU stay: 6.54 days
White
Chronic Kidney Disease;
3
Lifetime: 75.72 years
16
Single
Cerebral Embolism With Infarction;
Hospital Stay: 314.68 hrs
Male
Heart Failure
Episcopalian
Diabetes Mellitus (Uncontrolled);
ICU stay: 14.16 days
White
Chronic Kidney Disease (Stage V);
Lifetime: 77.55 years
36
Married
Heart Failure; Sepsis;
Hospital Stay: 512.29 hrs
Male
Hepatic Transplant Complications
Graft- Seaft-Versus-Hore Disease;
Not Specified
ICU stay: 7.05 days
Atrial Fibrillation;
White
Lifetime: 57.35 years
Acute Vascular Insufficiency
Single
Hospital Stay: 362.37 hrs
of Intestine:
Male
Implant Infection
Diabetes Mellitus
Catholic
ICU stay: 8.84 days Lifetime: 67.53 years
(With Ketoacidosis);
Black
22
Acute On Chronic Diastolic
Married
Hospital Stay: 563.22 hrs
Heart Failure;
Female
Sepsis; Thrombocytopemia
Not Specified
Secondary Malignant Neoplasms
ICU stay: 16.95 days
White
(Intra-Ablominal, Nervous System,
7
Lifetime: 70.73 years
21
Single
Bone, Other Sites);
Hospital Stay: 515.13 hrs
Male
Tuberculosis Of Ureter; Dementia
Atrial Fibrillation:
Protestant Quaker
ICU stay: 2.86 days
Congestive Heart Failure;
White
8 Lifetime: 41.29 years
31
Acute Kidney Failure;
Widowed
Hospital Stay: 341.52 hrs
Subdural Hemorrhage;
Male
Septic Shock
Malignant Neoplasm of
Catholic
Bronchus and Lung:
ICU stay: 8.80 days
White
Heart Failure;
Lifetime: 66.34 years
28
Married
Chronic Pain Syndrome;
Hospital Stay: 654.82 hrs
Female
Hepatitis C;
Severe Sepsis
Diabetes Mellitus:
Buddhist
ICU stay: 4.00 days
Chronic Kidney Discase (Stage V);
Asian
10
Lifetime: 80.60 years
13
Acute Diastolic Heart Failure:
Divorced
Hospital Stay: 136.56 hrs
Acute Respiratory Failure:
Female
Dysthymic Disorder
is worth noting that the number of diseases per patient is relatively high, since during preprocessing restricted the dataset ICU patients with at least 15 documented conditions. This reflects the severe health status typical of such cohorts. The demographic characteristies the generated patients also appear consistent with real-world ICU populations.
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To illustrate, closely examine the set of co-occurring diseases for the first generated patient pro- file. The disease combination observed is highly representative of older, medically complex ICU patients. Chronic conditions such as hyperlipidemia, hypertensive chronic kidney disease, coro- nary atherosclerosis, atrial fibrillation, cardiomyopathy. chronic systolic heart failure, and dementia collectively suggest substantial long-term cardiovascular, renal, and neurologic impairment. Su- perimposed acute
intracerebral hemorrhage, abdominal aortic ancurysm rupture, gastrointestinal bleeding, acute kidney injury, and embolic thrombotic events-are well- known, life- tife-threatening occurrences frequently encountered in critically ill patients with multiple comorbidities.
Importamtly, there is considerable overlap between pre-existing chronic conditions and their acute manifestations. Similarly, congestive heart failure and chronic kidney disease exacerbate one an- other (the so-called "cardiorenal syndrome"), and this interaction is frequently observed in ICU co- horts. Infectious complications (e.g., pneumococcal infection) and procedural complications (e.g., procedure-related hematoma) are also prevalent in this context. Respiratory comorbidities (such as chronic obstructive asthma or airway obstruction), gastrointestinal conditions, and prior malignancy further reflect the clinical complexity of elderly ICU patients with multimorbidity and high risk of adverse outcomes.
Taken together, the cocxistence of chronic diseases, acute organ failures, infections, and iatrogenic complications is highly characteristic of ICU populations, particularly among older adults. The gen- erated case therefore mirrors realistic clinical scenarios commonly observed in critical care cohorts such as MIMIC-III.
B.4.4 SENSITIVITY ANALYSTS
As addressed in Appendix B.7.1, in the real-data experiments, ReLaSH/ReLaSHe use Forest Dif- fusion, while Gau-Diff uses standard diffusion model. This inconsistency score learners may confound the comparison. To address this concern, we conducted two additional experiments on all three real-world datasets: (1) applying standard diffusion the joint latent space reconstruction step in ReLaSH/ReLaSHc. and (2) applying ForestDiffusion directly to the entire dataset. The error metries for these experiments are presented below. In the table, (S.D.) indicates that standard Gaus- sian diffusion was used for the joint latent space reconstruction step; otherwise, ForestDiffusion was employed. The results are shown in Table 30, 31, 32.
Ax. +
A X
Ax
FED +
ReLaSH-(5,0, 16)
2.355
1.533
1.112
0.766
ReLaSH.-(5, 0, 16)
1.847
1.533
1.112
0.180
ReLaSH-(5, 0, 16) (S.D.)
2.067
1.832
2.854
0.659
ReLaSH (5,0 0, 16) (S.D.)
2.462
1.832
2.854
0.473
Gau-Diff
2.375
2.154
4.256
0.802
ForestDiifission
1.886
8.211
2.073
0.848
Table 30: Results for the recipe generation task. Scales AH. A Ax... Ax,, FED are 10-3, -3 10-2, 10-2, 2, 10-1, respectively.
An.
Ax.
Ax. +
FED |
ReLaSH-(8,8,
1.626
8.608
1.887
1.454
ReLaSH.-(8,8,5
2.816
8.608
1.887
6.451
ReLaSH-(8, 8, (S.D.)
1.925
11.836
1,889
2.648
ReLaSH (8, 8, (S.D.)
2.816
11.836
1.889
6.027
Gau-Diff ForestDiifiusion
1.672
10.016
1.824
5.060
OOM
OOM
OOM
OOM
Table 31; Results for the co-citation hypergraph generation task. Scales of AH.. AXm. Ax,, FED are 10 10-3, 1, 10-1, 10-1, respectively.
The enror of the three generation tasks generally tends to increase slightly when the distribution- free generator used in the latent space reconstruction step is standard Gaussian diffusion rather than
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Az. +
Ax, 1
Ax,
FED
ReLaSH--77,0,)
3.260
2.989
1.435
0.532
ReLaSH,-(7.0.2)
27.794
2.989
1.435
0.013
ReLaSH- (7,0, (S.D.)
4.085
2.849
1.385
0.835
ReL.aSH.- (7,0, (S.D.)
17.266
2.849
1.385
0.548
Gau-Diff
4.268
3.497
1.719
39.731
ForestDiffusion
OOM
OOM
OOM
OOM
Table 32: Results for the patient profile generation task. Scales of AH,, A Ax, FED are 10-4, 10 10-3, 1001,01.1 1002 respectively.
ForestDiffusion. This is expected, as ForestDiffusion has more complex structure and typically achieves lower error across the entire process. Nonetheless, using the ReLaSH pipeline still outper- forms directly applying standard diffusion for the whole dataset, highlighting the effectiveness of the ReLaSH pipeline. However, it is important note that ForestDiffusion encounters out-of out-of-memory memory (OOM) errors on the patient profile and co-citation datasets due to their high dimensionality when processed as tabular data. In contrast, within ReLaSH, both standard diffusion and Fifest sion exhibit comparable, fast runtimes. ForestDiffusion is optimized for low-dimensional tabular data, with the largest tested feature size being 90 (Joliceer-Maartincaa et al., 2024), while the pa- tient profile has 2,237 features and the co-citation dataset has 3,039. These results also illustrate subtle difference in ReLaSH/RRe LaSC performance depending on the type of distribution- free gen- erator used for joint latent space reconstruction. Because the joint latent space is dol-dinensional, distribution-free generators such as ForestDiffusion tend to perform well. This finding also under- scores the role of our dimension-reduction technique in the pipeline, which greatly improves both efficiency and accuracy in our generative model.
B.5 FRÉCHET EMBEDDING DISTANCE (FED)
In generative modeling, Fréchet Inception Distance (FID) is a widely used metric for evaluating the quality of generated samples, particularly in visual tasks. FID compares the feature representations (e.g., from the coding layers) of two datasets (e.g., real images and AI-generated images) using pre-trained Inception model, such as Inception-v3. The difference between the two populations is measured by calculating the Fréchet distance, assuming that both distributions follow Gaussian distribution. This distance computed using the means and covariances of the respective distri- butions. While FID has proven effective for evaluating generative models in vision-related tasks, its applicability is limited in other domains, as it relies on pre-trained Inception models specifically trained on visual data.
To overcome this limitation, FID has been adapted into specialized variants for different domains. For instance, Fréchet Audio Distance (FAD) (Kilgour et al., 2018) has been proposed for evaluating music enhancement algorithms, Fréchet Video Distance (FVD) (Unterthiner et al., 2019) is used for generative models of video, and Fréchet ChemNet Distance (FCD) (Preuer et al., 2018) used for evaluating Al Al-generated molecules. These variants maintain the core idea of FID, but they are customized to better suit the unique characteristics of their respective domains.
In similar vein, we extend the idea of FID to our own task by introducing Fréchet Embed- ding Distance (FED). FED generalizes FID for generative tasks where no pre-trained models are available for evaluation. Specifically, for given generative task T, define "true" dataset pirue = {a.!te) Fie(m representing real data, and a generated dataset Xeee = {0ē Ji sampled from the generative model being evaluated. To compute FED, we introduce an embedding machine ET : - RR which maps the data points into a continuous embedding space that captures the essential features of the original data. Once the data points are mapped into this embedding space, FED calculates the distance between the distributions of the true and generated datasets as
FED X
|| tie
113 Tr(E.re + Egen 2(E ggen) 1/2),
where me s00 are the sample means of (ET )iEmmm))) {ET (CT(2e) (x Dii and time Egon are the sample covariance matrices of {ET (x="te) )ie ). {ET (2.t0) he(maal which provides a flexible and robust evaluation metric for generative models in various tasks.
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Thus, similar to how FID has been adapted for different domains, we introduce FED a flexible and robust evaluation metric for generative models across a variety of tasks. FED generalizes the concept pre-traine of FID by eliminating the dependence pre-trained models. Furthermore, it proves particularly useful in scenarios where data distributions are too complex to compare directly, such as the high- dimensional hyperedges together with attributes explored in this study.
To avoid the possible invalidity problem emerging from embedding the entire population hyper- graph (containing the training set) obtain reference parameters, we split the training dataset from the overall population and using the remaining data to train the embedding machine E², thereby ensuring that the evaluator and evaluated models are independent. splitting does not change the final selected rank of the embedding machine.
our experiments, this data
For the patient profile and recipe datasets, the a-FED values is very similar to FED, as the training sets were constructed by randomly sampling from the population. For the co-citation dataset, there slightly more variation in the FED values, since we constructed the training hypergraph using the top m = 2, 000 most cited papers and defined the population the top 5, 000 most cited papers. Nevertheless, our method consistently outperforms other baseline methods under these independent evaluation settings. Additionally, we observed that generally, the value of a-FED is slightly higher FED, as reducing the amount of information available for training the embedding machine leads to an increase in error.
Furthermore, the approach of evaluating synthetic data using a pre-trained embedding machine ET is inspired by the Inception Score (IS) and Fréchet Inception Distance (FID), both of which use an Inception model pre-trained on a large subset ImageNet to assess image generative mod- els (Szegedy et al., 2016). While using a Fréchet-style distance may introduce some dependency between the evaluator and the generator, this effect is mitigated when the training set is randomly selected. In hypergraph settings, hyperlink sparsity can limit the effectiivnesss of splitting the dataset into disjoint training and testing sets. Using a larger training set that includes the data used to train the generative models for the embedding machine may introduce some bias in evaluation, but also significantly improves its ability to capture the characteristics of the full population.
B.6 EVALUATION METRIC CALCULATION
In this subsection, we provide a detailed explanation of how we compute the quantities used in the evaluation steps for both the simulation experiments and real data analysis.
Given an observed hypergraph [[] where &m {ei, (e,,62,... .em] and Xm,. (2,,,2, ,m,) ). we encode each hyperlink as binary vector e, E [0, 13", where the i-th el- ement of ej indicates whether node is connected to the hyperlink. We then concatenate e, with the corresponding attribute vector I, to form an (n + p)-dimmessional vector h2 = (e, xy), which encodes the complete information of the j-th hyperlink in the observed hypergraph. Consequently, the dataset used to train the generative models (i.e. Gau-Diff, WGAN, RealNVP, VAE) compared with our method is Dine = (hy 1=1-
Similarly, for the generated hypergraph [[m)), Em. &a), where E.,. = fēr,".. ,.,) and S. (-i,... , }. encode the hyperlinks in the same manner the observed hypergraph, resulting in the dataset Dgen 1-
Also, for the random hyperlink E and its random attributes X, we denote = (E X) as the encoded random vector, in the same manner as above.
B.6.1 CALCULATION OF FED
To calculate the Fréchet Embedding Distance (FED), we first define embedding machine ET and optimize its parameters using the gradient descent method. For each hyperlink data point hy. we aim minimize the following loss function:
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n
= (,n(())) Nl5x((u,)
log(1 + exp(u) ( 21 + +a))) - -nj(lu) 22)) 21 + ail
(u) (12)T + +2)² xz ( bi 71)
(12)T
where 2;. b;, ag Y/ represent the true node embeddings and associated parameters. We optimize this loss using the gradient descent approach in Algorithm 4, which updates 1 during each iteration. The initialization steps are the same as in Algorithm 2. This procedure allows us to ob- tain the embeddings (ET (hy))?= for the observed hypergraph and (h;(); for the generated hypergraph. Finally, compute the FED between the two datasets according to its definition, i.e.
FED(D trues Dgen == FID((E" (h,)) (E" (h,)),
Algorithm 4 Projected Gradient Descent for Embedding Machine ET in FED Calculation
Require: Initialized U(o), known embeddings and parameters {a, 7, Z, B), generated hypergraph matrix H and according attribute matrix X, learning rate 7,, likelihood weight parameter X, maximum number of iterations T.
1: for t = 1 to I" do
2: Calculate 0 (1-1) = all T m
+ (=(23) (1-1) Z, and ((-1)) = T + + B
(t-1)
(1) = (1(1)
(t-1) "lua) Vowle- = -U() + Anooo (X [/(( (11(
3:
U(0) =U((- NuKa Voml(-1) = (t-1)) (1 + fA(X f1(0 (1/0)/11 12) (H
(2)
4:
( (1-1) (2,
5:
0/0) = -((0) Myr V Uro, ((-1) -
(t-1) + Tipe (H -H - ()))
6: Project the embedding U(o
to the constraint set, with the transformation in Remark
7: end for
8:
return U(r) as (ET (hş))sem for further FED calculation.
For the synthetic data analysis, we directly utilize the true node embeddings and associated pa- rameters (2,, b,, az (s) from the data generation procedure. In contrast, for the real data analysis, we construct the reference node embeddings and corresponding parameters (2,, b,, ag (x) by em- bedding the population hypergraph together with its attributes. dataset, we analyze a training set of 2, 000 papers, which is sampled from larger population of 5, 000 papers. To obtain a fair comparison, we embed the entire population dataset to derive the reference embeddings and associated parameters (z4, b,, a4, 7:).
instance, in the co-citationship
To select the latent space for the embedding machine, we use cross-validation to determine the optimal latent dimension. For each chosen dimension k. we split the population hypergraph into a training set and a testing set, with 80% of the nodes and attributes used for training and the remaining 20% for testing. We then perform the embedding procedure testing sets to estimate the hypergraph embeddings, denoted as 4 and U Next, we calculate the Fréchet Distance (FD) between 4i and U denoted as FDx, and compare the FD values across different latent dimensions kt The dimension that minimizes the FD is selected as the optimal latent dimension for the embedding machine ET, and the corresponding (&,, by di,, 9x) are used as benchmarks for the embedding machine.
both the training and
The cross-validation results for the three real datasets are shown Appendix B.4 respectively, in- dicating that & = 2 or b 22 K3- = 2 is selected for all three datasets. This choice is based on the fact that low-dimensional latent spaces yield the smallest error, thus it provides a much fairer benchmark for comparing different generative models.
B.6.2 MEANS AND COVARIANCES
As outlined in the setup of simulated data, we assume that the hyperlink embeddings {uy}m are drawn from distribution Py. Using this assumption, compute the expected occurrences of
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the nodes well the covariances between the occurrences of the nodes by considering (EE) and cov(E; E,), and the expected mean E(X) and covariances of attributes cov (X,, (X,, Xy).
For E(H) E (F
X). we have
E H Beg B(H.U)] Erg (B(E (isn) + Beg E [X; (i>m)
e a (2 U123) + aa)) + EP [6b (112) + 7) ()sm))
where H; is the i-th coordinate of H. Similarly, for cov( cov(E,, Ej) and cov(X, Xj), we have
Cov(E,, Eb) Ero [cov(E, By U)] + covee (Eru (B; U1, Erg [e, UT)
= covpo (Epp [ U}, Epu [Eg Ui)
= -or, U(23) + ai ao) (2) (2(33) [a)]
-Bpulo(zz U((3) + a: Epg (o( U(23) + aa)]
and
cov( (3; X,) Eru (cov(c X, U)] + coVr, (Erg [x; U]. Er, (x, U) co (Bpy [X; U1, E [x UT)
Epy [(b (112) ])6 (112) + 73)l
Er, (112) + 7:) Er, [(b] Tr(22) +;)]-
In our evaluations on real-world datasets, these expectations are approximated by their empirical counterparts, as illustrated below.
B.6.3 COMPARISON WITH BENCHMARKS OF MEANS, COVARIANCES, AND FEDs
Since directly computing the expectations HH,H)) and covariances cov( cov(H; H,) based on the distri- bution Py is generally challenging. in synthetic data analysis, we approximate them using Monte Carlo integration with = 20, 000 samples. Specifically, for given set of parameters and node embeddings (B, Z, a, 7), generate test set of embeddings te = {ulj") Jj=1+ and then generate hypergraph with attributes ([n), EN, according to the hypergraph generation model. We encode [(*s[(m]] , xxx) into hyperlink sets (1 and take the sample mean B(E), [XX) as an approximation of (E)),, B(X), while the sample variance cov (E, E), cov(o, X) provides an approximation of cov [H, H), cov(X, In real data analysis, directly utilize the sample mean and variance of population data as an approximation of cov H, H), cX,(X,
To calculate the RMSE of the means and variance-covarianee of node co-occurrence and attributes vector from the generated hypergraph [[n)), &ny &rx)) we first encode the generated hypergraph as vectors {ē, { then calculate the sample means and sample covariances of both popula-
tions as E(E), E(X) and cōv(E, E), (X, We then calculate the RMSEs for each population by computing:
(E E)le, Vn
[ - E(E)
n
cov (o, (X, X) xX)
[ -EX))2
p
Since comparing our method for mixed data types with other generative models that focus continuous data, we perform calibrations on such methods (e.g. RealNVP, WGAN, Gau-Diff, VAF, and on) to ensure fair comparison. Recall that the hyperlink data is encoded as binary vectors in the first dimensions, while the generative models produce (n + p)dimnnssional continuous vectors. To map the first n dimensions of the continuous vectors hyperlinks, we incorporate calibration step based on the observed hypergraph M([m), Em,, X.m) Let {[uf. 32,3 (22.11 1 represent the generated (n p)-dimensional vectors, where y Rn and y2.5 RP for each j [m]]
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gen Denote the i-th coordinate of Vi.j by Vi and let 7, be threshold for each = 1, ...., Define yf.ji 1 (vf where 1 is the indicator function. The thresholds {Ti)==1 are selected such that the following condition holds for each i [n]]:
1 771
gen P1.1
Cjj
This calibration step effectively decreases the error from the hypergraph part, as it mimics the distri- bution of node co-occurrence in the observed hypergraph. The rationale behind ensure that the node degree sequence of the generated hypergraph matches that of the observed hypergraph. Finally, we replace the yf. with their calibrated versions vf. s, which are binary-valued variables that rep-
resent hyperlink. The calibrated encoded vector becomes D (h } { V2.;l) mh representing a hypergraph with m hyperlinks, where the j-th hyperlink is associated with an attribute vector y2..
In this way, we can also introduce a variant of ReLaSH, denoted ReLaSH, (calibrated ReLaSH), which generates hyperlinks not by sampling from the connection probability matrix P = logit(@")) but by calibrating P with respect to the training data as described above. Consequently, ReLaSH, achieves performance comparable competing generative models in terms of the RMSE of hyper- link vector means, and in both synthetic and real data experiments, we observe that ReLaSH, attains state-of-the-ar results on certain error metries in specific settings.
For the FED, we generate Diest {E" (Aaat) ) as the ground truth. We then compute the FED between the test dataset and the generated dataset using the formula FED De
B.7 DETAILS AND IMPLEMENTATION OF THE GENERATIVE MODELS
In this subsection, describe the details of the implementation of the methods used in our simula- tion studies and real data analysis.
B.7.1 RELASH AND GAU-DIFF
A diffusion model consists of a forward process and reverse process. In the forward process, Gaussian noise is gradually added the original data, eventually transforming it into pure noise. In the reverse process, denoising neural networks are trained to remove the noise and recover new samples from the data distribution.
Specifically, for both the ReLaSH and the Gau-Diff, we consider the Ornstenn-Uhlenbeck process, which is described by the following Stochastic Differential Equation (SDE) the forward process:
dU. UU + /2dW..
where { J a standard Wiener process. Under mild conditions. as noise is gradually added to the data over time, the resulting perturbed realizations will approach standard multivariate Gaus- sian distribution for sufficiently large T. The reverse process, which generates new realizations from the noisy output, is given by another SDE:
aUF = (U, + 2 Pl dt V2dW,
(5)
where V log (.(.) is the score function at time t. Since direct sampling from this SDE is computa- tionally infeasible, we discretize the process with a step size
and train deep neural networks
to estimate the score functions at T/h discrete time steps.
The Orastein-Uhlenbedd forward process specified can also be expressed U, e + (0, Ta). The conditional score has closed-form expression, i.e., V., log Pt (w)))) = -z/a; with of = - - 222 Plugging this into the general objective, the | o, + =112 which is the form used in our implemen-
e with
loss simplifies to {() = Em.....
tation.
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In Gau-Diff, train diffusion model on the encoded (n + didmenssioa vectors { J with the Constein-Ulhenbeer process as shown above. In ReLaSH, diffusion models are trained on the k-dimensional continuous spaces, with the same architecture in Gau-Diff without the calibra- tion step. We refer to the diffusion model architecture as outlined in ht https: / thub. com/ rang-song sesorceesscee __dee. The score neural network is set 5-layer MI.P, which is enough to avoid overfitting during generation of tabular data. For optimization, we use Adam (Kingma & Ba, 2014) and follow the schedule therein. The models are all trained for fixed epochs for ReLaSH, while trained for a larger fixed epochs for Gau-Diff, with fixed batch size of 128. The difference in required epochs comes from the fact that the diffusion model in ReLaSH only works on a pop- ulation of k-dimensional vectors, while the diffusion model in Gau-Diff works on population of (n + (-pp)ddinn dimensiooll vectors with l < n +-+2 p. Therefore, the dimension reduction performed by our algorithm successfully boosts the speed of hypergraph generation and enhances performance in all error types.
In the application to real datasets, employ the Forest Diffusion method proposed by Jolicoeur- Martineau et al. (2024), which leverages XGBoost, a widely used Gradient Boosted Tree (GBT) technique, instead of neural networks to learn the score function. Owing to its substantial compu- tational cost, Forest Diffusion is applied only to the ReLaSH and ReLaSH, methods, as both train the generative model in the embedded latent space of relatively low dimension. Forest Diffusion also provides an R implementation, which ensures a smooth integration with our entirely R-based pipeline. In contrast, Gau-Diff requires training directly on the full dataset of dimension (n + p), for which we instead adopt the standard diffusion model described earlier with a simple architecture. This comparison highlights the scalability advantage of our dimension reduction framework.
B.7.2 GANS
Generative Adversarial Networks (GANs) (Goodfellow et al., 2020) are class of generative models based on a game-theoretic framework between two neural networks: the generator and the discrimi- nator. The generator aims produce synthetic samples from random noise, while the discriminator attempts to distinguish between real samples from the dataset and those produced by the generator.
However, this setup is often plagued by issues such as vanishing gradients and mode collapse, partic- ularly when the discriminator becomes too strong, leaving the generator with poor learning signals. To address these challenges during training, we employ the Wasserstein GAN (WGAN) (Arjovsky et al., 2017) in our experiments. WGAN introduces the Wasserstein- 1 distance as the divergence metric between real and generated data distributions, providing smoother gradients and significantly improved training stability, with the loss formulated as:
min max Ex--pan [f - -E-=pte)((((a))
G fELip,
where f is a 1- 1-Lipschitz function parameterized by the discriminator. Unlike the classic GAN discriminator, the WGAN critic outputs real-valued scores for how "real" sample appears, rather than a probability. To enforce the Lipschitz constraint, weight clipping (or, in improved versions, gradient penalty) is used during training.
For all simulation experiments, the WGAN is trained for a fixed number of epochs, ranging from 1500 to 15000, with a consistent learning rate and a latent dimension fixed at 50 to handle synthetic datasets of varying complexity across all settings. each real data experiment, we adjust the number of training epochs based on the dataset's complexity, considering limited computational resources, and have saved the specific settings for each dataset to ensure reproducibility. We observe that WGAN exhibits stable convergence under these configurations and is comparable to ReLaSH in terms of run time. We apply the early stopping mechanism which monitors the generator's loss, halting training if there is improvement for patience epochs to prevent overfitting.
B.7.3 VAE
Variational Autoencoders (VAEs) represent class of generative models that fuse principles from autoencoders and probabilistic graphical models (Kingma & Welling, 2013). VAEs learn an approx- imate posterior distribution over latent variables by maximizing lower bound on the ELBO. The model consists of an encoder network that parameterizes a variational distribution over the latent space, and decoder network that reconstructs data from latent codes. The loss function includes
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both reconstruction term and regularization (KL divergence) term encouraging the latent distri- bution to remain close to a chosen prior, commonly the standard normal distribution.
In our experiments real world datasets, observe that VAE training is significantly consuming compared to WGAN and ReLaSH. Furthermore, the quality of generated samples, as evaluated by our downstream metries, systematically lags behind that of the other generative ap- proaches in our comparative study. As a result, we exclude VAE from large-seale experiments involving scalable m, n, p, K to maintain feasibility and focus our analysis on more tractable and performant alternatives.
time-
B.7.4 FLOW-BASED GENERATIVE MODELS
Flow-based generative models provide exact data likelihood evaluation and support efficient sam- pling by learning a sequence of invertible (bijective) transformations between the data space and simple latent space. The central concept is to represent a complex data distribution by mapping observed data x to latent variable via an invertible function f. so that (=) The density of data points can then computed exactly using the change of variables formula
logp(x:) log p(2) + log (z)l
where - 1 is the Jacobian matrix of the inverse mapping. Typical flow-based models include NICE (Dinh et al., 2014), RealNVP (Dinh al., 2016), Glow (Kingma & Dhariwal, 2018), Masked Autoregressive Flows (MAF) (Germain et al., 2015), PixeIRNN (Van Den Oord et al., 2016) and so NICE and RealNVP affine coupling layers and can be flexibly applied continuous vector data of arbitrary dimensionality, while Glow, i-ResNet, PixeIRNN, and some hierarchical models are specifically designed for image generation and utilize architectural components that exploit spatial locality and dependencies within images. These designs are less suitable for plain vectors where such spatial structure is absent, making direct extension to general vector-valued data challenging and sometimes inefficient.
In this work, we select RealNVP because of its modular structure and its generalizability to vector- based data. For all experiments, the RealNVP is trained for a fixed number of epochs with a con- sistent learning rate. For optimization, we use Adam (Kingma & Ba, 2014) and follow the schedule therein. To enhance the robustness of the algorithm and prevent overflow of values, we apply stan- dardization in the training and generating process. Also, we apply the early stopping mechanism which monitors the generator's loss, halting training if there is no improvement for patience epochs to prevent overfitting. The run time of RealNVP is about double Gau-Diff, while the metric results quite similar the compared methods.
B.7.5 TABULAR DATA GENERATIVE MODEL BASELINES
To the best of our knowledge, existing generative models designed for hypergraphs do not handle data-generation tasks, whereas tabular data generative models can be applied. Therefore, in addition to including well-established models such as GAN, VAE, diffusion models, and RealNVP, we have also incorporated several recent and high- performing tabular data generative models in our comparisons.
As summarized in Table I of (Stoian et al., 2025), most state-of-the-ee tabular data synthesis models were previously tested on datasets with fewer than 100 features, and their training and sampling processes are generally time- and memory- intensive. Nevertheless, many of these models can handle mixed data types, which aligns well with the nature of our datasets. For our baseline comparisons, we include ForestDiffusion Poolicoeur-Martinee
al., 2024), TabPFGen (Ma et al., 2024), CTAB- GAN (Zhao et al., 2021), CTAB-GAT (Zhao et al., 2024), and CTGAN (Xu et al., 2019), all using their official implementations and default parameters.
To assess the computational resources required, we first trained each method on the recipe hyper- graph, which contains 205 features. TabPFGen and CTGAN require approximately twice the run- time of ReLaSH, while CTAB CTAA--GAN requires about four times long. CTAB-GANT GAB++ and ForestDif- fusion are the most computationally demanding, requiring several hours for training and generation. The error metrics for these experiments are shown in Table 3.
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Among the models, TabPFGen has good performance and is comparable to ReLaSH/ReLaSH, while the other tabular generative models perform similarly to the standard WGAN baseline. However, according to the official implementation, TabPFGen is limited to handling tabular datasets with at most 500 features, whereas both our patient profile and co-citationship datasets exceed 2,000 di- maissions-ddmosstraatiii the advantages of ReLaSH in terms of scalability. All the other reviewed baseline models fail to complete training on these larger datasets due to timeuut or out-of-memory errors, highlighting the efficiency of ReLaSH for hypergraph data generation tasks.
C PROOFS OF MAIN RESULTS
Proof of Theorem I. Suppose two sets of model parameters (PPv, 2, 0, 7, B), (Pu, L', .. B") yield the same distribution for hyperlinks and attributes, i.e.,
22(23) a a' ('U(23) and 7 BU(12) _ = B'U
First, by taking the mean on both sides, we obtain a' and y = 7/, thus we have
22) d 7'U and (U & B'
Next, we demonstrate that Iy and Z, B are identifiable up to sign permutations. Suppose there exists an invertible matrix A such that (23)) A (23)), and define Ep, [((33) = D and Er,,, [2322) 1 TT z' D', both of which are diagonal matrices with distinct positive diagonal elements. Thus, we have:
1 - n
ZA- 1 ZT z' -Ee (
A
71
A Er (23) (233) [2 = A DA.
which implies that
D DAA & (MDA)
where M AAT. By the Cholesky decomposition, conclude that MD1 = DåW for some orthogonal matrix W, meaning that W D- MD!, and 1 == WTW D MD! i.e. M² = I. and M is positive semi-definite. Therefore, M = AAT = 1, A must be an orthogonal matrix. Furthermore, since A DA D', where both D and D' are positive diagonal matrices with distinct elements, we deduce that A must also be a sign permutation matrix. Therefore, we conclude that (23))/ L (23) and Z' ZA-- i.c., Py and Z are identifiable up to sign permutations. To resolve the identifiability issue up to column sign flips during estimation, we fix the sign of all coordinates in the first row of Um-
Next, suppose there exists an invertible matrix F such that U(22 - FT FT and B' BF-T.
Fii
F12)
Let F
so that:
F21
F22)
uay 1 PLU + F and (22 FTU + 12)
Since have already shown that (2) d T2M it follows that
0 = Ep [ Ep, (10) (2)) + Ee, (122 U = F Ep, 2(2))
EP (1 ((2)T + F Br U0 (22) = Frg (2))
thus, F21 0bzxky, F22 Ta and F12 x
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For the loading matrices B and B', we have:
Bu
B21
B1
Bin
FlT
0
Bi Bi
B'y1 B3n
B = B'F =
B12
Buy
B12
B/n
0
1iz
=
Fiz
which implies that B/21
B21 and Blz2 -B22-
Bi1 Bi2
B11
Under condition (C3), we assume that Bi
and B1 =
both contain unit lower
B12
triangular matrices, i.e. Bi and Bi are both unit lower triangular matrices. We conclude that Fu is unit upper triangular matrix because B1 4- By checking each of the diagonal elements of B1 and fi we can conclude that all the off-diagonal elements of Fn should be 0, ie. Fu1 = Ix.- This leads us to conclude that U(3) and B are both identifiable, and the strict identifiability holds for all parameters.
On the other hand, if the identifiability condition is (C3) B Ig. it follows that Fi = 13- It's also easy to show that with (C3**), Fú will be a sign permutation matrix, and the identifiability can be resolved by arranging the sign of the first column in each matrix. In practice, we use (C3) or (C3**) instead of (C3*) because it offers a more natural estimation procedure.
LI
Remark 5. In the case where the true parameters (Um: Z, B, ox, ox)7) do not satisfy the conditions in Theorem 1, we cun apply the following transformation w.r.. the observations.
Firstly, compute e and @ as defined respectively. Define a² = a + 1 2U.23) 1m and 7
+ m BU Ims and let U' Um
Secondly, we transform (Um, 7, B) such that @x and Q remain unchanged and the orthogonality condition U 1U Un is satisfied by orthogonal transformation. Specifically, let
T( (Im -1( (10
T(3)" = (Im (U -U Ju(3y.
To preserve the matrix product, we update BATT = B + - BJ and Z
2 + (
U(3 Then combine (233)" = [U [ U(3)).
At this stage, we need to rotate (Um, Z. B) to satisfy the conditions (C2) and (C3). Let V
diag(o oh. where of > o > obg are the eigenvalues of
1 (Z (U (23)TT ((23)" 12212 mn
and let T be the (kg + ks) (he2 + k2) matrix whose columns are the corresponding eigenvectors.
Define U(23)* (233) and Z" Z'G-T, where G (Z" Z' / 1/2py-1//
Similarly, let W = diag(1 (57. 82,
52,) where of > 82 ... > o% are the eigenvalues of
1
B BJ)
BB
mp
and let D be the kı x ki matrix whose columns are the corresponding eigenvectors. Define
(1) = UO'" K and B = B,K-T where K = (B, /2 Finally, we take B== BJ}, and U [ [7(33)].
Thus, we obtain the transformed parameters (U*,B*, ,,",,",,",7"" and the transformed parame- ters of distributions (ox*, HH*) (ex G²), meaning that the distribution of hyperlinks and attributes remains unchanged after the transformation.
Remark 6. In general, if the true Py and Z, B, a,7y do not satisfy these constraints, a transfor- mation of Pu, Z, a, can also be made so that the constraints met, and the distribution of hyperlink and altributes remains unchanged.
Firstly, define a + ZTZ (U(23) and 7 BER (U(2))) and let U' - UFEro (U].
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Secondly, transform (Pu. Z, B) such that the distribution for hyperlinks and attributes remain unchanged and the orthogonality conditions Exu (0 0k2xk, Be, (U()) 0k2xks are satisfied by orthogonal transformation. Specifically, let
UQ = =U( Erg (U(1)) [[ - U U Epc ( (EEr (
To preserve the matrix product, we update B3J B.j (Brg ( ( [
and 2 Z +Eee (Uee)v(e Eeey I1
At this stage, we need to rotate (Pu. Z, B) to satisfy the conditions (C2) and (C3). Let V =
diag(of .z...., T )g)) where of > 02
ohzkks are the eigenvalues of
1 -Z zt) 2 (Eru (U u (23y)T DIZ" Z²)
and let I be the (h2 + k3) (A:2 + k2) matrix whose columns are the corresponding eigenvectors.
Define ((23) G U(23)" and Z Z'G where G (Z' Z' /n) 1/44
Similarly, let W = diag (8. 82 . where of > 83
(2,)
5% are the eigenvalues of
(B B1) (Ee (0"' D
()B (2)
and let D be the 11 x k1 matrix whose columns are the corresponding eigenvectors. Define ( KJ UT)) and B₁ B,K- where = (B, B1/p) DW-1/4
Thus, we obtain the transformed model parameters (Pú. B, 2, 7), keeping the distribution of hyperlinks and attributes unchanged after the transformation.
Proof of Lemma 1. In the following, slightly abuse the notation by letting p(x) denote the count- ing measure when æ is discrete, the Lebesgue measure when æ is continuous, and the product of the counting measure and the Lehesgue measure when involves both discrete and continuous compo- nents.
We can that
dP (EXX,) E.U
(E, X,)
dKL (F = BP
log
(E. d
(,,XU)
dP' (( (X B.rl (E, E, U)
(U)
dp( (E),
du(U)
BP.E.X.D
log
(B). Zn.a..B.
(E, X,U)
(U)
dp(H(())
dp(U))
(P) (U)
log
dp(()
dRL (Pu (Pu))Po)
(U)
(U)
dp(U)
Proof of Theorem 2. We're going to show that
dKL (IP (E,,),) || (B,x,))) (z,,, estimation where the three components given by:
A Ae-estimation
Jassnttrooossrruction
dP Reih (8), ( B.ain)
F,X,UU
[p [E) (X)
A (2,, B,057) estimation
E
log
dP
(E) 8.6.5)
E, (F,XU)
dp(H((m)) (EE,.E))). (E)
d'' dP An(U)
dP tim (U)
dFu AutUy
(U)
APc estimation
ErG
log
+ Er
log
Bra
log
dy((t)
oF du(UT)
(U)
An(UF) (U)
de'
(u((F) (U))
de dp(())
(U)
Masont-reconstuciionn
Epan
log
dPps
(U)
du((T)
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Here, Pú... denotes the marginal distribution of the estimated attribute embeddings (ār,-.. Um } given the observed and we assume absolute continuity of all log-ratios with respect to common base measure μ..
Note that
dRr. (Ec.(
((EXX.U)
(E, X,
dp(E.X,D)
(2
log
(6.8.0) dp(E,XX0) (E X,
(E) (XIM)))),,.)).....
U,) (U)
(e..0
log
(B), (X (X))U 2, 6,6 6,6,5)
(E,X,U)
du(() (U)
dP
dj(H[((m), (E)((())
dP (1) KIII
(E,X,U)
du(Hi (M(H(())E)E...C (X)
(,,00
log
dP
(B) (8
(E,X,U)
(E),
dPn d dP du(() (U)
(U)
+ (e..0
log
Here, we denote
dl' d['iill. (2) B.a.))
(F,X,U)
du dM(H(
( Zn, R,0,))
E EP(E,x,0)
log
(1 8.4. (E,X,U)
dy( (m) (B),
since (X (E,X,U) and PP (S 0...1 (,,X, U) only differs in ( (E),({))
(()
the estimated version and the true (Z,, B, a, 7) with the same E. plug-in.
To decompose the second term, we consider
deu du(UT dPF du(U)) (U)
(U)
|
deu
(U)
iin (U)
UptUy dP tm dp((i)
du(l)) dt' du((i) (U)
log
EE
log
Ep
log
(U)
dP (U) du(U)
(U') dp(U))
dPu
Erg
log
+
EPG
log
dl am dp(U))
(U)
dF' dp(U) (U))
de tyg
dP Pim
(U)
(U)
du(O) dP du(U)
du(U) S
Eran
log
Ee
log
im
du(U))
The first three terms primarily depend on the distance between Py and 12,.... and are collectively denoted as Ppy-etination The final term corresponds the diffusion error observations of Pit..
and is denoted as Alscnttccoossnnctionn
Proof of Theorem 4. Denote H (mn and X {x the hypergraph connection matrix and the attribute matrix, respectively, where hy: = 1 (KEe;) and zy represents the i-th attribute associated with hyperlink ej- We consider minimizing the objective loss function
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B,,,,,,a,1) and the optimal solution (U.2,B.a, implies that 0 < L(U", T B", a" ,7) -0,, a, fn((
M 12
+ A
3y (6) - (x(0)
/=1
66 0 f (hye - 11 - #0
1
m "
+
Σ j=1
( (2x - - 0331
where 6), = a, + (1 0 and 6 = 6 + +1- for some azi. bys E [0, 1]. Here, we have (x) = log(1 + exp(x)), and fx(x) _/2- Denote Eμ H f'1 () Ex X fx (0x)). Since we require that both true parameters and estimated ones should belong to the feasible parameter space F(O), we have the above equation transferred into:
1
2 min
min (1
(6 A) 10
1
S [6 - 11 A
min
2
2
11 > +X < Ex, Ox Ex, - -0x'> << En, 6" --
S |E |Enllzrank(( (6 - 6 @ ||
1F + A| |2rank -xx))//x-- -1x11|1
S 22kk2 + Ka + E 11 + 2(k1 + 12 1) Ex 1 0H + +110x - 113
/2 + Ka + 1) 113 2(k1 k2 + 1x -- ||
Note that ,, f// (6 > exp(-Om.n). we should consider the bound of ||E Exlle.
For [HHl2, we prove that EH 2 Vmm V exp))xp((e) (m
S Let
Omxm
Eн
Σн =
EN
0nxa
we have Amax (ET,E)) = Amax ( [ET EH), so that | la = |HH12 We introduce Lemma 4 to bound it.
Lemma 4 (Theorem 5 from (Chung & Radcliffe, 2011)). Let X1.X2, Xm be independent
random n X n Hermitian matrices. Moreover, assume that |Xx, E(X,)ll S M for all and put 2 = Σ Var(X.)ll2. Let X = E, then for any >> O,
a
P(IX B(X)122 > a) < 2n
2v2 2Ma/3
Let E be the (m + n) x (m + n) matrix with in the (j.2) and (i.j) positions and 0 elsewhere. Therefore, we represent E. = Σx= Ei= yim+ii E/e En (hze () Note that jyj.mii. = 0[m+n)x(mtn)) yymttill S 1, and im (o(0))))0(())))))) Em+im++i) thus
R(y max max jE[m] vE(n)
Σ a(0)!1) ? 7E(m) je(m
Σ -
max
< (m V maxo 3,
53

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For any 6 > 0, we take a = 33 V n) max,, ( log(4(m Vn)/e), and note that exp(am.n) 2 (m A (og(m An) implies that there exist large enough m, s.t. explōm.n) Z (m A n) log((m (m A n)/e), and max;,1 ( 2 (m A n) lo A n)/e). Therefore, (m V n) m ( 2 3(m V n) maxy, (0)) log(4(m V n)/e)/3, and Lemma 4 im- plies that
P(|EHH12 2 3(m V n)) max log(4(m
3(m V n) max,, o(0 log(4(m vn)aa)
<2nexp(-
2(m V n) max (0 + 2 3(m V n) maxx, (()) log(())) n)/s)/3 S 4(m Vn) exp( log(4(m V n)/e)) =
Moreover, note that exp(am,n) x maxj, (00 there is
P(| 2 / (m n) exp((m (lg S,,
then the desired result follows.
For || E22, note that the noise term is sub-Gaussian with zero mean; therefore Theorem 4.4.3
from (Vershynin, 2018) implies that
Ex l2 2 CK(Vm++ + + + +)) < eexp(-t²)
for some constant C, and we also have K max,, | = maxtelp) ||e ll as bounded constant, since the noise term is assumed to be Gaussann,, therefore we have Ex ll2 Op(ym + vp).
In all, with the assumption that exp(am,,)) x
we arrive at the result that
- 0' = =0 (m Vn) exp(ammn) log(m V n) 4 V p)
A A)
Proof of Theorem 3. By taking the hyperlink and attribute E, X, and the corresponding embedding U as random parts of the random function, we have:
E (s) = Ei(zi, 0z E. UU) 1. 24 az) log(1 + (23(U 2 + a;)),
1
(2( 1x) = lo 7 E,X,U) 21(11 + ((2)T b.)
マタ ( + (12)T
where b; denotes the i-th row of matrix B, so as the b. and denotes the i-th element of the attribute vector X.
Then we expand two error terms to the second order:
24 2 a,
1
2 - 24
2: 22- & a; a; à:)
(1 oz)) (1( = Ver
+
à;
2 a; à;
b: 0
b 6,
bi - b Pi 9/3
E2(bi, m)) - l2( 7/) = (b,
22
%1)
7/ R/ 2 7/ 7%
for some (2,, āi. being the linear combination of (zi, os)) and (É:, â.)). Note that
(23)
(23( (23) T 2 + a;)) 1 (U)))
(233)
(ei ās) = lieF
& + à.)
exp(U 24 + āz)
(123)
(233)
T
(() = (1 + exp(U(23)T b + āz))²
1
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for the & error, and also
(Ura))
Vl2(be, = -xx; - (fx + +.(2
(UG2)) (U02)
(112(b,,%)
1
To consider the
A (,,n) estimation
dP (x)
(E, XU
(F),
log
dP (() 3r, B.6.9)
(F,XUU)
(E),
dP (((0),(1)11222) Zz mo)
(,, (E,U)
[X n.n)
(X(H(()) (X,U)
=BP(R,X,0) log
+ +log
(1100)18110020 2.,
(E,U)
(X, U)
du(H((m),EE))
dp(H((m).(X))
= =EP(s,0)
(1(2),))) (1(2,, (E(n)
Σ
+ Ex
(2) - ( 91)
(E(p)
we should analyze the expectation over (E, X, U) by first conditioning on U, i.e.
EP 16 (zi, a.) - (1(2)
-Ee Ez ((( ((( - (1 a.)l
exp(U((2)T 24 + 0:))
exp(UU(23)T 24 dz) + (23(( 24 + a.)
(U(23)
2 2
< Eee
1 + exp(U 2 + a:)
a2; à
1 exp(U 2 + 6:) 2 (1+ + exp (23)
2 - 2/
T (U(23)
(2(33)
T
2; - &
+ Ezv
2 + )2))²
ai - à
1
< (C + 1)1/2
21 2/ [1
(23) 2j + as)
exp(U (23) T 24 + 6:) 1 (Xp(( 2 + 6:)
EFU
as à
1 (Xp(((23) 2i + a;)
2
(T(23))
(T(23)
2 & (or - as
+ C1
LBFU
(1) 1
where we can further expand
(23( 24 + a4))
exp (U33) 2; + 6.)
Epg
1 e exp(U((3)) (U(33T + 03))
+ exp(U(23) 2, + 6.)
(233) 2/ + a()
(1(3))
2: &
(1 + (23( 2f + a{))²
a; à.
( ||
((23)
4 &
){s
S C1 exp(amm.))
a4 az
Therefore, by combining Assumption 2, we have
12 (1 - (2,2 S exp(amm))
2 ( 21 a; 19--0 à:
55

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And also, for the attributes part, we have
Bp Frxx,,112(br n - E2 %))
((12))
h3 bj
E2g BB
(x, (8x b.))
x(0(12)
1
7/ 5%
b:
6,
(U(2))
U(22)
by b
+
7/
9t)
1
1
7i 9/)
bi
be
T
(001))
(U(22)
bi 0z
Bre
2
1
11 9/)
(102))
(
T
bb de
|
b;
b %
Amax
RFF
2
1
7/2
7//
7/
2
By combining the results above, we have
A (Zn. BBa,m) estimation
(E.))
Σ
(s) e1 di))
+ BFixo
Σ Ea(bis -2( 76)
VElD]
Er Sttm,,/C((ss
l1 dssl +
Σ
Ep(x,o) (2( 76) E2(br, 9:)1
iE|m]
Ve pl
|
b; 03
2 eepp(ām.n)
Σ
2
+ Σ
0i 0:
7
9i)
2
= exp((mma ) [11
1 + l a + B BI? + 119 -711),
then by combining Theorem 5, the desired result follows.
Proof of Corollary 1. First, due to the identification conditions, we have
[x'
θ1 m.
0''m.
7
thus
a
1m 1. (6 -7)|
1m
1m
i.e.
[ [
a
Ô 0r1lF Op(6m.n.).
< mm
As a consequence,
([0 2 - (223) z" zz 1|F
= =1((" - ) - 1m(a - d1) < 11@0" -100"0000]11 + 11 aaa)) T1) 1|F 2 - *H**)| |||r Opl Op(vmomm,..)
Let 02, rkg and ,2, ... .kgtky be the singular values of - U1233) zTT and (23) 2T respectively. Further let 61, "., Ckatks and é1, ê2, Ckktks be the corre- sponding left singular vectors. Under the identification constraints, the k-th column of ((23)+ is m and the k-th column of (23) is emōgé..
mm
According to Theorem 3 of (Yu et al., 2015), we have
22
|lex - êxll2 S
(23) 2T - (23) (TT)
min - of o
mm
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and by Assumption 2, there is |lex êxl12 Opl vi mm, (p
for any S k S l:2 + Kss
Also by Weyl's inequality, lox - oxl < |1 / ((:(23) 2 ( z
Mlop ZTT Opl 1 pm Let
Vmn
U=(23)= (((23) denote the k-th column of (1(28) (2(3)) respectively, we have
tka
[| - 1123) S
10 (2223))
||2 = m (123)) b
lones - ōRēxl12
tks
S vm
(Ivos - Voxl + VoR - ēxll2)
k=1
tot
10x -xxl + Voullex - -Eallz 2allz)= =Op(Vm//nmm mm/mmm..p)
S vm
Vox + Vōx
Similarly, we can prove that space.
1 1= = Op(6mmnp) by considering about the right singular
On the other hand, we have
(002) BT (12))* BaT
|| =1(@0* -0x1) - 1m 135---7) 7-)
$ =1(@a~ - 0x*)]==== + +111m (9- )²) 1=== = Op(Vmóm...))
To consider about the error between B = [B] Bg] and B" = [Bi Bý}, we construct the error rate for B1, B2 respectively. According to the orthogonality between ú()) (2) and U(1) U(2), we have
|B2 B2 = (1 BT - (1 (1
(12))) B ||
$ 110100001))) - - (1
(112)) -
(112) B* |F + (* 1|2
| BT (12)) B ||P
bath mmxx
+Cill(((0T (U: -1 (2)) |F
= Opp6mmpp):
because the rows of (((()) ( are
62
mony kg and the rows of
Vmnd
Vmdz
((()) are v Vmooi
Denote 01(2) -(2)- the k-th
maa
row of (22)),2(2)0 k = 2, ... k2 Therefore, with Theorem of (Yu et al., 2015), we have ( T (22)))) (2()) ((+2) ) ((((2) |F
h2
(2) b
(2(2)* 1|2
1
1
1
Ek
vm k=1
Vax
Vox
2
1
1
|lex - Exll2
v m
Vok
Vok
Vok
k=1
10x
0x|
_ lle éx1l2
1
+
m
VOLōv Th + Vox)
Vok
Opp
v Vm pm,pp)
Therefore, with |12 B3 || F, we have
110((2) BJ (2 B,_ || s ||0(2) - 12/1 + 10 - BB ||F
||2
S V -00
+ max kE(k2]
/ Vmook BT - B3 T ||
= =mmmmmmm...))
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and finally arrive at
1 - u B = 11(0112 - -U(R)BBT) - (0
- U12) BB }l
< 110002) - U112) }||F + (1(012) B3 U1 B2 )|
= Op(Vmmmmmm.)
Similarly, we denote σ1, 02, .kk and 81, 22,02, .. Ok be the singular values of U BT BJ respectively, and further let /2, Sk. and fi, f2.... Súr be the correspond- ing left singular vectors. Then under the identification conditions, the k - th column of ((1)* is v vnd lu. and the k - th column of (111) V mmôrfi. By applying Theorem (Yu et al., 2015),
/mp
and vmp
we have
f1 fella S
V2
( - U -UWBB] BT)
=pp( 8mm)p)
min 18 82 8-1 - 1%
Vmp
vp
for any k 1,2,... ... .1. And by Weyl's inequality, 16 8u1 S 11 Tmp
(T BT U BTD
Op( 1 /p pm) Let "² (( denote the k-th colunm of U ( respectively, we have
k2
1000--001 -
10 t-20 122
1 - 8alall2
v
m
k==1
$ vm
(1 5b -Vael Voallfa -fal2)
k2
18 8A1
+ Voolll fulla = =Opl (0ppVm/pSmmm.pp
m
Voh + Vōa
Similarly, we can prove that | Bille = Op(ōm, by considering about the right singular space. By combining the results together, we have
(vm(u + np
- U = Op
=0p
kmmm)) | - Br = -pp(5m
Proof of Theorem 5. According to Assumption 2, we first prove that || O (( for both 1,3. By directly calculating
1
Σm
-= uj
(2)1 (3)TT
(2)
(1)T
F|
2
Σ
7 T 1|
m
m
m
j=1
1=1
m
m
m
Σ
u 0
(2) T
(
u
(2)
m)
||
m j=1
j=1
1=1
m
<
(2) (8)
[F +
Σ
Uj
(1)
/m
Σ
u
(2)
m
= O
j=1
==1
#=1
there is also
1 (22T (( m for any 1,3.
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Denote (U*, Zz, BB, ."") as the transformed version of (U, Z, B, a, accordng Remark 5. Then we can analyse the errors as follows.
m
a|a a212 = m
(2))
1m
S ||
||2 O(Vrio m..)) "j
(23)
m
j=1
m
11 - 7/12 =
(
m 112 -
B||2
(12) 1 0
((pommpp)
m
m
j=1
||U(23) U(230 P < ||:23|11 (2)
||u(23) ((23)²1
< || G (23))) |F + ||0 - ((233)
||F,
1000) - 10 | $ | - U101 |0 00'
< | || r+ ( -U(1) ||
[ Bıll 1 B - -B1 5 |1llr| K- --11,r
| r| - 1 - 2 Z|r = || T
|| p
1 - -|
Note that G, K can be decomposed into
|| 11(2' |I(z'Zz/nn) 1//=
$ |l
+ 1(Z z'/n) 1/2 1/21 1111/
|||| [[B] B1/1) /2
$ [[(B] B1/D) D 1W- + [I B1/p) 1/2 |/ 1-1/1 |r.
We first analyze the error bounds of G, K related terms.
With Assumption 2, we have
(Z 22)2/2(( (1135))7((223102 21)2/22 - -Z z' /n)3
(Z 22)1/2(1 ( /mmm 2 Z/n + T Z/n -T z' (T) 21)1/2
|1(ZT Z'/ 22/n)1/213 || ((33"' ( /m- 7. 7/n|l 1
Z/n 7. 7. /nllF
-| z' \1/1112 u - Z F + | Z/n - z Z1/nal
By Weyl's inequality, we know that the 2-norm of TT Z'/n can be bounded by both -| Z/nl|22 and permutation term, i.e.,
|| 7' Z S I Z'/n Z 2/nl12 + || Z/nll2,
where
Z/nl|l12 O(1). Note that 1
1 -T
ZT 2/
Z3 L2 (72 22)
(2 2Z
2" z
n
n
n
7g3
0
1
[ng 7 22 + T22 232[23222 73. Zo T22 Z3. ZgT
T22 I. Zg Zg
23
n
Zg Lzl 23l
0
where we denote 122
= 222=
UT3 U12 hence
11 - TT z'
1
Z||F ZTIl= < -
zT
111 Zzl 22
+
1
22 23 2g 23T22TT2x 22122
11
n
n
n
4
1 1zz Zs
_
ZallF
T
T
22122
||
[ 23 2
Tza
n
2 )
according to Assumption 2. Also, here we bound ||222122 2 by Weyl's inequality, i.c..
|| ll2 = Tmax ((U (12 (119-)))))))))) = mmin (U(2)T Omin ( (2) U122)
- m
o
m.m..
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Therefore, Z'/ll2 (1). Next, we consider about the error bound of [[ ( /m- z T z/nl| by decomposing it into
[| ((23)" /m - -- 2/ 2/nllF & |1 ((23)" //m U/23)T (23)) /mllF
+ ( U(28)) /m 2 z/n|FF,
where the latter is of (( For the former, we have
U ((23)T U (23)) /m (23yyT ((23) /m
0
U - (U(8M-U113)
m (1(3)) - -U( U(2)
U ((3)" - U(3NT u(3y)
1
0 -U(3)) U@y -U TU
U
whose F-norm can be bounded by considering
|| (1(3)) /m - U223)TT ( /mllr
(2nayin) "l' + |10 1 0 (0mm.n..) m
To sum up. we have
(Z" (2)1 ( (2 z1)a/22 - (ZTTZ//n)
= -O(o
Therefore, by Weyl's inequality, have
10 oz z' 22/n)²)1 S O(omm..))
and accordingly,
10t 0i (//T///))) ( pmmp))
|p: 1o=-o((2TT (ZT)/////)
Pi 0,' 81/n)
lp: o,(ZTT 22/n)] 0(6mmm.))
1/2 - (2 Z1/n)
P. 01((Z Z'/ /n)2222)
lp; 1/2 - (1 Zz/n) 112)] 1p, 111p (( Z'
22/n)-1/2) -((mm
Regarding the singular vectors I, we further denote I as the singular vector of (Z z' /n)²² and apply Theorem from (Yu et al., 2015), then there
[ - F| < || - |' + || - 1|FF
= =
(T (1
2'T22/n)
+ |I(zTZ'/n)) (z 2/n)llF)
0 ).
Regarding the singular value matrix V, we have
|V- 11-1/4
$ 1--1/4 (Z' 21/n)/ ||- + 11 2' 2//nn-11/ 227 + (2T 2/n) 11/2
Vkz 12 + Ky xax
lp;
11/2 (zZ' (("TTz//)'"aaa (//m)~1131 || 22/n) (2 2/n) 2//n-1/211F
+ +1 2/n) -1/2
-()).
To sum up. we have
| < |1(ZT 22/1 72/m))
////n///11 1/ [I| /|p + +1)111-1/411
=|[(Z' 1 (Q| - -1]1= -1
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On the other hand, should estimate the error bound of ||z G- and ||G I| in order to bound || - Z| and ||U - - U..... Note that
|| G 1 1|
G
G ||-
where
12/n///)-1/11 - |[[||
| /|1> + 1)) 21/n))1/2111 =11))=/== = O(1),
I G|| = -Z'TT
+ (1 -|
[1(ZZZZ) 222 1/2 F -1- |
|1
Therefore, we reach at the result that
|l Ie
G||r (
Similarly, we estimate the error bounds of || k|lF and || - K-T
|| by considering about
B (1)0))(000000000M00 B1) - (B) B1/1)
mp
(B) B1)
(ay) B₁) 1
mp
[ B1 (1 BJ Bi (B B,) 1/21
1
+
p
1 |B B/BB/ppalluuu - -
m + B.plelllu/ E- UAUDTTI 1|
where
| 10 ( (010 (
S 100NT U12) 1191 mmmom.n.)))
Therefore, we have
- (U (BT (0,))2-BBBBB///)) (1/)) =0 m.p mp
By Weyl's inequality, and note that W = diag(of, , ,2) is the eigenvalue matrix of
(BT B,) (2) ( U 12) we have
15 - ((B] B1/1) O
10? ((B] B1 B1/p))
16: - (B B1 B1/p)al=
O(om,np))
8; + + BI/P)a
16: (BT B1/P)aa
- B1/) 13)al
Ol.m...)
6./2 + ((BT B1/p)//2)
Similarly our analysis on G, we use Theorem 3 in (Yu al., 2015) and obtain the error bound of singular vectors as
|| DI[F = ( mp
Regarding the error bound of || K] and || K ||,, consider
||W-1/*|l < |W-1/4
111/44(BB) (BJ B1/p) 11/2 || + | B1/plle O(1),
||||| < 11/
M|| + [| B1 ||/" 00));
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and then
| - KllF
11 (Bi B2/D) DD
- |1 - 1B ( B1/p) DD - /||1||11 1/
< IIBBT (1/1) - [|
S O0
Similarly, there is
| II (B. B1/D)
S ||W [ |
|| B1/1)-1//11 |/
O(1),
and then
|K-K F KK |||||| 11=111 K||F
0 m..p))
Now, we go back to the analysis of error bounds for embeddings and parameters. According to the process of transformation, there is
||0:(23)= U(223) | S (23) (2 || || ((23)² 1|
I I G||=|||((3))11 || + ||U(23)) - U1(23) ||
< [| Gll ((1,((1) || [|3 U (|=) + [| -U((3) ||
where
||3( - m 1/
|| U 1/ || - 1/2
< 11 - 1 + (11 - U(3)) IF + || - m
- 1|)
11 U 11 + |0 U 1/- + (((3) - U 1/ 2 - U || 11 U |
< 110
U 13 + (3yT | |1221 + 13 (3) - 11 + 2 U (3y (3y 1 -
11
= Ommo m)
thus
(1123)--U((2) ||r - T [|- ||2 [|00(33)0 (0) ||])+|| (23) -U(23) || = 00 OVVmoom.p))
Similarly, we have
||0 - 12 I || - (T1) F + [1 - U())
< T K /1I=||00000 + | r0> 0( OV
therefore, the overall error bound of hyperlink embedding U is
U Un || |r v / U [T(23)) U23 12 O DVmmommm.)) For the node embeddings, have the following:
1| - - < - Z'll= + Iz' 2 1 G |F 1|' Z1IF,
where
11 - 2 = -1 - 12 < (0 / -1 ((3) 2all F
< [ 22 2| [ |
therefore
2 - Z||F F = O O(Vmom...)))
Regarding the parameters B, note that
| Billr < [B, K- Bill || S B |r IT
K-Tl|r 0 mm
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and similarly |B2 B2ll O n.p) during transformation, have
|B - B|ir v I Bill + Ball = O O0 m..
To up, have proved that for the transformed version (U*, 77" B 0², y²",, the enror bounds are as follows:
||U UmlIF O( m(Tmmmm...)) 1 Z|FF O B BlI O (Vmoom..p)
||a" all2 = ( m(. 11 - 7/12 OlvToommm..pp
Combine the above result with Corollary I, then the desired result follows that l|á - a²112 Op(6mmpp) I10 - Ur = Op (0mmmmrz mmn p)/(n))mm..)) || Z = Op(0m,p.))
11 - 11 Op(mm.pp)s | 1 || F Op(6m.a.p)
Proof of Lemma 2. Note that pr (u) according to the definition, then only need consider about the integral, i.c.
J Sappotes Pun (u)du(u)
{m / (welh())) zmmmp with (u)+ 2mm.n.p
-mmpptt))
pu pu ddp(u)))))))) (dj(()))))))))))))))))))))))))))
{ (aut(()) pimm..
Tmmmp tmmo
Pu
w A.E.c,mmm c.mmpa.p
2.m.n.p
Σ
(u (ul)ddd(()
(wda (x)) semimm (M) smmmm
whEA
Csmman.
Pu (w()d((())) 1,
supp(Fu)
then the result follows.
Proof of Theorem 6. According to the mean value theorem and the condition of Lipschitz continu- ous, we have
A PU (u) = Tmm
Po (u)du(u)
((a 2mm.n.p wi (1) 2Tm.i...
mmm Pm Pu (c) = (U(C)
for some E [u 2ym.n.p" (u) + 2zm.n.p) Note that |/u -u--c cll S Virmm since they lie in the same hypercube, then we have
Ipun (u) - (u)) (c)(c) po (w)l Dllu cll S LVky
according to Lipschitz continuity.
Proof of Lemma 3. Let Cm = # (occurrences of u in Um Then
Cm (u) ~ Binomial(m, Pra (u)).
By Chernoff's bound for binomial random variables, for any 8m,n 1. 8mm
Perate (u) (w)) )
Pl Pua (u) Prrcia (u) 2 8m,n,p Perau (u)} < exx
3
Let
Pmax == max Piraio (u);
Pmin == min Peraio (u),
Sm,n. == 8m,n.p Prnax-
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Since Pinxx 2 Peraiu (u) Pmin for all u, obtain
8 mm.ppp m Pmin
P([pogs(ul) - Pera (u)) 2 Em...p S exxp (-
Em. Pmin
2 exx
3
3p
Applying a union bound over u E Ac and noting that | S (m...... we have
Pf3u E Ac
s.t. [poyja (u) Ptraia (u) 2 Emmpp
< (m .
2exp (
mm m Pmin
3p
[-
(mmm. m Pmnin
k klog (2C mm..p) }}
< exp
3p
The failure probability thus converges as m, n, oo whenever
k logglo (m00m))))) and Em.n.p < Pmax
Em.n.p
2 Pmmxx
mpmin
(since Em,n.p 8m,n pPmax with dm 1), which concludes the proof.
64


Reviews:
[{"review_id": "RGCgxpiggn", "reviewer": "Reviewer_ELf7", "summary": "This paper introduces ReLaSH, a generative framework for hypergraphs with hyperlink attributes. Its core contribution is a novel integration of a likelihood-based joint embedding model with a distribution-free latent space generator. The method first embeds the high-dimensional, sparse hypergraph and its attributes into a structured, low-dimensional latent space, which is then reconstructed and sampled from. This approach effectively circumvents the challenges of high-dimensional generative modeling, is supported by theoretical analysis, and demonstrates superior performance across multiple real-world tasks.", "questions": "1.Regarding the selection of k1, k2, and k3:\n1.1. Could you provide ablations to demonstrate the effectiveness of the proposed tri-partite partitioning strategy compared to a simpler, unified latent space?\n1.2. Please supplement experiments comparing the equal partitioning method with non-equal alternatives to show its (non-)optimality.\n1.3. Could you offer practical guidance on how to determine these values for a new, unseen dataset?\n\n2. To strengthen the experimental validation, it is crucial to include more recent and stronger baseline methods to ensure the timeliness and competitiveness of the reported results.\n\n3.The paper would significantly benefit from a systematic sensitivity analysis of its key hyperparameters (e.g., k1/k2/k3, λ, diffusion steps). Please show how the performance metrics trend with variations in these parameters to better understand the model's robustness and tuning requirements.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "1. The proposed joint latent space reconstruction framework is highly innovative. By unifying the embeddings of both hypergraph structure (hyperedges) and attributes into a cohesive low-dimensional space, it effectively circumvents the challenges of direct high-dimensional generation. \n\n2. The method is inherently domain-agnostic, as demonstrated by its seamless and successful application across highly heterogeneous scenarios—from synthetic medical records and recipes to co-citation graphs. \n\n3.ReLaSH is fortified by solid theoretical contributions, including identifiability proofs and a decomposition of the generation error.", "weaknesses": "1.The paper proposes a novel latent space partitioning strategy but provides no principled or systematic method for selecting the dimensions k1, k2, and k3 for a given dataset. The experimental setup relies on an ad-hoc equal partitioning scheme or setting k2 = k3 = 0 for specific cases, which lacks justification.\n\n2.The baseline methods compared in the experiments are not up-to-date, with the most recent being from 2020.\n\n3.While the method introduces several key hyperparameters (e.g., k1/k2/k3, $\\lambda$) and the appendix includes some empirical studies, a systematic sensitivity analysis is still missing. It remains unclear how the model performance varies with changes in these hyperparameters.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "zvJC7CsxXz", "reviewer": "Reviewer_Suk1", "summary": "The paper introduces ReLaSH, a three-stage framework for hypergraph generation. (1) A likelihood-based joint embedding maps the observed hypergraph into a low-dimensional latent space. (2) A distribution-free generator is trained in this space to model and reconstruct the latent distribution. (3) The trained likelihood model decodes sampled latents into hyperlinks and associated attributes to synthesize new hypergraphs. On the theory side, the authors argue, and provide supporting analysis, that operating in the latent space mitigates the curse of ambient dimensionality. Empirically, experiments on co-citation, recipe, and symptom co-occurrence hypergraphs show consistent performance across domains.", "questions": "See Weaknesses", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper targets hypergraph generation with hyperlink attributes, going beyond structure-only models. The likelihood-based joint formulation explicitly links structure and attributes, which is a well-scoped and meaningful contribution.\n\n2.  Decoupling (i) likelihood-based joint embedding, (ii) distribution-free latent modeling, and (iii) likelihood decoding is clean and practical. Each component is swappable (e.g., alternative generators or attribute heads), which favors reproducibility and future extensions.\n\n3.  The analysis provides an error decomposition and argues why operating in a low-dimensional latent space can mitigate the curse of ambient dimensionality. This gives readers an interpretable lever for understanding where performance gains originate.\n\n4. Experiments on co-citation, recipe, and symptom co-occurrence hypergraphs indicate consistent improvements across heterogeneous domains, suggesting the approach is not tailored to a single benchmark and generalizes reasonably well.", "weaknesses": "1. Your FED metric relies on an embedding machine $E^T$ trained on the full population: you first embed the entire population hypergraph to obtain reference parameters, then perform an 80/20 split to choose $k$ by minimizing train/test FID, and finally reuse this $E^T$ to embed real and generated samples for FED. This makes the evaluator and evaluated models non-independent and is a threat to validity. Could you add a fully independent evaluation, for example training $E^T$ on a disjoint subset of the population or using a fixed external encoder such as a frozen autoencoder or a random orthogonal projection to compute FED, and compare the results and sensitivity with your current setup?\n\n2. In the real-data experiments, ReLaSH/ReLaSHc use Forest Diffusion, while Gau-Diff uses a standard diffusion. This inconsistency in score learners may confound the comparison. Could you provide a small matched-architecture ablation?\n\n3. In B.6.1, you select the latent dimension $k$ by minimizing the train/test embedding FID. Is there a clear theoretical connection (e.g., consistency or error bounds) between this selection criterion and your generative likelihood? If the data deviate from the model assumptions, is this criterion still reliable? Could you provide a sensitivity curve (FED, reconstruction error, and downstream metrics vs. $k$) to show that the selected $k$ aligns with overall performance?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "5Yb94fNJoT", "reviewer": "Reviewer_MqMV", "summary": "The authors present a framework for hyperedge-attributed hypergraph generation.\nThey combine various generation techniques that were successful in diverse domains to achive this goal, and demonstrated its effectiveness in certain hypergraph datasets, compared to several baseline methods.", "questions": "See Weakness", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "S1. The paper is grounded on several theoretical results.\n\nS2. The authors leverage diverse datasets and put the results in the Appendix.", "weaknesses": "**W1. [Key novelty]** While the authors present various theoretical results, I cannot understand what is the key novelty of the work. Is this the first work that generates hyperedge attributes? If so, in hypergraphs, changing node to hyperedge, and hyperedge to node, is trivial in many cases (e.g., if node: author and hyperedge: a paper's author list, then node: paper and hyperedge: a set of publications published by an author). There are certain works on hypergraph generative models with features [1], and authors should compare their work with such methods.\n\n**W2. [Implication of theoretical results]** Could the authors present more intuitive results regarding their theories? For instance, Lines 354 - 355 state: \"Theorem 3 implies that if m ≍ n ≍ p, the error introduced by estimating (Z, B, α, γ) is asymptotically negligible ...\" Does this mean that the model can capture the ground-truth data distributions under certain assumptions?\n\n**W3. [Comparison with tabular generative models]** To my knowledge, the MIMIC 3 dataset is often processed as tabular data. In my opinion, the effectiveness of the proposed method should be compared with the SOTA tabular data generative methods.\n\n**W4. [Dataset construction]** I think the description regarding how the datasets are transformed into hypergraphs is not sufficient. Could the authors elaborate on it?\n\n[1] Attributed Hypergraph Generation with Realistic Interplay Between Structure and Attributes", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "GDzusBEDmu", "reviewer": "Reviewer_hHTL", "summary": "The authors propose **ReLaSH**, a hypergraph generative model that produces random hypergraphs together with _hyperedge-level features_. The key idea is to construct a **joint latent space** of hyperlinks and their attributes and learn it in a probabilistic, likelihood-based manner.\n\n- **Data-driven latent space learning:** Build a joint latent representation that maximizes the likelihood of observed hyperedges and their associated features.\n\n- **Latent reconstruction and decoding:** Reconstruct this latent distribution via a generator, and decode new samples from it to synthesize realistic hyperedges and their attributes.", "questions": "- **\\[Q1]** _Missing references._ It seems that several recent works also tackle hypergraph generation with attributes. Could the authors clarify the novelty of ReLaSH relative to these and, if feasible, include them as baselines?\n\n  - \\[r1] Gailhard et al. _Feature-aware Hypergraph Generation via Next-Scale Prediction_, arXiv:2506.01467 (2025).\n\n  - \\[r2] Chun et al. _Attributed Hypergraph Generation with Realistic Interplay Between Structure and Attributes_, arXiv:2509.21838 (2025).\n\n  - \\[r3] Badalyan et al. _Structure and Inference in Hypergraphs with Node Attributes_, _Nature Communications_ 15 (2024): 7073.\n\n- **\\[Q2]** The proposed framework currently handles _hyperedge features_ only, while _node features_ are at least equally important and more common in real applications. Could the authors discuss whether and how ReLaSH could be extended to incorporate node features?\n\n- **\\[Q3]** I have concerns about the _evaluation metrics_, which do not appear to be standard. Could the authors justify why these particular choices are reasonable compared with alternatives, and clarify what properties of the generative distribution are expected to be captured if these losses are ideally minimized?\n\n- **\\[Q4]** Relatedly, is it conceptually possible to design a model that _directly optimizes_ these metrics (or differentiable surrogates) instead of using them only post hoc? What practical or theoretical challenges would such a design entail?\n\n- **\\[Q5]** Several configurations (k = 6, 12, 24, 48; with/without calibration) are reported, but the paper does not discuss how to _choose_ an optimal configuration for new datasets. It would be useful if the authors could provide practical guidelines or an automatic criterion (e.g., based on validation FED or structural RMSE) for selecting the latent dimension and deciding whether to apply calibration.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- **\\[S1]** The problem of _hypergraph generation with features_ is timely and important.\n\n- **\\[S2]** ReLaSH introduces, to my knowledge, the first _probabilistic and likelihood-based_ joint generative framework for hypergraphs with hyperlink attributes.\n\n**\\[S3]** The experiments span datasets with diverse feature modalities (textual, numerical, categorical), showing the model’s potential generality.", "weaknesses": "See “Questions” below.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)