Directional Sheaf Hypergraph Networks: Unifying Learning on Directed and Undirected Hypergraphs
Authors: Emanuele Mule
Stefano Fiorini
Antonio Purificato
Federico Siciliano
Stefano Coniglio
Fabrizio Silvestri
Venue: ICLR 2026 Poster (2026)
Keywords: hypergraph neural networks
hypergraph
sheaf
higher-order
directed graphs
Primary area: learning on graphs and other geometries & topologies
TL;DR: We introduce the Directed Sheaf Hypergraph Laplacian, which generalizes several Laplacian formulations previously introduced in the graph and hypergraph learning literature.

Abstract: Hypergraphs provide a natural way to represent higher-order interactions among multiple entities. While undirected hypergraphs have been extensively studied, the case of directed hypergraphs, which can model oriented group interactions, remains largely under-explored despite its relevance for many applications. Recent approaches in this direction often exhibit an implicit bias toward homophily, which limits their effectiveness in heterophilic settings. Rooted in the algebraic topology notion of Cellular Sheaves, Sheaf Neural Networks (SNNs) were introduced as an effective solution to circumvent such a drawback. While a generalization to hypergraphs is known, it is only suitable for undirected hypergraphs, failing to tackle the directed case. In this work, we introduce Directional Sheaf Hypergraph Networks (DSHN), a framework integrating sheaf theory with a principled treatment of asymmetric relations within a hypergraph. From it, we construct the Directed Sheaf Hypergraph Laplacian, a complex-valued operator by which we unify and generalize many existing Laplacian matrices proposed in the graph-and hypergraph-learning literature. Across 7 real-world datasets and against 13 baselines, DSHN achieves relative accuracy gains from 2% up to 20%, showing how a principled treatment of directionality in hypergraphs, combined with the expressive power of sheaves, can substantially improve performance.

Published as a conference paper at ICLR 2026
DIRECTIONAL SHEAF HYPERGRAPH NETWORKS: UNIFYING LEARNING ON DIRECTED AND UNDI- RECTED HYPERGRAPHS
Emanuele Mule 1
Stefano Fiorini² Fabrizio Silvestri
Antonio Purificato 1.3f Federico Siciliano
Stefano Coniglio
Sapienza University of Rome, Rome, Italy
2 Independent Researcher
3 Amazon Research University of Bergamo, Bergamo, Italy
ABSTRACT
Hypergraphs provide a natural way to represent higher-order interactions among multiple entities. While undirected hypergraphs have been extensively studied, the case of directed hypergraphs, which can model oriented group interactions, remains largely under-explored despite its relevance for many applications. Re- cent approaches in this direction often exhibit an implicit bias toward homophily, which limits their effectiveness in heterophilic settings. Rooted in the algebraic topology notion of Cellular Sheaves, Sheaf Neural Networks (SNNs) were intro- duced effective solution circumvent such drawback. While generaliza- tion to hypergraphs is known, is only suitable for undirected hypergraphs, fail- ing to tackle the directed case. In this work, we introduce Directional Sheaf Hy- pergraph Networks (DSHN), a framework integrating sheaf theory with princi- pled treatment of asymmetric relations within a hypergraph. From we construct the Directed Sheaf Hypergraph Laplacian, complex- complex-valued operator by which we unify and generalize many existing Laplacian matrices proposed the graph- and hypergraph-learming literature. Across 7 real-world datasets and against 13 baselines, DSHN achieves relative accuracy gains from 2% up to 20%, showing how principled treatment of directionality in hypergraphs, combined with the expressive power of sheaves, can substantially improve performance.
INTRODUCTION
Learning from structured, non-Euclidean data has been dominated by Graph Neural Networks (GNNs), which propagate and aggregate features along pairwise edges (Scarselli al.] 2009 Asif ct al. 2021). Sheaf Neural Networks (SNNs) (Hansen & Gebhart 2020 Bodnar et al. (022)) extend GNNs by leveraging the algebraic concept of a cellular sheaf. They assign vector spaces nodes and edges, along with learnable restriction maps propagating information between them. By operat- ing in digneralimensional feature spaces, SNNs mitigate oversmoothing and enhance performance on heterophilic graphs, where neighboring nodes exhibit dissimilar features (Purificato et al.) 2025)
While effective on graphs, both GNNs and SNNs are inherently limited to dyadic relations. Many real-world systems such as social networks (Benson et al. 2016 2018a). biological sys- tems (Traversa et af., 2023), and protein interactions (Murgas et al.) 2022) exhibit multi-way rela- tionships that cannot be captured by pairwise links alone. Hypergraph Neural Networks (HGNNs) address this limitation by modeling hyperedges as sets nodes, enabling the learning of multi-entity dependencies (Murgas al., 2022 Chen al. 2022). However, traditional HGNNs face two main limitations. First, they inherit fundamental drawbacks from their graph-based counterparts: many architectures assume homophily, which is often violated in heterophilic settings, and prone to
"Corresponding author: emi mum e200 1 i com.
Work done outside of the company.

---

Published as a conference paper at ICLR 2026
oversmoothing, where deep message passing causes node representations to converge and lose dis-
criminative power (Li et al., 2025 Telyatnikov et al. 2025 Chen et al. 2022 Nguyen et al. [023) Second, most HGNNs are tormulated for undirected hypergraphs, treating hyperedges symmetr- cally and neglecting orientation even when such hyperedges encode asymmetric or causal relation- ships such as chemical reactions, metabolic pathways, and causal multi-agent interactions (Mann &
Venkatasubramanian 2023: Traversa et al. (023).
Sheaf Hypergraph Networks (SHNs) (Duta et al. 2023) address the first of these challenges by ex- tending the principles of SNNs to hypergraphs. They assign vector spaces to nodes and hyperedges and propagate information via learnable restriction maps naturally mitigating the issues associated with oversmoothing and heterophily while generalizing message passing to higher-order, multi-way interactions, which provides a more expressive framework than traditional HGNNs.
Despite these advantages, SHNs have two key limitations. (i) They only model hyperedges as undi- rected, limiting their ability to capture asymmetric directional relationships. Indeed, while direc- tionality has been recently incorporated in GNNs via, e.g., specialized Laplacians such the one by Tong et al. (2020) and complex-values operators (Zhang et al.] 2021 Fiorini et al. 2023). exten- sions to hypergraphs remain limited (Fiorini et al.) (024) and often task-specific (Gatta et al. 2023
Zhao et al. 2024). To our knowledge, no SHN methods which can handle directed hypergraphs are known. 21) Second (as we show in this paper), the Laplacian operator proposed in Duta et al. (2023) fails to satisfy the spectral properties required of well-defined convolutional operator, such as posi- tive semidefiniteness, contrarily to what the authors report (and claim to have proven) in their paper.
In this paper, we introduce Directional Sheaf Hypergraph Networks (DSHN), a principled extension of SHNs to directed hypergraphs. Specifically, we define Directed Hypergraph Cellular Sheaves, equipping hyperedges not only with the notion of tail and head sets (source and target nodes, respec- tively) but also with asymmetric restriction maps that respect orientation within hyperedge. From this, we derive the Directed Sheaf Hypergraph Laplacian, a novel complex-valuee Hermitian opera- tor whose phase naturally encodes direction while preserving essential spectral properties, including admitting spectral decomposition with real-valued, nonnegative eigenvalues. We evaluate DSHN on real-world datasets, as well as on synthetic benchmarks specifically designed to test DSHN's ability to capture directional information within a hypergraph. Compared to 13 state-of-the-arr base- lines, our method achieves relative accuracy gains from 2% up to 20%, demonstrating that explicitly modeling orientation via proposed asymmetric and complex-valle valued restriction maps improves predictive performance. Our contributions can be summarized as follows:
We introduce the concept of Directed Hypergruph Cellular Sheaves, a framework that ex- tends directed hypergraphs by providing a principled representation of directional interac- tions. This is achieved by assigning complex- peredges, capturing the node-to- tode-to-hyperedge relationships within each directed hyperedge.
complex-valued valued lincar maps between nodes and hy-
We introduce the Directed Sheaf Hypergraph Laplacian, a novel complex-valued Hermi- tian matrix that satisfies the key properties required of a well-defined spectral operator. Our formulation generalizes existing graph and hypergraph Laplacians, providing a unified framework for learning on hypergraphs with both directed and undirected hyperedges.
We introduce Directional Sheaf Hypergraph Networks (DSHN), a model that combines Sheaf theory with a principled treatment of directional information, enabling state-of-the- art performance on directed hypergraph benchmarks
BACKGROUND AND PREVIOUS WORK
Sheaf Neural Networks and Sheaf Hypergraph Networks. Sheaf theory provides a principled
framework for modeling local information flow across structured domains. In algebraic topology, sheaf associates data to open sets together with restriction maps ensuring consistency (Curry) (0144). Cellular sheaves adapt this idea cell complexes by assigning vector spaces to cells with linear maps along face relations. Building on this, Sheaf Neural Networks (SNNs) (Hansen & Gebhart
2020 Bodnar et al. 2022) assign vector spaces to graph nodes and edges and learn restriction maps
We provide the code reproduce the results
https: / //gi thub. com KmaM: l
Di rect i oo
2

---

Published as a conference paper at ICLR 2026
for each node-edge incidence relationship, generalizing message passing. SNNs particularly effective in heterophilic settings and help mitigate oversmoothing, a common drawback of deep GNNs. As described by Hansen & Ghrist (202I), attaching cellular sheaf to graph can be inter- preted through the lens of opinion dynamics. Unlike traditional Graph Neural Networks (GNNs), each node's s representation "lives" in its own vector space, representing private opinion, while the restriction maps, linear transformations between node and edge vector spaces, govern how this opin- ion expressed along each incident edge. In this way, node can maintain own distinct repre- sentation while still having different "opinions" across the various edges they participate in, allow- ing for more expressive and flexible representations compared standard spectral-basee methods, where adjacent nodes tend to have similar representations.
Expanding on this idea, Duta et al. 2023 extend SNNs to the hypergraph setting. However, as it will be shown in this work, their formulation suffers from two key limitations. First, it does not capture directionality in hypergraphs. Second, although the Sheaf Hypergraph Laplacian is presented as a Laplacian operator, as shown in paper, it fails to satisfy fundamental spectral properties expected of such operators, most notably positive semidefiniteness.
Laplacian matrices for Directed Graphs Classical spectral methods define convolutional oper- ators through the graph Laplacian (Biggs) 1993; Defferrard et al., 2016b] Kipf & Welling 2017).
While effective, such methods require to either work on inherently undirected graphs or sym- metrize the graph's 's adjacency matrix, thereby discarding edge directionality. Drawing inspiration from the Magnetic Laplacian introduced by Lieb & Loss] in the study of electromagnetic fields, spectral- methods have been extended to incorporate edge directionality. In particular, Zhang al. (2021); Fiorini al. (202)) developed operators that encode orientation in the imaginary part of complex-ves valuue Hermitian matrices. This construction preserves the desirable spectral proper- ties required for well-defined convolutional operator while embedding directional information, en- abling convolutional operators to faithfully capture the asymmetry of directed graphs.
Undirected and Directed Hypergraphs The hyperedge weights are stored in the diagonal matrix W E R The vertex and hyperedge degrees are defined as D., = Eee Exuee [wel for V and de = |e| for e E E. Hypergraphs where 5, = for some E N for all e E E are called k-uniform. Graphs are 2-uniform hypergraphs. Following Gallo al. 1993), we define a directed hypergraph as hypergraph where each edge e E E is partitioned in a tail ser (e) and a head set H (e). If H (e) empty, e is an undirected edge. Research on learning on directed hypergraphs remains limited, with most existing works either constrained to task-specific scenarios (Luo et al. 2022 Gatta et al.)
2023) or restricted to 2-uniform directed hypergraphs (Zhao et al.) [024] Ma et al.] 2024). This gap has been recently addressed through the Generalized Directed Laplacian (Fiorini et al. 20024) a complex Hermitian operator that unifics directed and undirected hypergraphs, and extends several popular methods for directed graphs to the hypergraph domain. However, their method, while being suitable for directed hypergraphs, is still implicitly biased towards homophilic settings and can be prone to oversmoothing.
3 DIRECTED SHEAF HYPERGRAPH LAPLACIAN
3.1 DIRECTED HYPERGRAPH CELLULAR SHEAF
In this work, we introduce the notion of Directed Hypergraph Cellular Sheaf, which assigns directed hypergraph complex-valued restriction maps designed to capture and encode directional information contained in the hypergraph's 's underlying topology.
Definition 1. The Directed Hypergraph Cellular Sheaf of a directed hypergraph H = (V, E)) is the tuple (S (F(u))eev (F(e))eER, {F (Fugehuer(e)) ) consisting of:
1. A complex-valus valued matrix s(9) e Cmxn with E TR, defined entry-wise for each hyperedge E and node V as:
if u E H(e) (head set)
(t9) de
25iq
if E T(e) (tail set)
otherwise
2. A vector space F(u) C Cd associated with each node E V;
3

---

Published as a conference paper at ICLR 2026
3. A vector space F(e) _ C associated with each hyperedge E;
4. A restriction map Fuse F(u) - F(e) with Fuse 809) F Caxd where Fuge Idd is a real-valued, directionless, restriction map.
uge
The idea associate to each node-hyperedge incidence relationship linear restriction map Fage which can either be real- or complex-valued hased on the directional matrix S specifying whether node within hyperedge belongs to the tail or to the head set. In line with Zhang et al. (202I), although their work focuses on directed graphs and hence fails to model many-to-man interactions, the parameter q associated with the matrix S(r) serves as charge parameter that con- trols the relevance of the hypergraph's directional information. In fact, when == O, the restriction maps are all real-valued independently of the hypergraph's ' directions, and we come back the def- inition of a (Hypergraph) Cellular Sheaf as introduced in Duta et al. (2023). While prior work has incorporated directional information hypergraphs using complex- -vauued coefficients, these ap- formulation introduces a tunable complex-valuee coefficient, allowing the model to flexibly adjust the contri- bution of directional information. Moreover, by equipping the topology of the hypergraph with a Cellular Sheaf, our method provides a more expressive framework for representing hypergraph structures than existing directed hypergraph neural networks. To visualize of Directed Hyper- graph Cellular Sheaf associated to a directed hyperedge, see Fig.
proaches typically rely fixed complex phase (Fiorini et al. 2024). In contrast,
12
F2
F.
Xi Sydy F)
Frysles
Figties
Friiee ae Sai Fors
122
Y1
41
Directed hyperedge
Directed Cellular Sheaf
Figure 1: Visualization of sheaves over directed hyperedge, illustrating the incidence relationship between nodes and the hyperedge, together with the restriction maps Fuge- The tail node is encoded via the e -2rig coefficient which pre-multiplies the directionless restriction map Fu
3.2 DIRECTED SHEAF HYPERGRAPH LAPLACIAN
Given directed hypergraph and its corresponding Directed Hypergraph Cellular Sheaf F. let ( Cddxnnd be complex-valued incidence matrix which, for each pair E and u E V, reads:
B
(
Fuse
s(g) uggn
Fast
fu
0
otherwise.
When factoring in, for each pair e E and u E e, whether u belongs the head tail e, we obtain:
Fuge = 5(()) Fugee Fuge "dh
if E H(e)
(head set)
(
Fuge =(9)) -
Fusic =e 2ni
if E T(e) otherwise.
(tail set)
(1)
We define the Directed Sheaf Hypergraph Laplacian L associated with Directed Hypergraph Cellular Sheaf as follows:
L $
- DD
OF with Q F :
D D
(2)
where Q is the Directed Sheaf Hypergraph Signless Laplacian.
4

---

Published as a conference paper at ICLR 2026
In the formula, D is the block-diagonie hyperedge degree matrix DE diag(01 (ogg(ó1 ii.............................. SmLs) Rmdxmd, where òg := |e| is the degree of hyperedge e E E, and Dy E Rudxnd is the block-diagonal node degree matrix defined as Dv == diag (D D. uzs..... Du.). with D.. Fl Fage Fuge E Raxd, U4 EV.
EeEEuEE
To clarify how L encodes the hypergraph structure, let us examine its entry corresponding to a pair of vertices u, V:
Du
Σ e:uCe
F
Fuge==
Σ
(1-
FuggeFuge
8
(L (we
cuuEr
(3)
Σ e:u
Fi
Σ C.R
1
(S use
(5
uge (F Fuge
u 7 V.
de
uue
8e
Note that the block-diagonal entries L are always real. contrast, its off-block-diagonal entries complex-valued when the hypergraph is directed and q 0, and real-valued if the hypergraph is undirected. By setting g O, the hypergraph directions are entirely disregarded and L coincides with the Laplacian matrix of its undirected counterpart. The off-diagonal products in L strongly depend on the interaction between the two components of the directional matrix S(4) associated the restriction maps. The product of these contributes to the real and imaginary parts of I. as follows:
v, T(e) (tail-tail),
(8 )
sso) usle
0, u (ee) (head-head),
(4)
e 2mig
v E [e)), u E (e) (tail-head) (equal to if q = 7)
e 2srin
Hee), T(e) (head-tail) (equal to -i if = 2)
We can expand Eq. (3) by considering the special case where 4 = 4. In this case, each entry of I. can be expressed explicitly highlight the impact of the directionality of each hyperedge:
Σ e:uEe
F
Furee
u = V,
dc
(IFF).
(5)
Σ 5e
FT
Σ de eEE (ET(c) (VEH((e)
FT Fuge
FT FuseFuse
Σ 8e
F
Fuse
u u v.
eEE
((VEH(e) Va,uET(()
WE NE()( AveT(e)
The entry (L uu determined by all hyperedges E that contain both nodes u and From the second case in Eq. (5) whenever both and v are both heads (u, v E He)) or tails (u, v e T())), the contribution to the real part D2
(I, Ju is negative and given by the opposite of the normal- ized weight - F In the undirected case, this is the only possible contribution, which matches the expected behavior of undirected hypergraph Sheaf Laplacians, where u, T(e) (or, equivalently, u, v E (e)). Hyperedges where u, v take opposite roles contribute to imaginary part with their weight either negatively (if T(e) and v E H(e)) positively (if u H(e) and
E T())). Due to this, in the special case where 9 = 1.
(L Juv depends on the net contribu-
tion and e across all the directed hyperedges that contain them. This is in line with the "net flow" behavior observed in directed graphs (Fiorini al. 2023) and hypergraphs (Fiorini et al. 2024).
When interpreted as a linear operator acting on complex signal x E And, F reads:
L (x)),,
Σ
Ft UsE
Σ
Fuse,, X,, F Foge x, x.).
(6)
5e
eEE:
uge
5

---

Published as a conference paper at ICLR 2026
We define the Normalized Directed Sheaf Hypergraph Laplacian L as:
L == D D
Using Eq. 22. this yields:
F L
N
D Dv
2
D
Ind -QN: where Q = DD
B D (7)
3.3 SPECTRAL PROPERTIES
We now establish that our proposed Normalized Directed Sheaf Hypergraph Laplacian L satisfies all the spectral properties required for a principled convolutional operator. Specifically, we show that Ly is diagonalizable, has real, nonnegative cigenvalues, is positive semidefinite, and has bounded spectrum. These ensure that the Fourier transform well-defined and that polynomial filters of Lý implement localized, stable convolutions, in direct analogy with classical spectral- based approaches (Shuman al., [2033] Kipt & Welling] 2017 Detterrard et al. 2016a). Failure preserve them can result in a complete breakdown of the connection between message passing, the Fourier transform of boyper-)graph and its connection with signal theory. The proofs of the claims in this and the next section are provided in Appendix A
We begin by showing that Lý admits an eigenvalue decomposition with real eigenvalues:
Theorem 1. L is diagonalizable with real eigenvalues.
Next, we derive the formula of the Dirichlet energy function associated with L which provides measure of the global smoothness of a signal x E Cnd across the entire hypergraph:
Theorem 2. The Dirichlet energy induced by L for a signal x E Cnd is:
Ex(x) x X
- Σ
Fuge
D X Foge D
X
2
de
u.uEe: uzfu
By leveraging the two previous theorems, we show that the spectrum of the Directed Sheaf Hyper- graph Laplacian L only admits (real) non-negative eigenvalues:
Corollary 1. L is positive semidefinite.
Finally, we prove that the spectrum of L upper bounded by 1:
Theorem 3. Amax (L S
3.4 GENERALIZATION PROPERTIES
Beyond its spectral properties, Directed Sheaf Hypergraph Laplacian provides unified defini- tion of Laplacian matrix that recovers and extends several existing Laplacian matrices.
First, we discuss the relationship between e and the Sheaf Laplacian introduced by Hansen & Gebhart 2020) and highlight its connection with the classical graph Laplacian defined as L := D A where A is the graph's adjacency matrix and D its degree matrix (see (Biggs 1993) for reference)
for the undirected case:
Theorem 4. For a 2-uniform hypergraph without directions, the Laplacian operator L reduces to the Sheaf Laplacian (Hansen & Gebhart 2020) (up to a scaling factor of 2) and, when considering 1), it coincides with the classical graph Laplacian (up to
the case of a trivial Sheaf (where Fuse scaling factor of 2).
With Theorem 5 we show that L generalizes several Laplacians designed for directed graphs like the Magnetic Laplacian (Zhang et al. 2021 and the Sign-Magnetic Laplacian (Fiorini al. 2023

---

Published as a conference paper at ICLR 2026
Theorem 5. For directed 2-uniform hypergraph with unitary edge weights (i.e., We 1,e E) with directed and undirected edges, L recovers, as a special case, the Magnetic Laplacian Zhang et al. 2021 for any E Il and the Sign-Magnetic Laplacian (Fiorini et al. 2023) when q T.
In the context of hypergraphs, our operator naturally recovers existing hypergraph Laplacians. We begin by showing that recovers the undirected hypergraph Laplacian of et al. (2006):
Theorem 6. Given hypergraph H (directed or undirected), the normalized Directed Hypergraph Laplacian L recovers, as special case, the undirected hypergraph Laplacian of Zhou 2006
We show that L generalizes the Generalized Directed Laplacian proposed by Fiorini al. 2024
Theorem 7. Given directed hypergraph H. with unitary weights associated to each hyperedge (i.e., We 1), the Normalized Directed Sheaf Hypergraph Laplacian L recovers, as special case, the Generalized Directed Laplacian Ly of Fiorini et al. (2024).
Crucially, our Laplacian operator does not recover the (linear) Sheaf Hypergraph Laplacian proposed in (Duta et al. 2023). This difference is intentional: the formulation proposed in their work, defined for any pair nodes u, v E V as:
Nm = Σ
8 f Fuge
udles
D = Σ
FuseFusee
(8)
C: uEe
6:0 W'e
fails satisfy several key spectral properties required to construct well-defined convolutional operator, most notably positive semidefiniteness. Comparing this definition with Eq. 3 it is evident that, for undirected hypergraphs
when g 0, the operators differ in the diagonal term: formulation introduces the factor (1 1) rather than à. Their definition suffices to guarantee 6.
such properties only in the special case 2-uniform hypergraphs (i.e, standard graphs), however this discrepancy can produce negative eigenvalues for general cases, preventing the matrix from being interpreted as a diffusion operator and precluding the definition of a stable Fourier transform. Polynomial filters based on the Laplacian of Duta al. (2023) therefore cannot guarantee localized and stable convolutions, limiting its general applicability. A more detailed analysis, including an example that demonstrates this drawback, is presented in Appendix E
In contrast, when setting q = O or considering undirected hypergraphs, our operator satisfies all the properties required of a principled Laplacian matrix and, the best of our knowledge, provides the first definition of Sheaf Hypergraph Laplacian suitable for undirected hypergraphs.
4 DIRECTIONAL SHEAF HYPERGRAPH NETWORK
In this section, we describe our proposed Directional Sheaf Hypergraph Network (DSHN), Inspired by Hansen & Gebhart (2020). we define the sheaf diffusion process on a directed hypergraph H as an extension of classical heat diffuuion,, which plays central role in spectral-basee GNN convolution operators KKipf & Welling 2017). Starting from the differential equation:
X, Xt.
and applying a unit-step Euler discretization, we obtain:
X - (Ind ( )
By introducing learnable parameters and nonlinear activation function this discrete diffusion process leads to the following equation of the convolutional layer of DSHN:
X (And L (In WT) X.) =o (ax (I, Wi),, E Cndxf
where W, E IR ddd and W₂ E R are trainable weight matrices, and & denotes the Kronecker product. X, is obtained from the input matrix of node features of size X f. to which linear projection is applied to produce an n x (df) matrix, which is then reshaped into (nd) x f matrix before applying the diffusion process. Finally, since the convolutional layer operates in the complex

---

Published as a conference paper at ICLR 2026
domain, we transform the output of the final convolutional layer into a real-valued representation right before feeding it to the classification head. Following Zhang et al. (2021): [ioriii et al. (2023), we do so by applying an unvind operation, concatenating the real and imaginary components of the complex features as follows:
unwind (X) = R(X) || (X) E R=x2f
Restriction Maps The expressive power of sheaf-theoretical approaches lies in their ability to define a diffusion operator via a learnable d x d restriction map associated to each node-edge pair. Following Bodnar et al. (2022), we learn each restriction map as a function of the corresponding node and hyperedge features. In particular, for each hyperedge E E and each node u E e, each directionless restriction map Fuse is parametrized as Fuse = 4(x, x.) E Rd where zu is the node feature and x is the hyperedge feature and & is MLP. If hyperedge features are not explicitly provided, they are computed by aggregating the features of the hyperedge's 's nodes via a mean sum. Given that the node and the hyperedge features x, and x, complex-valued due to Eq. (9), we employ the same wnwind operation to map them into a form suitable for input to Q. Activation functions can be either sigmoid or tanh.
DSHNLight A key objective in our model design is to balance predictive performance with com- putational efficiency. Constructing an nd x nd Laplacian matrix from d x d restriction maps in- herently increases complexity, an issue already noted in the graph setting (Bodnar et al. 2022). To mitigate this, we introduce DSHNLight, a variant of DSHN that achieves competitive, and in some cases superior, results across several datasets (Tables I and 2 at significantly lower computational cost (see Appendix C.T). In it, we detach the gradient computation during the Laplacian's 's construc- tion: this way, the model continues to rely on the predicted restriction maps, but avoids costly gra- dient propagation. In DSHNLight, the parameters of the MLP responsible for predicting the restric- tion maps (which are encoded in (-)) remain fixed throughout the training process. The model's adaptability arises from the initial projection layer, which embeds the inputs into a shared feature space where they can be more effectively processed through these parameters (see Appendix for a better visualization). This phenomenon aligns with insights from the literature on overparameteri- zation (Arora al. (019) and extreme learning machines (Huang et al.) 2006), where fixed random projections can still yield strong generalization due to the expressive power of the input embeddings. Further details on the difference between the two approaches are discussed in Appendix
Computational Complexity We provide an estimate of the asymptotic complexity of our model inference time. Let n denote the number of nodes, m the number of hyperedges, d the stalk dimension, the product of input and output feature dimensions in the linear transformation, é the average hyperedge size, and e the average number of hyperedges a node participates in. Summing the contributions from the feature transformation, message passing, learning of restriction maps, and Laplacian assembly, the overall complexity is O(n(c² + d) + mm(d + (d & + bc)) for diagonal maps, and O(n(c² + +f) + m(ēd + (d + de) + od)c)) for non-diagonal maps, sharing the same asymptotic complexity as SheafHyperGNN by Duta et al. (2023). For a comprehensive analysis of the contributions leading this asymptotic complexity, refer the reader to Appendix
5 EXPERIMENTAL EVALUATION
We evaluate DSHN and DSHNLight against 13 baseline models from both the directed and undi- rected hypergraph literature on real-world datasets (Section 5.1 as well as synthetic datasets (Sec- tion (.2) for the node classification task. From the undirected hypergraph-learning literature,
we include HGNN (Feng et al.) 2019), HNHN (Dong et al. 2020). UniGCNII (Huang Yang 2021). LEGCN (Yang et al. (022).0222222 HyperND (Tudisco et al. 2021). AllDeepSets and AllSeiTrans-
former CChin et a (022). HD- (Wang et al. |l.] (023a), SheafHyperGNN (Duta et al.) 2023). and PhenomNN (Wang et al., 2023b). From the directed hypergraph-leariing literature, we con- sider GeDi-HNN (Fiorini et al. 2024 and DHGNN (Ma et al. 2024), along with a variant, as base- lines. Model performance is measured in terms of classification accuracy. Following the standard practice in the literature (Chien et al. 2022 Wang et al., 2023a Fiorini et al. 2024). we adopt a 50%/25%//25% split for training, validation, and testing, respectively, and, for each model, we port the average test accuracy and the standard deviation over 10 independent runs. Details on the baselines, hyperparameter tuning, and the experimental setup are provided in Appendix
8

---

Published as a conference paper at ICLR 2026
5.1 REAL-WORLL DATASETS
To evaluate our models real-world datasets, we follow the pre-processing procedure intro- duced by Tran & Tran (022) and Fiorini et al. (2024). and apply it to a suite of publicly avail- able directed graph benchmarks to obtain their directed hypergraph counterparts for performing the node classification task (see Appendix D.3). The considered datasets are: Cora (Zhang et al, 2022), ema MomallEEnoon, ema i E-EU (Benson et al. 201 Te relegram, (Bovet & Grindrod 2020), Chame Squirre and Roman-emp.l empire. Due to space limitations, Table I lnncludes only the datasets that yield the most interesting insights. Additional results can be found in Table while additional informations on the datasets are provided in Appendix D.4
Table 1: Mean accuracy + standard deviation on node classification datasets. For each dataset, the best result is shown in bold, and the second best is underlined.
Roman-empire
Squirrel
email-EU
Telegram
Chameleon
email-Enron
Cora
HGNN HNNIN
38.11
18.91
8.11
51.73
87 25 1.01
16.07
1.22
35.62
1.30
29.68
1.68
38.22 1: 6.95
35.81 3.23
18.64 6.90
78.16 1: 0.98
UniGCNII
78.89 + 0.51
38
2.56
44.98
2.00
51.73 + 5.05
39.85 + 3.19
47.43 + 7.47
87.53 + 1.06
LEGCN
32.91
1.80
45.38
4.23
37.00
74.96 0.94
HyperND
68:31 + 0.69
40.13
1.85
32.70
2.90
44.62
+ 5.49
44.95 + 3.20
38.11 + 7.69
78.48 1.02
AllDeepSets
81.79 0.72
10.09
1,90
37.37
6.29
19.19
+ 6.73
42.97 + 3.60
37:29 + 7,90
86.86 0.85
AlISeTTransfommr
0.65
4.36
1.13
ED-HNN
83.82 0.31
39.85
1.79
68.91
4.00
00.38
+ 3.88
44.67 2.33
51.35 6.04
86.94 1.25
SIealHHyperGN
74.50==0.55
42.01
1.11
62.78
9.13 37.89
70,00 + 5.32
41.06 4.94
63.51 5.95
87 15 0.64
PhenomMN
71.22
1.40
47.69 6.59
47.02
GeDi-HNN
0.66
43.02 3.001
52.31 + 2.84
77.12 4.82
30.20 + 2.04
10.54 11.44-+100 5.80
85. 0.94
DHGNN
0.51
39.85
1.79
32.35
2.93
79.62 4 5.78
41.08 1.11
12.16 8.04
83.16 1.33
DEGNN (w/ cmb.)
22.50
55.10
3.48
80.58
3.80
7.57
73.12 1.04
DSHN
00M
13.55
2.87
78.62
2.50
88.65 4: 5.54
47.02 #: 4.35
75.68 4:
87 0.90
DSh
89.24 0.57
44.09
2.36
82.67
1.29
81.13
4.19
46.50
4.09
76.76
2.48
88.02 1.11
DSHN, and its variant DSHNLight, which both leverage the theoretical advantages of associating Directed Cellular Sheaf to directed hypergraph, consistently outperform the 13 baselines from both the undirected and directed hypergraph learning literature on out of 7 real-world datasets. The largest relative gains are observed on the mmail-Enron and email-El datasets, where DSHN and DSHNLight improve over the best baseline by up to 20%. A substantial improvement is also achieved on Te 1 egram, confirming the importance of directional information in this benchmark where all directed methods perform strongly. More moderate but consistent improvements are found on highly heterophilic datasets such as Roman coman-empirz Chame leon and Squirrel while on highly homophilic datasets such as Cora performance is on par with the strongest baselines. As shown in Table 3] the charge parameter q selected by the apprraaraetter-seeectioo procedure for our models on highly homophilic datasets is consistently 0.0. This observation is in line with the findings of Zhang et al. (202T). who report that, in such settings, directional information behaves as noise for node classification.
better visualization of the impact of the charge parameter q on the predictive performance of our model can be found in Fig. 2 where we highlight the positive impact of directional information on the Telegram dataset and how direction is detrimental on the Cora dataset. These results not only demonstrate the effectivvness of our models in highly heterophilic settings, but also show how integrating the concept of directionality in hypergraphs can substantially improve performance. Moreover, unlike GeDi-HNN and DHGNN, which are based on Laplacian formulations that embed directionality without any degree of freedom, in our models one can flexibly choose the relevance of directional information by suitable choice of the charge parameter q.
5.2 SYNTHETIC DATASETS
We additionally evaluate our models on the synthetic datasets introduced by Fiorini et al. (2024), built over n 500 nodes and split into c classes. Each class contains 30 random intra-class hyperedges, while inter-class directed hyperedges, consisting of multiple tail and head nodes, are added between class pairs with sizes drawn uniformly from (3..... 10). By varying the number of inter-class hyperedges Io {10,30 30, 50}, we control the strength of directional connectivity. This design provides a clean benchmark to test the models' ability to capture directionality; further details given in Appendix D.4

---

Published as a conference paper at ICLR 2026
Table Mean accuracy + standard deviation on the synthetic datasets.
Figure Effect of the charge parameter g on Telegram and Cora.
Method
1.
1. 30
1.
50
Telogram datasct
HGNN
47.12 5.37
6.63 43.44
37.76 7 7.72
Accuracy (3)
86
HNHN
20.10 2.993
28.88
9.45
19.76
3
80
UnIGCNH
21.44 4.33
21.1
19.84
LEGCN
17.60
20.72=3.48
2.82
75
HyperND
20.49 2.93
20.61 31.1
70
SetDeepSets
6.81
32.32
4.82
51.70+5.58
AllSeTTrasffomer
13.08 3.79
31.84
0.05
0.1
0.15
0.2
0.25
ED-HNN
34.00 +: 6.05
18.88 = 2.56
32 48 11388-+600 6.17
Charge
SheafHyperGNN
30.64 + 5.39
27.28 = 7.31
26.00 + 9.59
Cora dataset
PhenomNN
22.24 + 4.73
4.20 22.08
18. 3.22
BeDEIINN
71.44 1: 3.14
71.84
3.31
78.24 1 5.64
Accunccy (%)
86
DHGNN
40.72 +: 4.55
51.68
= 3.97
35.76
3.70
DHGNN (w/ emb.)
84.48
+: 3.22
85.28
3.32
81.12 3.22
85
DSHN
94.96
+ 1.75
97.84
+ 1.86
95.84
2.17
84
DSHNLight
95.60
+ 2.15
97.01 = 2.79
99.04 + 0.86
0
0.05
0.1
0.15
0.2
0.25
Charge
The results in Table demonstrate the advantage of our models DSHN and DSHNLight over existing baselines. Classical undirected hypergraph methods are unable to capture the directional structure that dominate these benchmarks, and as result their performance is limited. Directed methods such as GeDi-HNN and DHGNN achieve stronger results, confirming the importance of explicitly incorporating directionality into the convolutional process. Yet, DSHN and DSHNLight, which provide a principled and more expressive treatment of directional structure, yield consistent improvements across all synthetic datasets, outperforming the strongest directed baselines by up to 18 percentage points and reaching 99.04% accuracy on the third synthetic dataset- this highlights the expressive power that the notion of Directed Hypergraph Cellular Sheaves unlocks.
CONCLUSION AND FUTURE WORKS
We introduced the concept of Directed Hypergraph Cellular Sheaves for directed hypergraphs and derived the corresponding Directed Sheaf Hypergraph Laplacian, which we integrated into our pro- posed framework DSHN. By encoding hyperedge direction via a topology-awan complex-valued inductive bias, our method naturally accommodates both directed and undirected hypergraphs while also unifying and generalizing several operators from the graph and hypergraph learning literature. Across a broad set of benchmark datasets, DSHN consistently outperforms methods from both the directed and undirected hypergraph learning literature. As future work, natural step forward is to evaluate our framework on larger and natively directed hypergraph datasets such as protein-pun prottii interaction networks to further test the scalability and expressivity of the method, possibly employ- ing Language Models (LMs) to generate features. Finally, an intriguing direction to make the charge parameter g directly learnable, allowing each layer to adapt its diffusion process dynamically.
ACKNOWLEDGEMEET OF SUPPORT
Antonio Purificato, Federico Siciliano and Fabrizio Silvestri acknowledge projects FAIR (PE0000013), under the MUR National Recovery and Resilience Plan funded by the European Union - NexIGenerationEU, and project NEREO (Neural Reasoning over Open Data), funded by the Italian Ministry of Education and Research (PRIN) Grant no. 2022AEFHAZ. Stefano Coniglio's work was partially supported by the European Union under Next Generation EU - the Italian Na- tional Recovery and Resilience Plan (PNRR), PRIN 2022 PNRR (project code P20227CTY3, CUP D53D2301 8800001), project title "HEXAGON: specialspee EXact Algorithms for Grid Op- erations at the National level".
10

---

Published as a conference paper at ICLR 2026
REPRODUCIBILITY STATEMENT
We provide all the necessary information to facilitate the reproducibility of our results. Our code repository code can be found here The README contains all that is needed up the Python environment and run the experiments with the different configurations. Further details on the Ex- perimental Setup can be found in Appendix D
ETHICS STATEMENT
All datasets employed in this work are publicly available for research and contain no personally identifiable information or harmful content (see Appendix [D.4] for further details). The methods introduced in this paper have societal impact comparable to that of other graph neural networks.
LLM USAGE STATEMENT
All technical content presented in this paper is entirely our own work, with LLMs serving only as editorial tool. No scientific content research findings generated using an LLM.
REFERENCES
Sanjeev Arora, Simon Du, Wei Hu, Zhiyuan Li, and Ruosong Wang. Fine- anneggaiied analysis of op- timization and generalization for overparameterized two-layer neural networks. In International conference on machine learning, pp. 322-332. PMLR, 2019.
Nurul A. Asif, Yeahia Sarker, Ripon K. Chakrabortty, Michael J. Ryan, Md. Hafiz Ahamed, Dip K. Saha, Faisal R. Badal, Sajal K. Das, Md. Firoz Ali, Sumaya Moyeen, Md. Robiul Islam, and Zinat Tasneem. Graph neural network: A comprehensive review non-euclidean space. IEEE Access, 9:60588-60606, 2021. doi: 10. 1000
Jose Agustin Barrachina, Chengfang Ren, Gilles Vieillard, Christele Morisseau, and Jean-Philippe Ovarlez. Theory and implementation of complex-valued neural networks, 2023. URL https:
abt///aa/2222222 08286
Austin R. Benson, David Gleich, and Jure Leskovec. Higher-order organization of complex networks. Science, 35366295; July 2016. ISSN 1095-9203. doi: 10. 126/science. aad9029. URL.sttp://dx..dno
wwwen//////eeeee.ee///eeeee aad9029
Austin R. Benson, Rediet Abebe, Michael T. Schaub, Ali Jadbabaie, and Jon Kleinberg. Simplicial closure and higher-order link prediction. Proceedings of the National Academy of Sciences, 115 (48), November 2018a. ISSN 1091-6490. doi: 10.1073/pnas. 180006811...00 URL http://dx.
doi www.org///go /t.zg/11... 1073 1800056115
Austin R. Benson, Rediet Abebe, Michael T. Schaub, Ali Jadbabaie, and Jon Kleinberg. Simplicial closure and higher-order link prediction. Proceedings of the National Academy of Sciences, 115 122) 2018b. doi: 10.1073/pnas.18000 1800683115. URL https www pnas.
org/ /tt//oo/abs//0..o
www.naass..0...0 1800683115
Norman Biggs. Algebraic graph theory. Number 67. Cambridge university press, 1993.
Cristian Bodnar, Francesco Di Giovanni, Benjamin Chamberlain, Pietro Lio, and Michael Bronstein. Neural sheaf diffusion: A topological perspective on heterophily and oversmoothing in gnns, 2022.
Alexandre Bovet and Peter Grindrod. The activity of the far right on telegram v2.11, 12 2020.
Guanzi Chen, Jiying Zhang, Xi Xiao, and Yang Li. Preventing over-smoothing for hypergraph neural networks, 2022. https::/:/ www./br.////ooooooor/o...o...ere http:/aab/222....../
Eli Chien, Chao Pan, Jianhao Peng, and Olgica Milenkovie. You allset: A multiset function framework for hypergraph neural networks. In International Conference on Learning Represen- fations, 2022. /tt.nntppss/pgeeeeeeeeeeeeeeoeoo

---

Published as a conference paper at ICLR 2026
Justin Michael Curry. Sheaves, cosheaves and applications. University of Pennsylvania, 2014.
Michaël Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks on graphs with fast localized spectral filtering. In D. Lee, M. Sugiyama, U. Luxburg, 1. Guyon, and R. Garnett (eds.), Advances in Neural Information Processing Systems, volume 29. Curran Associates. Inc., 2016a. URL https: http::/ss/////roeedeesoo new. www.eewwwws.....ppeeeeeeeeeeeeee cc/paper files files/ paper/2016/ 44440//44444444444455544b2bbbb 6
pdf
Michaël Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks on graphs with fast localized spectral filtering. In Proceedings of the 30th International Confer- ence on Neural Information Processing Systems, NIPS' 16, pp. 3844-3852, Red Hook, NY, USA, 2016b. Curran Associates Inc. ISBN
Yihe Dong, Will Sawin, and Yoshua Bengio. Hnhn: Hypergraph networks with hyperedge neurons. ICML Graph Representation Learning and Beyond Workshop, 2020. URL https: //arxiv.
org/abs/2000 2006. 12278
Julia Duta, Giulia Cassarà, Fabrizio Silvestri, and Pietro Lió. Sheaf hypergraph networks. In A. Oh, T. Naumann, A. Globerson, K. Saenko, M. Hardt, and S. Levine (eds.). Advances in Neural Information Processing Systems, volume 36, pp. 12087-12099. Curran Associates, Inc., 2023. URL |ttpps://proeeeinnnnn neurips cc wappa files/paper/202 E5122722223320000222256111333333 oog.ppaaeeeeaaaaaaaaaaaaaoereeer www.rerrreooooeeee...ooo pdf
Yifan Feng, Haoxuan You, Zizhao Zhang. Rongrong Ji, and Yue Gao. Hypergraph neural networks
In AAAI, 2019.
Stefano Fiorini, Stefano Coniglio, Michele Ciavotta, and Enza Messina. Sigmanet: One laplacian to rule them all, 2023.
Stefano Fiorini. Stefano Coniglio, Michele Ciavotta, and Alessio Del Bue. Let there be direction in hypergraph neural networks. Transactions on Machine Learning Research, 2024. ISSN 2835-
6pmvi
Giorgio Gallo, Giustino Longo, Stefano Pallottino, and Sang Nguyen. Directed hypergraphs and ap- plications. Discrete Applied Mathematics, 42(2):177-201, 1993, ISSN 0166-218X. doi: https:
URL https: www. www.scedirenecceeecceeeeeeeeeeee com/
Valerio La Gatta, Vincenzo Moscato, Mirko Pennone, Marco Postiglione, and Giancarlo Sperlí. Music recommendation via hypergraph embedding. IEEE Transactions Neural Networks and Learning Systems, 34(1344((0))))) 34(10):7887-7899 2023. doi: 10. N0.1109/NNIS
Jakob Hansen and Thomas Gebhart. Sheaf neural networks, 2020. URL htLttt:///axx org/
abs/2012.06t 06333
Jakob Hansen and Robert Ghrist. Opinion dynamics on discourse sheaves. SIAM Journal on Applied Mathematics, ((5):2033-2066) 2021. doi: 10. 10.1137720M1344108. URRlhttps:://ct orr/
10.1137/2 2017/0000141188
Guang-Bin Huang, Qin Yu Zhu, and Chee-Kheong Siew. Extreme learning machine: theory and applications. Neurocomputing, 70(1-3):489-501 2006.
Jing Huang and Jie Yang. Unignn: a unified framework for graph and hypergraph neural networks. In Proceedings of the Thirtieth International Joint Conference Artificial Intelligence, IJCAI- 21, 2021.
Steven M. Kearnes, Michael R. Maser, Michael Wleklinski, Anton Kast, Abigail G. Doyle, Spencer D. Dreher, Joel M. Hawkins, Klavs Jensen, and Connor W. Coley. The open reaction database. Journal of the American Chemical Society, 188(4 le09.00. URL https: : //o....../////// htt///di.gggg/////// 100 1001 1c09820]
2021. doi: 10. 1021/
PMID: 34727496.
Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization, 2017. URL
httpss://ars www.///g///ooo....oo.oooo. 6980
12

---

Published as a conference paper at ICLR 2026
Thomas N. Kipf and Max Welling. Semi-Supervised Classification with Graph Convolutional Net- works, February 2017. URL [cs, stat].
02907 arXiv:1609.0299
Jure Leskovec and Andrej Krevl. Snap datasets: Stanford large network dataset collection. http:
//snap.stan edu/ data 2014.
Ming Li, Yongchun Gu, Yi Wang, Yujie Fang, Lu Bai, Xiaosheng Zhuang, and Pietro Lio. When hypergraph meets heterophily: New benchmark datasets and baseline. In Proceedings of the AAAI Conference Artificial Intelligence, volume 39, pp. 18377-18384 2025.
Elliott H Lieb and Michael Loss. Fluxes, laplacians, and kasteleyn's theorem. In Statistical Me- chanics: Selecta of Elliott H. Lieb, pp. 457 Srrinerr..
Jose Mogo-Martinez, Daniel Zeiberg, Thomas Gaudelet, Noël Malod- Natasa Przulj, and Predrag Radivojac. Classification in biological networks with hypergraphlet kernels. Bioinfor- matics, 37(7):1000-10000 2021.
Xiaoyi Luo, Jiaheng Peng. and Jun Liang. Directed hypergraph attention network for traffic fore- casting. IET Intelligent Transport Systems, 16(1):85 98, 2022. doi: https://do..or//00.000 1049///t.. 12130. URL hattts:://etteeee orr www.y. http:///aaaaata......aaaaaaaaaar 1049/itr2. 12130
Zitong Ma, Wenbo Zhao, and Zhe Yang. Directed hypergraph representation learning for link prediction. In Sanjoy Dasgupta, Stephan Mandt, and Yingzhen Li (eds.). Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, volume 238 of Proceed- ings of Machine Learning Research, pp. 3268-3276 PMLR, 02-04 May 2024. URL https: 7 //procediiggs.o mll wtt.:/mmw..ppra..eeeeeeeeeeeeeee svreess/v238 ma24m. html
Vipul Mann and Venkat Venkatasubraamaiian. Ai-driven hypergraph network of organic chemistry: network statistics and applications in reaction classification. React. Chem. Eng. 8:619-635, 2023. doi: 10.1039/D2RE0030KK URL ottp///..//..........oo.//////./
Katarina A. Murgas, Emil Saucan, and Romeil Sandhu. Hypergraph geometry reflects higher-order dynamics in protein interaction networks. Scientific Reports, 12(1):20879, 2022. doi: 10.1038/ s41598-022 222--24544-w
httl::/tt::/aa http://ooooorg//////// 10. /ep////42299--255022222
Khang Nguyen, Nong Minh Hieu, Vinh Due Nguyen, Nhat Ho, Stanley Osher, and Tan Minh Nguyen. Revisiting over-smoothing and over-squashing using ollivier-ricci curvature, 2023.
Oleg Platonov. Denis Kuznedelev. Michael Diskin, Artem Babenko, and Liudmila Prokhorenkova. A critical look at the evaluation of GNNs under heterophily: Are we really making progress? In The Eleventh International Conference on Learming Representations, 2023. URL https: / //operreview.o wastt/fforummddtt.bboeeeee 5wv
Antonio Purificato, Giulia Cassarå, Federico Siciliano, Pictro Liò, and Fabrizio Silvestri. Sheaf4rec: Sheaf neural networks for graph-based recommender systems. ACM Transactions Recom- mender Systems, 4(2):1-26, 2025.
Jörg Reichardt and Stefan Bornholdt. Statistical mechanics of community detection. Physical Re- view E, 74(1), July 2006. ISSN 1550-2376. doi: 10. 11.100//hhaareee74..111111 URL http: 77dx. doi. ww..or//// 10. htt.g//...1/////a...... 1103/PhysRevE.....m 01.011110
Brian J. Reizman, Ying M. Wang, Stephen L. Buchwald, and Klavs F. Jensen. Suzuki-miyaus cross-coupling optimization enabled by automated feedback. Reaction Chemistry & Engineering, 1(6):658-666) 666, December 2016. doi: 10. (0.103//66ee00155.
Guillermo Restrepo. Spaces of mathematical chemistry. Theory in Biosciences, 143:237-251, 2024. doi: 10.7// 220777/41220----0444423333
David Rogers and Mathew Hahn. Extended-ccomeetiityy fingerprints. Journal of Chemical Infor-
mation and Modeling, 50(5):742-754, 2010. doi: 10.1021/c11000500 URL http:// doi.
org/ 1011:111111111111110
13

---

Published as a conference paper at ICLR 2026
Franco Scarselli, Marco Gori, Ah Chung Tsoi, Markus Hagenbuchner, and Gabriele Monfardini. The graph neural network model. IEEE Transactions on Neural Networks, 20(1):61-800 2009. doi: 2001009TNNN20000000000
David I Shuman, Sunil K. Narang, Pascal Frossard, Antonio Ortega, and Pierre Vandergheynst. The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains. IEEE Signal Processing Magazine, 30(3):83 98, 2013. doi: 10.1 10.1199//PP200222235522
Lev Telyatnikov, Maria Sofia Bucarelli, Guillermo Bernardez, Olga Zaghen, Simone Scardapane, and Pietro Lio. Hypergraph neural networks through the lens of message passing: A common perspective to homophily and architecture design. Transactions Machine Learning Research, 2025. ISSN 2835-8856. URt.ttts:: URLAttpps///pperrvieee
Zekun Tong, Yuxuan Liang, Changsheng Sun, Xinke Li, David S. Rosenblum, and Andrew Lim. Digraph inception convolutional networks. NIPS '20, Red Hook, NY, USA, 2020. Curran Asso- ciates Inc. ISBN 978171382954
Chiheb Trabelsi, Olexa Bilaniuk, Ying Zhang, Dmitriy Serdyuk, Sandeep Subramanian, Joao Felipe Santos, Soroush Mehri, Negar Rostamzadeh, Yoshua Bengio, and Christopher Pal. Deep com- plex networks. In International Conference on Learning Representations, 2018. URL https:
//openzeview. wett//oroummii--11222mm22m22
Loc Hoang Tran and Linh Hoang Tran. Directed hypergraph neural network, 2022. URL https: //arxxv.org///////./l...// abs/ 2008 63626
Pietro Traversa, Guilherme Ferraz de Arruda, Alexei Vazquez, and Yamir Moreno. Robustness and complexity of directed and weighted metabolic hypergraphs, 2023.
Francesco Tudisco, Austin R. Benson, and Konstantin Prokopchik. Nonlinear higher-order label spreading. In Proceedings of the Web Conference 2021, WWW '21, pp. 2402-2413, New York, NY. USA, 2021. Association for Computing Machinery. ISBN 978 450383117. doi: 10.1145/
9442381.3450035 URL htLhhttp:://ddt /ttL/https///60...r//111 http:/1..111... 11w.:11114/44444 3450035
Peihao Wang, Shenghao Yang, Yunyu Liu, Zhangyang Wang, and Pan Li. Equivariant hypergraph diffusion neural operators. In International Conference on Learning Representations (ICLR), 2023a.
Yuxin Wang, Quan Gan, Xipeng Qiu, Xuanjing Huang, and David Wipf. From hypergraph energy functions to hypergraph neural networks, 2023b.
Chaoqi Yang, Ruijie Wang, Shuochao Yao, and Tarek Abdelzaher. Semi-supervised hypergraph node classification on hypergraph line expansion. In Proceedings of the 31st ACM International Conference on Information & Knowledge Management, CIKM '22, pp. 2352-2361, New York, NY, USA, 2022. Association for Computing Machinery. ISBN 978 7781450392365. doi: 10.1145/
URL
http://oi..oo..//// 10. 1111/1511110000 3557447
Jiying Zuang,Fuyang Li, Xi Xiao, Tingyang Xu, Yu Rong. Junzhou Huang, and Yatao Bian. Hyper- graph convolutional networks via equivalency between hypergraphs and undirected graphs, 2022.
URL httttt:s////xiii/e httpsss//wwwvvvor.....e/ abs/ 000b5/22000 2203. 16939
Xitong Zhang, Yixuan Hc, Nathan Brugnone, Michael Perlmutter, and Matthew Him. Magnet: A neural network for directed graphs, 2021.
Wenbo Zhao, Zitong Ma, and Zhe Yang. DHMConw: Directed hypergraph momentum convolution framework. In Sanjoy Dasgupta, Stephan Mandt, and Yingzhen Li (eds.), Proceedings of The 27th International Conference on Artificial Intelligence and Statisties, volume 238 of Proceedings
of Machine Learning Research, pp. 3385-3393 PMLR, 02-04 May 2024. URL https:// proceedings w
ngs .mlr. / w233//2222//// zhao24c. html
Dengyong Zhou, Jiayuan Huang, and Bernhard Schölkopf. Learning with hypergraphs: Clustering, classification, and embedding. In B. Schölkopf, J. Platt, and T. Hoff- man (eds.), Advances in Neural Information Processing Systems, volume 19. MIT Press, 2006. URL ht pttp://es////pr..eeddn.ss... http:iiwnnn..uiiiiiioor.. http://aaaaaaeeeeeeeeeeeeeeeeeee f.less/appeee/00///// fi Flle
14

---

Published as a conference paper at ICLR 2026
A THEORETICAL RESULTS
A.1 SPECTRAL PROPERTIES
The following Lemma (which state for clarity even if is not reported in the paper lemma)
derives the expression of our proposed Laplacian matrix L when applied as linear operator on signal:
Lemma 1. Let ø E Ond be a complex-value signal. Component-wise, the application of L and to its normalized counterpart reads:
(I)
Σ
usle Σ (Fuse - -
DgE vifl
(L)())) = Σ
1
D
Fuug Du "x, X1
FF De x, xx)
de
u
e.NEC
oEc u7fu
Proof. We start by applying the definition of the L component-wise as in Eq. 3
II (x))
EL
F
Juo Xx
DEV
Σ
be
Fl uge
uge Xx
ΣΣ
BEC 8e
Fose
uge
Σ 5c
16 usee usle X Σ
F
uge usee use X,:
e: uée
DEE vguu
Σ de
wele
(6r -
ug Xx
Σ DEE
F Fust X
uEE
uyfu
Finally, notice that the coefficient & - is exactly the number of vertices in different from u. Thus, the term (6e - 1) Fuge X., can be written as sum of Fusl X over all E e, v + u. Substituting this back, obtain:
L
F X,, E Xu
de
usee
DEE Upfie
Σ 6: uEE
Fugex - Fuge xo)
5e
UCE
vigfu
The linear expression for the normalized case can be derived analogously.
In the remainder of this section, we report a proof for each of the theorems we stated paper.
Theorem 1. l is diagonalizable with real eigenvalues.
Proof.
The claim follows rather directly since, as is not hard to see, L is Hermitian by construc- tion.
Theorem 2. The Dirichlet energy induced by
L for a signal x E Cnd is:
(x(x) x L
1
Σ
Σ
Fuge D X.
F.
D X
2
de
F
15

---

Published as a conference paper at ICLR 2026
Proof. By definition of the Dirichlet energy as the quadratic form associated with Lý have:
Ex(x) x Σ (Lx((xx))
By substituting for (L (x)) (see the previous lemma), we have:
Ex(x)= Σ E Dms. x xx)) (Fuse Du Xu Fugle D (xx
de
vēe
Distributing the product, we obtain:
Ex(x) Σ 3
LF Fi xxx)
FasoDD
Da =x,
(
(FuseDir Du F
de
CE ucc WEC
CE
u,aEe ust
uzu
Σ
Fr Fugie Du ΣΣ
(F xx) FF FegeD 2xx
Xu
EE
uEe DEc eiffus
R Nigh
Since Ly is Hermitian, the second inner summation can be rewritten as:
Σ
(FugDDD
usg D. 2 xx)
F.g. D.
2 xx,
M,DEE uFF
1 (F (FugeDDy Dr 3,x 3x)' F.ge D 2x, + (Fuge D2 (xxx) 1 FrueeDe D. xx xxx)
1
Σ
Σ 2
(F (FugeD Dr xxx) F Dr. xx
uÇV
R
(F uuge Dr xx,) FesteDD Xx
E.
Substituting back and doubling both terms of the summation, we obtain:
(x)
1
Σ
(IFrgeD
D Ex
| F.geeDD D
xx,12 23 (Fuge xxx) " FogeDo2x,) x))x
cEE
u,rEc uyff
Thanks to the identity ||a lla-bl² ||a|12 + ||b1|2 - 2R(ubb). we conclude:
(x
Σ Σ cEB
Fr xn,
Fr D X
5e
u,vÊE ufo
Notice that the constraint * be dropped from the inner summation w.l.o.g..
Corollary 1. L is positive semidefinite.
Proof. This follows directly from the previous theorem.
Theorem 3. Amax (L S 1.
Proof. By definition, have L Ind O with Q
D
(
D E
D
16

---

Published as a conference paper at ICLR 2026
Q can be factored
(D
B( DE
(D
(D
It follows that
x 2
xx
B
D x||2 2 0,
which implies that its spectrum is nonnegative.
Since Lý Ind Q follows that the spectrum of Ly is upper-bounded by 1, which concludes the proof.
A.2 GENERALIZATION PROPERTIES
Theorem 4. For a 2-uniform hypergraph without directions, the Laplacian operator L reduces to the Sheaf Laplacian (Hansen & Gebhart 2020) (up to a scaling factor of 2) and, when considering the case of a trivial Sheaf (where Fage 1), it coincides with the classical graph Laplacian (up to scaling factor of 2).
Proof. In the 2-uniform case, every hyperedge e contains exactly two nodes (i.e., 6, - 2). Consider the general expression of the unnormalized Laplacian given in Eq. 3 Since the graph has no directions, Si0e for all u E V,e E E, and for any choice of the charge parameter q. As a result, the off-diagonal terms of L are real-valued (the diagonal ones always are).
In particular, when de 2 for all E, L reads:
Σ FugeFuuee E RRdxd u = v,
3
Juv
e: c:uEc
- FI E Rdxd
u + v,
Thus, L precisely coincides with the Sheaf Laplacian of Hansen & Gebhart (2020) up to the multiplicative constant 2-
When considering the case of trivial Sheaf (i.e., when Fast = 1). L coincides with the definition the adjacency matrix and D is the node
of the classical graph Laplacian L D A, where degree matrix.
Let us note that, in both cases, this constant factor is immaterial in practice, as it can be absorbed by the learnalle parameters of the associated neural model.
Theorem 5. For a directed 2-uniform hypergraph with unitary edge weights (i.e., We = 1, E E
with directed and undirected edges. L recovers, special case, the Magnetic Laplacian (Zhang ct al. 2021 for any E IR and the Sign-Magnetie Laplacian (Fiorini et al. 2023 when q =
Proof. The Magnetic Laplacian proposed by Zhang al. (202) is defined as
(4) D.
H (0) D. A. O e
where (4) denotes the phase matrix defined
(4) 2 (A A
and A, is the symmetrized adjacency matrix defined
A 1 (A + AT)
and D, is diagonal matrix defined as
D (D.)mn
Σ (A (A,)av for all V.
17

---

Published as a conference paper at ICLR 2026
Entry-wise, H can be written as:
Ze 2
(u, v) E
2 2miq
(v, E
H
(u, E
0
otherwise.
In the directed, 2-uniform case, every hyperedge contains exactly two nodes (6, = 2). For every
E E, the product (S ((() can take one of the following three values:
1. Undirected edge e fu, v):
-(9)) uge
(5 udge
f S(9) =1
=1
2. Directed edge e = (u, r):
S -2mig
S(9) uSe
=1
(S 1 s(())
e+2mig
uige
3. Directed edge = (v, u):
S sg 1,
S(4) esle
e 2miq
(S
(
)
t [5(8)
Doge e 22igg
Letting (w.l.o.g., as the restriction maps are learnable)
Fede: == V2, Fuse == V2 if = {u, v)
Fuge:: = 1, Fuse= =1
if f (u, v) or e = (v,
we have:
1
1
2 usle uSe =
Ft
2
et
22ti F-
uge
F =
2
e
+xrg
i = (u, v),
(Q (Q²) Jun
1
Ft F
1
2 F
usle
F
u
2miq
if (v,u),
2
2
1 Ft F
1
2 uge Juse
FT
Fu
=
if = (u,
2
Hence, by construction, we have:
Q BB DEl B(a)
H())
with Dv = D,-
This implies:
L 5 = D.
H (0) = (0)
Lastly, noticing that, by construction, the Sign- Magnetic Laplacian proposed in Fiorini et al. (2023) coincides with the Magnetic Laplacian when q = 4 we conclude that our operator also generalizes the former.
Theorem 6. Given a hypergraph H (directed or undirected), the normalized Directed Hypergraph Laplacian L recovers, special case, the undirected hypergraph Laplacian of Zhou et al. 2006
Proof. In the unit-weight case, the Laplacian matrix proposed by Zhou et al. 2006 for undirected hypergraphs is defined follows:
A QN with QN := D. B
Since any undirected hypergraph be regarded every hyperedge consists solely of tail nodes (or, equivalently, solely of head nodes), as shown in
special case of directed hypergraph in which
Eq. (4), in our proposed Laplacian matrix L each product of two restriction maps reduces a real weight of 1, therefore contributing only to the real part of the operator. In particular, for a trivial sheaf where Fugn = 1, the incidence matrix ()) in Eq. (1) reduces to the transpose of binary
incidence matrix B of al. (2006).
18

---

Published as a conference paper at ICLR 2026
Theorem 7. Given directed hypergraph 7. with unitary weights associated to each hyperedge (i.e., We 1), the Normalized Directed Sheaf Hypergraph Laplacian L recovers, as a special case, the Generalized Directed Laplacian Ly of Fiorini et al. (2024).
Proof. Let's consider special case of trivial sheaf (i.e. Fuge 1). By setting q = 1 we have:
if (()) (head set)
{
S s(0.25) use
if T(e)
(tail set)
otherwise
Now, for each pair u, v belonging to the same hyperedge e:
Ft Foge ($122)) 5(0225)
Whose contribution, according to the four cases in Eq.
is given by:
u, v É H(e),
u, u,
Fuge
u
i
T v VE H()),
WE H(e),, e T(e).
Our Normalized Directed Sheaf Hypergraph Laplacian L component-wise reads:
1
Ia D
use
(L Juu
Du
5e
Fl
uge
USc
D
+ c.
Which reduces, in the considered scalar special case to:
Σ
D &e
v,
(I
(10)
Σ de EEE (.EEM(e) v (.eET(e
-1-
Σ
u + c.
de R (EH AvET(e)
de DDvDD,"
EEE uET(e) AvEH(e)
Such an expression coincides with the definition of the Generalized Directed Laplacian when con- sidering W I.
B EXTENDED EXPERIMENTAL EVALUATION
In this section, we include further experiments and details that did not make the cut in the main paper due to space limits. This includes:
The optimal value of the charge parameter found for DSHN and DSHNL ight during the hyperparameters optimization process.
The impact of the stalk dimension d and the number of layers the method's performance.
The complete results on 12 real-world datasets.
B. IMPACT OF CHARGE PARAMETER
The charge parameter g controls how much each directed hyperedge contributes to the real and imaginary parts of the Directed Sheaf Hypergraph Laplacian. Larger values of q place more direc- tional information in the imaginary component, whereas smaller values reduce the directional con- tribution, emphasizing orientation-agnostic interactions in the real part. Because the dataset differ
19

---

Published as a conference paper at ICLR 2026
in how informative directionality is, the optimal q is inherently data-dependent. In practice, care- ful tuning of it is needed q to select the value that yields the best performance, allowing either a partial or full contribution of directional information to be encoded needed. values chosen by our hyperparameter tuning procedure. As one can seen, for most datasets the hy- perparameter tuning procedure dataset, particularly for Telegram, Roman Roman-empil empire re and Synthetic datasets.
the
relatively high importance directional information for each
Table 3: Optimal / values for DSHN and DSHNL.ight across all real-world and synthetic datasets found by hyperparameter tuning.
Method
Roman-empire
Squirrel
email-EU
Telegram
Chameleon
email-Enron
DSHN DSHNLight
0.05
0.25
0.25
0,20
0.05
0.20
0.05
0.20
0.20
0.15
0.15
Method
Cornell
Wisconsin
Amazon-ratings
Texas
Citeseer
Cora
DSHN
0.25
0.25
0.25
0.00
0.00
DSHNLight
0.15
0.25
0.00
0.15
0.00
0.00
Method
Io = 10
30
Io = 50
DSHN
0.25
0.10
0.10
DSHNLight
0.10
0.10
0.10
B.2
IMPACT OF STALK DIMENSION AND NUMBER OF LAYERS
90
90
89
Accuracy (%)
80
Accuracy (%)
88
70
87
60
86
85
50
84
40
HGNN
83*
82
2
6
7
2
3
6
#Layers
Dimension of stalk (d)
(a) Accuracy of DSHN and HGNN the number of (b) Accuracy of DSHN as the stalk dimension in- layers increases.
creases.
Figure 3: Influence of architectural parameters on accuracy. (a) Effect of the number of layers on DSHN and HGNN. (b) Effect of stalk dimension d on DSHN.
As noted in Section standard HGNNs are prone to oversmoothing: as network depth increases, node representations become indistinguishable and accuracy degrades. In Fig. we study how depth and the stalk dimension d affect the accuracy of DSHN. DSHN shows no signs of oversmoothing, as accuracy improves we add layers. Performance also increases with a higher stalk dimension d, underscoring the additional expressive power associated to cellular sheaves. This stands in clear contrast to HGNN, whose accuracy steadily deteriorates with depth. This is in line with the observa- tions in Bodnar et al. (2022) for graphs: leveraging our Directed Sheaf Hypergraph Laplacian, built with d d restriction maps transport features between nodes and hyperedges, enriches local vari- ability rather than collapsing it. By projecting node features onto hyperedges (and back), the model retains discriminative power across neighhorhoods.
20

---

Published as a conference paper at ICLR 2026
B.3 EXTENDED RESULTS
In this subsection, we report the complete table of results for this work, which were not reported in the main just due to space limitations. As can be observed from Table 4 DSHN, and DSHNLight consistently outperform the baselines taken from both the directed and undirected hypergraph learn- ing literature on 10 out of 12 considered real-world datasets.
Table 4: Mean accuracy and standard deviation on node classification datasets (test accuracy = std). For each dataset, the best result is shown in bold, and the second-best is underlined.
Method
Roman-empire
Squirrel
email-EU
Telegram
Chameleon
email-Enron
HGNN
38.44 + 0.44
35.47 1.44
48.91
3.11
51.73 + 3.38
39.98 2:28
52.85
7.27
HNHN
46.07 + 1.22
35.62 + 1.30
29.68 1: 1.68
38. 18.22-2 + 6.95
35.81 + 3.23
18.64 6.90
UniGCNII
78.89 + 0.51
38.28
2.56
41.98
2.69
51.73 = 5.05
39.85 0 3.19
47.43 7.17
LEGCN
65.60 0.441
39.18 + 1.54
32.91
1.83
45.38
4.23
39.29 + 2.04
37.03
7.16
HyperND
68.31 + 0.69
40.13 1.85
32.79 + 2.90
44.62 + 5.19
11.95 3.20
38.
7.69
DeepeeSSsts
81.79 0.72
40.69
1.90
37.37
6.29
49.19
42.97
3.60
37.29 7.90
AlnSetTransformer
83.53 0.64
40.53 + 1.33
38.26
3.37
66.92 +: 4.36
43.85 + 5.42
63.78 3.66
ED-HNN
83.82 0.331
39.85 - 1.79
08.91
0. 1.00
4.00
60.38 3.56
11.67 2.33
51.35 6.04
SheufHyperGiNN
74.50 0.57
42.01 1.11
9.13
70.00 +: 5.32
41.06
63.51 5.95
PhenomNN
71.22
39.15 - 2.19
37.69
:: 1.40
47.69 + 6.59
47.02 6.75
GeDi-HNN
83.87 + 0.63
43.02 + 3.00
52.31 + 2.84
77.12 + 4.82
39.29 + 2.04
50.54 = 5.80
DHGNN
77.58 0.51
39.85 - 1.79
32.35
2.93
79.62 + 5.78
41.08 1.11
42.16 8.04
DHGNN emb.)
+2.00+00.81
40.33
1.42
55.10
3.48
80.58 3.89
40.85 2.76
58.38 7.57
DSHN
0OM
43.55 +: 2.87
78.62 + 2.50
88.65 + 5.54
47. + 4.35
75.68 3.42
DSHNLight
89.24 + 0.57
44.09 + 2.36
82.67 + 1.29
81.15 + 4.19
46.50 + 4.09
76.76 2.48
Method
Cornell
Wisconsin
Amazon-ratings
Texas
Citeseer
Cora
HGNN
43.51 # 6.44
51.56 = 6.68
46.20 + 0.45
52.77 + 7.48
76.02
0.81
87:25 = 1.01
HNHN
43.51 6.09
49.60 4.96
42.29 + 0.34
58.11 3.87
71.24
0.66
78.16 0.98
UniGCNII
73:24 1: 5.19
86.86 4.30
49.12
81.35
5.33
77.30
1.15
87.53
LEGCN
75.14 5.51
84.71 # 4.00
47.02 + 0.59
81.35 + 4.26
72.62
1.09
74.96 = 0.94
HyperND
75.14 5.38
86.67 5.02
17.33 - 0.51
83.51 5.19
75.21
1.37
78.48 1.02
SeeDeepSets
77.83 11: 3.78
87.84 3.69
51.91 + 0.68
82.76 +: 5.74
75.78
0.94
86.86 0.85
AlIS:CTransformer
75.94
86.27 = 3.92
52.28 - 0.67
82.76 5.07
75.61 0:.11:10 1.44
86.73 = 1.13
ED-HNN
76.49 -: 4.53
85.09
- 51.58
80.00
74.95
1.27
86.94 =
SheadHyprGNN
71.59 = 1.39
1.74 85.29
18.90 = 0.59
80,00
77.21
1.41
87.15 0.61
PhenomNN
72.16 4.19
80.58 6.10
48.81 + 0.37
81.40 + 4.95
1.32 77.21
88.12 + 0.86
GeDi-HNN
78.37 3.19
87.45 3.41
49.30 = 0.52
82.55 166
75.94
85.16 0.94
DHGNN
77.30 4.05
87.45 3.84
52.48 + 0.50
83.24
74.67
1.24
83.16 1.33
DHGNN (w/ cmb.)
51.08 + 1.43
59.80 = 5.63
53.64 + 0.52
63.51 9.81
56.78 = 1.32
73.12 = 1.04
DSHN
79.19 +: 4.37
88.63 + 3.49
OOM
83.78 +: 5.13
77.39
1.04
87.84 = 0.90
DSHNLLggt
79.19 1199::0 3.20
87.25 4.90
50.94 + 0.68
82.43 + 5.44
77.45
1 0.74
88.02 1.11
Additionally, we evaluate method real-world directed hypergraph dataset for molecular
reaction reframed as hyperedge classification task, results are provided in Table 5) These datasets
the result of the merging of data from different sources such as Kearnes al. (2021): Reizman
eet al. (2116); Lugo-Martinez et al. (202I) and are built inspired by Restrepo (2024). which proposes
novel way of modeling molecular reactions through directed hypergraphs. Dataset-1 contains
100,523 nodes and 50,016 hyperedges, with a total of 10 classes. Dataset-2 contains 956 nodes and 3,021 hyperedges
classify among 6 different classes. These datasets consist of inherently directional hyperedges as they contain the molecular reactions expressed as set of reagents (the tail set) and set of products (the head set) composing a molecular reaction. The nodes" features are built based on Morgan Fingerprints (Rogers & Hahn (2100,, which one of the most widely used molecular descriptors. We employ the Fl-score metric since the data has an imbalanced amount of samples for each class shown in Appendix D.6
As shown in Table 5 DSHN consistently outperforms all competing methods from both the undi- rected and directed hypergraph learning literature. On Molecular-1, it achieves Fi-score of 82.32%, improving upon the strongest baseline, GeDi-HNN, by 1.98%. On Molecular-2, DSHN at- tains 89.09% exceeding Transrrran by relative margin of 1.37%.
21

---

Published as a conference paper at ICLR 2026
Table 5: Mean Fl-score and standard deviation for hyperedge classification two molecular re- action datasets (test Fl-score + std). The best score is shown in bold, and the second-best is underlined.
Method
Molecular-1
Molecular-2
HGNN
69.38 0.48
81.40 +: 2.68
HNHN
32.27 + 1.30
45.69 + 7.48
UniGCNII
72.00 + 0.59
85.61 + 2.63
LEGCN
0OM
84.75 + 2.68
HyperND
44.16 1.27
82.86 = 3.17
AleDeepSets
79.17 0.53
85.78 + 3.01
AllSeTTransfrmee
79.24 + 1.08
87.89 = 2.87
ED-HNN
66.37 + 2.62
87.05 = 1.96
SheafHyperGiNN
57.99 + 2.75
80.25 + 2.20
PhenomNN
47.71 + 2.90
86.27 = 2.40
GeDi-HNN
80.72 + 0.78
85.64 + 2.42
DHGNN
OOM
85.93 + 3.49
DSHN
OOM
89.09 = 3.08
DSHNLight
82.32 + 0.56
86.52 + 2.68
C IMPLEMENTATION DETAILS
We provide additional details regarding the implementation of our models, with a particular empha- sis the computational complexity of DSHN and DSHNLight and the architectural choices that contribute their stability and expressiveness.
C.I COMPUTATIONAL COMPLEXITY
Comparison between DSHN and DSHNLight Tabeelpjpresents comparative analysis of DSHN and DSHNLight, across various datasets, measuring their performance in terms of average FLOPS per epoch and average step time. The results averaged over 10 runs. Over all the 12 datasets, DSHNLight always appears to be more efficient, consistently requiring fewer computational re- sources while maintaining faster processing times. By applying the aforementioned detachment op- eration through backpropagation, DSHNLight achieves similar and sometimes better results, as can be seen from T
Table 6: DSHN vs ASHNLight-- FLOPS and Step Time (in ms) Analysis Across Different Datasets
(Mean + Standard Deviation)
Avg FL.OP\/poch(L)
Avg step time (L)
Dataset
DSHN
DSHNLight
DSHN
DSHNLight
Cora
267.070,765. 386 1
196.828.716,921 +: 3,250
2635.02 1 112.51
973.18 1: 164.18
Citescer
315.705,637,19 192 1
310,747,699,333 = 5.239
263183 + 146.54
958.34 +: 159.00
962,025, 184 1: 172
696.069 022 1: 364
2559.10 1 115.97
932.18
152.09
email-EU
35,930,593.693 1.176
25,798.032,76 1,183
4170.54 1 105.64
1018.13 1: 155.71
Telegram
2,628,033,910
1,858, 200,420
2702.33 150.89
965.40
161.12
Cornell
180,,,544340 220
1,851,871 1,,66 ,,60
220
2467.84 1 132.16
886.07 164.73
Texas
2,228.554.459 +
1,876,832,188 187 0
2480.09 = 130.18
888.02 + 164.15
Wisconsin
3,684,554, 183 201
3,035,436,853 + 454
2547.95 + 116.24
923.18 + 161.33
Chamelcon
115.734 + 123
27.033.570.34 123
2629.45 - 132.52
959.08 + 155.75
Squirel
189,607,210,489 = 5,694
140,531,198,787 3,557
3870.93 119.79
1046.00 = 169.09
Roman-empire
0OM
12,898,1477 01.99,144.9966.999 43,606
1050.30 1 152.04
Amazon-ratings
OOM
15.066..77.374..2888
OOM
1080.26 159.91
Comparison between DSHN and other models Fig. reports the average test accuracy of five representative models under approximately the same parameter budget. The results indicate that model size alone does not explain the performance of DSHN. For instance, although SheafHyper- GNN and ED-HNN have comparable number of parameters, their accuracy is significantly lower, being these undirected methods. In contrast, DSHN achieves improvement of about 8% over the
22

---

Published as a conference paper at ICLR 2026
02
Gesa
mecae
Shaph
Figure 4: Comparison between models under the same number of parameters (~ 80k) on the Tele- gram dataset.
strongest directed basclines, despite having the same number of parameters thanks to the expres- sive power associated to complex-valuee and directional restriction maps. On the other hand, Ta- ble 7 shows comparison of DSHNLight with other models. Both DSHNLight and SheafHyper- GNN generally incur higher computational costs than traditional hypergraph neural networks, though achieving, at least in the case of DSHNLight, substantially better accuracy. This overhead stems directly from the requirement learn and apply restriction maps at every node-hyperedge in- eidence detailed in the next paragraph.
Table 7: FLOPs and Parameter Count Across Datasets and Methods.
Datasct
ED-HNN
SheafHyperGNN
DHGNN
DSHNLight
FLOPs
#Params
FLOPs
#Params
FLOPs
PParams
FLOPs
#Params
Cora
x 10
125,959
2 10
404,576
7 7 1010
4,077,051
2.0 10a1
437,447
Citeseer
x 102
271,174
3 10a1
985,440
3 10
12,751,318
3.1 10S1
1,018,502
email-Enron
x 10a
34,311
6 x 10
37,984
10"
13,334
7.0 10a
70,855
email-EU
10P
34,506
3 10
38,752
10"
14.372
2,6 T0
71,050
Telegram
10P
34,116
10"
37,216
10"
13,241
1.9 10P
70,660
Cornell
x 10
143.109
10"
473,184
10
542,566
1.9 10
506,437
Texas
x 10
143.109
1 10
473,184
X 10
542,566
1,9 10
506,437
Wisconsin
x 10
143.109
X
473,184
1 10"
658,370
3.0 10
506,437
Chameleon
2 x 10P
182,917
2 1030
632,416
10
2,379,783
2.7 10
632,416
Squirrel
x 10P
167,813
1 10
672,000
10
4,924,172
1.3 10TH
605,253
Roman-empire
2 10
54,162
1 1033
117,344
1 x 10
6,850,778
1.3 10
117,344
Amazon-ratings
3 10
53,317
2 x 10
114,016
x 10
7,398,933
1.5 10
114,016
Asymptotic Complexity inference time.
We provide an estimate of the asymptotic complexity of our model at
1. Linear Transformation. The feature transformation is defined
X' (I, X W W))XW,
where W, E Baxd and W. E / The resulting complexity is O(n(Pf + df2)) On((ed + ef)) = O(ne), where e df.
2. Message Passing, Once the Laplacian operator has been assembled, message passing re- duces to a sparse-dense matrix multiplication of the form
X X
The sparsity pattern of Q comes directly from the incidence matrix: each hyperedge of size |e| induces le|2 nonzero blocks through the outer product (e,) (B(()) (e.:). Sum- ming across all hyperedges gives total of S2 Ecse le|2 ((m²)) nonzero blocks,
23

---

Published as a conference paper at ICLR 2026
where is the average hyperedge size Applying the Laplacian then requires O(mê?)) for diagonal maps and Ome de) with non-diagonal maps,
3. Learning the Sheaf. Restriction maps are predicted as
(xx, x = o (V llx.))
where V is learnable transformation and nonlinearity. The resulting f-dimensional vector is then used as input to V for every node-edge incidence. Indicating with the average number of partecipations of node to an hyperedge, the computational complexity is O(ūmc) in the diagonal case, and (ōmmdec) in the non-diagonal case.
4. Constructing the Laplacian. In the hypergraph setting we assemble
N D $ B DE D
(
The work naturally splits into two steps:
(a) Degree normalization. This involves computing the node and hyperedge degree ma-
trices, D and DE'- For vertex degree normalization each node requires aggregat- ing contributions from its incident hyperedges, giving (mmēd) operations in the diag- onal case and (mmēd") in the non-diagonal case (since each block is x d), to which it must be added the cost of inverting the block-diagonal matrices, adding to the com- plexity O ((nd) in the diagonal case and O (nd2) in the non-diagonal case while since DE is obtained by expanding to matrix for the scalar hyperedge degrees 6 this cost adds a trascurable term to the asymptotic complexity.
(b) Sparse product. Forming the term
Q DV ( D D
requires, for each hyperedge e, generating block interactions among all pairs of nodes it contains. This gives total of S2 = Eees le|2 block products. The cost is O(S2d) and O( in the non-diagonal case. Since the normalization
in the diagonal
terms D block-diagonal operations they do not contribute substantially in the overall complexity. Since S2 O (me²), the dominant cost becomes ((m²2d) for diagonal maps and Ommē*d*) for non-diagonal maps.
m Ed + (d + ūc)) in the By summing the overall contributions we get: On (c² + d) + m diagonal case and (n (c² + dd) + m (ed + 2² (d + de) u (ddc) in the non-diagonal case.
Considerations the Asymptotic Complexity The leading cost arises from the Laplacian sembly step, which scales as O ((médd) in the diagonal case and (mm?2d) in the non-diagonal case. This quadratic dependence on the average hyperedge size E² makes the method particularly sensi- tive to hypergraphs with densely populated hyperedges. In practice, this means that even when the number of nodes and hyperedges moderate, the presence of densely populated hyperedges can dominate the computational cost.
C.2 ARCHITECTURAL CHOICES
Layer Normalization Each layer may optionally include layer normalization, with this choice considered a tunable hyperparameter, since it improves training stability and overall performance. Since the input signal to each convolutional layer is complex-valuee we adopt a complex normal- ization strategy proposed in Trabelsi et al. (2018); Barrachina et al. (2033), where each complex feature is treated as two-dimensional real vector (())),(())) Specifically, we compute the full 2 x 2 covariance matrix:
Oor Ori Σ
x = 1x -)),
Øri
dii
"nn could also upper bound the S2 term with (mm?), however, that approximation would be highly pessimistic, considering fully-dense representation of the hypergraph, where each hyperedge connects all nodes.
24

---

Published as a conference paper at ICLR 2026
where (1rr)44 p4) is the vector of the real and imaginary parts. The whitening transform > ensures that the two components are jointly normalized and decorrelated. To enhance flexibil-
ity, we apply an optional learnable affine transformation in the complex plane:
N = T + -,
with trainable parameters R and B E R These are initialized as 7 vz 72 and / thereby preserving the norm of unit-modulus inputs while maintaining the identity mapping initialization.
Following observations from Bodnar al. (2022), we optionally include residual connections in our convolutional layers, which we found to help the architecture in certain datasets. The of residuals treated tunable hyperparameter (see Appendix D.3) addition, convolutional layer takes the form:
Residual Connections
With this
X (o (I, & Wi) W2 + x.) E Cndx1,
Activation Function For the activation function, adopt the complex ReLU commonly em- ployed in related works (Zhang et al. 2021 Fiorini et al. 2023 2024). defined as:
2; if R(()) > 0,
ReL ((x)
0, otherwise.
DSHNLight The architecture of DSHNLight is illustrated in Fig. 5 The model takes as input node feature matrix Xinput- which projected into a digher-dimensional stalk space via a learnable linear transformation. This representation is then used both in the pesssae-passsngg pipeline and as input to the MLP that predicts the restriction maps Fuse. Unlike DSHN, the Laplacian operator built outside the computational graph, so the MLP parameters are not updated during training. Nevertheless, the initial projection layer remains trainable, which allows the model to indirectly influence the restriction maps: by shaping the input embeddings, the network can still control the outputs of the MLP. In this way, still able to predict good values of embeddings and restriction maps, as confirmed by the empirical results in Tables 2 and
though the restriction map MILSs frozen, the model is
4iz.
(Paavall mao euuueee pazsing for layer))
Xoqa
Sook Dineecion
Projecioo
Xack
Piwlicion of Restrction Mapo
Xs. (aftz as Wy)x.a)))
Daaa
Figure 5: Illustraiion of the DSHNLight architecture. The Laplacian construction is detached from the computational graph, but the initial stalk projection layer remains trainable, allowing the model to indirectly influence the restriction maps.
D EXPERIMENTAL SETUP
D. HARDWARE DETAILS
All experiments are carried out on two different workstations: one equipped with two NVIDIA RTX 4090 GPUs (24 GB each) and an AMD Ryzen 7950X 6-core processor, and another fea- turing Intel Core 19-10940X 14-core CPU (3.3 GHz), 256 GB of RAM, and single NVIDIA RTX A6000 GPU with 48 GB of VRAM. We utilized the WandB platform to monitor training pro- cedures and to carry out hyperparameter tuning for each model.
D.2 SELECTED BASELINES
We compare our models against twelve state-of-the thootffhaaan art methods from the hypergraph learning lit- erature. From the undirected hypergraph-leaming literature include HGNN (Feng al.) 2019
25

---

Published as a conference paper at ICLR 2026
HNHN Dong et al. 2020). UniGCNII (Huang & Yang 2021 LEGCN (Yang et al. 2022
HyperND Tudisco et al. (021), AetseepSets and AllSet
(Chien et al. 2022). ED- HNN (Wang et al. 2023a), SheafHyperGNN (Duta et al. 2023) and PhenomNN (Wang et al.
2023b) From the directed hypergraph-learnnng literature, we consider GeDi-HNN Fioriiii et al, baselines. DHGNN was originally designed for link prediction on directed graph datasets and relies on learnable embedding table to represent node features. In our evaluation, report the model's performance using both this original embedding approach and an alternative setup with explicit node features.
2024) and DHGNN (Ma et al. 2024
D.3 HYPERPARAMETER TUNING
For tuning all the models, we employ a Bayesian optimization method. All models are trained for up to 500 epochs with early stopping set 200 epochs. We employ Adam (Kingma & Bal 2017) for optimizing the model parameters with Ir E {0.02, 0.01, 0.005, 0.001J, wd E (0, 5 x 10 ,, 10-4). For all the models, we adopt a dropout {0.1, 0.2. ................................ 0.9}. and for each model that has selectable number of layers for the final classifier we fix it to 2. For each baseline, we select range of parameters consistent with those investigated in their respective original works:
AltDeepSets, ED-HNN- basic blocks {2, 4, 8); MLPs per block {1, 2, MLP hidden width (64, 128, 256, 512}; classifier width {64, 128, 256}.
AllSerTransformer:: basic blocks {2, 4, 8); MLPs per block {1,2}; hidden MLP width {64, 128, 256, 512}; classifier width {64, 128, 256}; heads {1, 4, 8].
UniGCNII, HGNN, HNHN, LEGCN: basic blocks {2, 4, 8); {64, 128, 256, 512].
MLP hidden width
HyperND: classifier width 164,128,256).
PhenomNN: basic blocks {2, 4, 8); hidden width (64, 128, 256, 512}; Ao E 00 Å₁ {0.1, 50, 20}: propagation steps {8, 16).
GeDi-HNN: convolutional layers {1,2,3}: MLP hidden width (64, 128, 256, 512]: classi- fier width {64, 128, 256].
DHGNN, DHGNN (w/ emb.), basic blocks {2, 4, 8); hidden width {64, 128, 256, 512), classifier width (64, 128,55)..
SheafHyperGNN, DSHN, DSHNLight:
sheaf dropout E {false, true}
convolutional layers (1,....5)
MLP hidden width (64, 128, 256, 512) classifier width (64,
d E (1.
sheaf actvation (siggoidd d, Lanh,
left projection {false,
residual e (faise, true)
dynamic sheaf E (false, true}
E {0.00, 0.05, 0.10, 0.15, 0.20, 0.25) (for DSHN & DSHNLight only)
D.4 DATASETS DESCRIPTION
We follow the data splits proposed by Zhang et al. (2021) for the Telegran, Texas, Wisconsin, Cornell. For Chamel leon and Squirrel we adopt the splits proposed by Platonov et al. (2023). For Roman-empixe re and Amazon-Rat ings ngs we adopt the splits proposed by PPlttonov ct al. (202)) and adopt the splits of Chien et al. (202)) for the remaining ones. all cases, the datasets are partitioned into 50% training, 25% validation, and 25% test samples. For the imail-Enron and email-EU datasets and for all synthetic datasets, node attributes are not avail- able. In these cases, resort structural features, representing each node by its degree. The statis- tics of the real-world datasets as well as synthetic ones are provided Table g The datasets used for the experiments are:
26

---

Published as a conference paper at ICLR 2026
Cora, Citeseer Standard citation benchmarks in which vertices represent research pa- pers and directed edges encode citation links. Node attributes are constructed from text us- ing bfg-of-words representations of the documents.
email Mall1-EEron, email EU A corporate email communication network built from En- ron's message logs. Nodes correspond email accounts and edges record sender inter- actions. As ground-truth labels are unavailable, we derive node classes via the Spinglass community detection method Reichardt & Bornholdt (2006).
Texas, Wisconsin, Cornell WebKB datasets collected from university computer science departments. Each node a webpage, hyperlinks are edges, and features are bag- of-words page content. Pages annotated into five categories: student, project, course, staff, and faculty.
Telegram interaction network extracted from Telegram, capturing exchanges among users who propagate political content.
Squirrel, Chame leon The Squirrel and Chameleon datasets consist articles from the English Wikipedia (December 2018). Nodes represent articles, and edges represent mutual links between them. Node features indicate the presence of specific nouns in the articles. Nodes are grouped into five categories based on the original regression targets.
Roman-empi re The dataset is based on the Roman Empire article from English Wikipedia, which was selected since it is one of the longest articles on Wikipedia and follows the construction proposed by Platonov et al. (2023). Each node in the graph corre- sponds to one (non-unique) word in the text.
Amazon minazonrraiinge The dataset, proposed by Platonov al. (2023), is based the Amazon product co-purchasing network metadata dataset from SNAP Datasets Leskoved & Krevl (2014). Nodes are products (books, music CDs, DVDs, VHS video tapes), and edges connect products that are frequently bought together.
Synthetic Introduced in Fiorini al. (2024) by following the methodology adopted in (Zhang et al. (202T), these datasets are built as follows: vertex set V is partitioned into с equally sized classes C1..... C.. For each class C, we sample I intra-class hy- peredges that are undirected. The cardinality of each hyperedge is drawn uniformly from {Amin Amaax and its nodes are sampled uniformly from C.. For each ordered pair of distinct classes (C,, C2) with j, create To inter-class directed hyperedges. For ery such hyperedge e, the tail set Te) is sampled from C; and the head set (e) from Cj; the sizes [T)) and (e)) are drawn uniformly from (imin..... m
Table 8: Statistics of the datasets used in our experiments. Reported are the number of nodes, features, hyperedges, and classes, as well as the average hyperedge size (le)), the average node degree (lv)), and the clique-expansion (CE) homophily computed in Wang al. (2023a).
Dataset
Nodes
Features
# Hyperedges
Classes
avg
avg [u]
CE homophily
Roman-empire
22,662
300
22,662
18
2.73
2.73
0.2363
Squirrll
2,223
2,089
2,060
5
23.81
22.07
0.2448
email-EU
986
787
43.36
34.61
0.2608
Telegram
245
49.70
37.12
0.2854
Chameleon
890
2,325
5
12.11
10.84
0.3221
email-Enron
143
139
7
19.58
19.03
0.3251
Cornell
183
1,703
96
5
4.07
2.14
0.4200
Wisconsin
251
1,703
170
5
3.94
2.67
0,4398
Amazon-ratinn ratings
24,492
300
24,456
5
5.63
5.62
0,4460
Texas
183
1,703
110
5
3.81
2.29
0.5049
Citeseer
3,312
3,703
1,951
6
3.35
1.98
0.7947
Cora
2,708
1.433
1,565
7
4.47
2.58
0.8035
I. 10
500
250
5
9.05
4.53
0.6233
I. 30
500
450
5
10.79
9.71
0,5020
I. 50
500
650
5
11.63
15.12
0,4528
27

---

Published as a conference paper at ICLR 2026
D.5 DIRECTED HYPERGRAPH FROM DIRECTED GRAPH
Given a directed graph G = (V, E), let the out-neighborhood of E be
Nowr (v) = { w E E (v, w) E E.
We build a directed hypergraph H EV, by creating one hyperedge ey for each node with its outgoing edges and setting
T(e)) {v).
H(e,) (v).
Thus every hyperedge has a tail consisting of a single node and a head set containing all nodes belonging to the neighborhood of that tail. A clear example of this construction procedure can be visualized Fig.
Transform
é1
Figure 6: Example of the creation of directed hyperedge from the out-neighborhood of node. Suppose we have a graph where node i1 connects to nodes 12, U3, and 14, so that (V1, U2), (01, va). and (1i, 14) belong to E. The construction procedure yields directed hyperedge e1 with tail T(e)) } and head set (e1) = {12, V3, J4).
This formulation preserves the source-target semantics of the original graph by expressing them as a higher-order relation. Such hyperedges are often referred to as forward directed hyperedges (aalo al., [193). When every hyperedge is forward directed, the structure forward directed hypergraph, which is the case for all real-world datasets considered in this work.
D.6 HYPEREDGE CLASSIFICATION FOR MOLECULAR REACTION TYPE PREDICTION
In Table 9 and Table TO we show the distribution of labels for the directed molecular reaction pre- diction datasets employed for the hyperedge classification task. As mentioned in Appendix B.3 evaluate all models on additional real-world molecular prediction datasets (see Table 5 from the perspective of a hyperedge classification task. To do so, before feeding the output of the last convo- lutional layer to the classifier, we perform an aggregation (sum) of all node representations belong- ing given hyperedge. Specifically, if X the final feature matrix of shape N F, where N the number nodes and F is the feature dimension, we can compute hyperedge-leve represen- tations using (real-valued and binary) incidence matrix H of shape N E as follows:
Xmaie Xadge E EXP.
The resulting matrix Xedge is then fed to the classifier which will output a matrix of shape E x C, where C the number of classes.
E ON PREVIOUS PROPOSALS OF THE SHEAF HYPERGRAPH LAPLACIAN
In this section, we revisit the definition of the Sheaf Hypergraph Laplacian proposed in Duta et al. 2023), noting that it fails to satisfy basic spectral properties expected of a Laplacian operator, most notably positive semidefiniteness. This shortcoming motivates our formulation, which, as discussed in Section 3.4 constitutes (to our knowledge) the first definition that is fully consistent with the spectral requirements of convolutional operator also in the undi- rected setting. For comparison, we recall the (called linear in the paper the nonlinear one is, in essence, the Laplacian of 2-uniform hypergraph) Laplacian of Duta al. (2023
Sheaf Hypergraph Laplacian
28

---

Published as a conference paper at ICLR 2026
Class # Hyperedges
Percentage
15,151
30.29
11,896
23.78
5,662
11.32
909
1.82
672
1.34
8,237
16.47
6
4,614
9.23
811
1.62
8
1,834
3.67
230
0.46
Table 9: Label distribution for the Molecular-1 dataset.
Class
# Hyperedges
Percentage (%)
0
960
31.78
1,536
50.84
2 213
7.05
1.79
226
7.48
1.06
Table 10: Label distribution for the Molecular-2 dataset.
Definition 2. Let 7 = (V, be hypergraph with hyperedge degrees 8 and let Foste Rd - Rd be linear restriction maps from node to hyperedge The Laplacian L E xnd has d x d blocks indexed by u. V:
IL (L) FT www
Σ esu,VEe
1 8n
FI
C: C.ue
Definition 2 essentially coincides with Signless Hypergraph Laplacian, except for the fact that the off-diagonal entries are flipped from positive to negative Such a sign-flip suffices to build a positive semidefinite Laplacian matrix exclusively in the 2-uniform case, where the Laplacian operator for a graph can be obtained by assigning an arbitrary orientation to each edge. Notice that, in the undirected case, Laplacian differs from theirs due to featuring coefficient of (1 1) in the diagonal term, rather than 5. Considering the proposed definition, we can compute the equation of the Laplacian
linear operator for signal Rnd as follows:
(L (x)) = Σ(L
Jun X
VEV
Σ
FT de
uste Fuge XT,
ΣΣ
FT Fuge X.
de
e: uEE
e: C: DEE
Σ de
FT FTgerFnee Σ
FT For Xyi
e: UEE
DbE
-
Fugexa Inga Xx:
uuse
de
VEE
"This consistent with their implementation.
29

---

Published as a conference paper at ICLR 2026
Which substantially differs from the expression reported in their respective work, which reads:
(L*(x))
E BEE
FT
[(Fuse X - Foosex.)
de
Crucially, the latter is the expression that is obtained with our operator in the undirected case, as reported in Eq. (6).
Let us illustrate the issue with a numerical example. Let us consider a hypergraph with node set V ( U2, V3, U4 and = {(i,ee) with hyperedges e1 {u, 122 23 e2 {12, U3 U4 in the case of a trivial Sheaf (i.e. Fuse = 1). Let 6, denote the hyperedge size and let Fuge E IR be the (scalar) restriction on incidence cu.
By Definition the entries of the Laplacian are:
Jaln
dug
F
Fe
Sei
F 2e1
dag (L 0202
FI
F
Bog ooSez
FT
Josss (L
FT
F
FI
F veger
dag
Sej
beg
(L
Jouns
FT F S62
Oog
(L (o)es
1 _T F u2 4e1
801
11 Sep
(L Dojes
8og S F Se
1 FIF
(L Jores 0,
(L Doees beg
FT Forse, F Fugger dog FesserFeseen
1 FF
(L Jeres bug
FT F FusSez
Feg
( Dogas
For Forges.
FT
buz
Numerically, we have:
1
1
3
3
3
22
0
The spectrum of the above Laplacian is:
eig (L 3
1+v17
1-v17
3
Since a negative eigenvalue appears, L is not positive semidefinite in this example.
30


Reviews:
[{"review_id": "8scsYZDMjq", "reviewer": "Reviewer_rWec", "summary": "This paper extended Sheaf Neural Networks to work with directed hypergraphs. The authors define Cellular Sheaves for directed hypergraphs and then the Laplacian matrix associated with it. The authors show that the Laplacian matrix associated with directed sheaf hypergraphs has some favourable properties such as being diagonalizable with real eigenvalues, positive semidefinite, and has bounded spectrum. The authors also show that the new Laplacian matrix generalizes several previously considered special cases, including the classical graph Laplacian, the Magnetic Laplacian for directed graphs, and some existing hypergraph Laplacians. By parameterizing the heat diffusion induced by the new Laplacian operator, the authors obtain a neural network architecture which they call Directional Sheaf Hypergraph Network (DSHN). The authors show that DSHN outperforms several baselines over both real-world and synthetic datasets.", "questions": "None", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The Laplacian operator for directed sheaf hypergraphs is new and generalizes several previously considered special cases (but the extension is straightforward).\n\nThe experiments are reasonably comprehensive and well presented. Compared with existing baselines, overall there is a notable improvement in accuracy. Figure 2 clearly demonstrates the benefits of taking directionality into account and having a hyperparameter q which adjusts the level of directionality to be used.\n\nOverall, the paper presents a simple yet seemingly effective idea to solve node classification tasks on directed hypergraphs.", "weaknesses": "The presentation and clarity can be improved. In particular, for readers who are not very familiar with cellular sheaves and Sheaf Neural Networks, the notations are a bit heavy and difficult to parse. The authors should try not to overload the text with unnecessary and complex notations. Maybe using different fonts for matrices/vector spaces/functions/maps could help. The choice of notations should be clearly stated at the beginning.\n\n(This is not necessarily a weakness of this paper, but my rating is definitely affected by this) For several years I have been reviewing papers that propose new GNNs to solve node classifications tasks on standard benchmarks, and over time I start to get bored at reading papers that aim to improve node classification accuracy on standard benchmarks only. Indeed, with new methods like the one introduced in this paper, you can improve the performance on benchmarks. But really, how many of these new methods are practically relevant at all (i.e. being implemented in practice to solve actual tasks)? The benchmarks are so poor that many of the node features are still bag-of-words one-hot encodings. Even if a new method performs better on these benchmarks, I am not sure how well this will generalize to practical settings, where you can at least use a language model to generate much better features from raw texts. Because of this, I think the practical relevancy of this work is relatively low. Also see the recent position paper [1]. This work would have been great if the authors also develop some new benchmarks that are more aligned with the current ML practice (e.g. use better features).\n\nRefs:\\\n[1] M. Bechler-Speicher et al. Position: Graph Learning Will Lose Relevance Due To Poor Benchmarks. ICML 2025", "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": "rLgYGu5Ja6", "reviewer": "Reviewer_SDBh", "summary": "The paper introduces the concept of directed hypergraph cellular sheaves, extending sheaf neural networks to directed hypergraphs. The paper proposes a directed sheaf hypergraph Laplacian, a principled spectral operator. Finally, the paper introduces DSHNN, combining sheaf theory with directional information, enabling strong performance on homophilous/heterophilous graphs.", "questions": "My questions are integrated with the weaknesses section.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "### Strengths\n\n- **Interesting technical problem.** Extending sheaf neural networks to directed hypergraphs is an interesting technical problem.\n- **Principled operator design and theoretical foundations.** The work provides a directed sheaf–hypergraph Laplacian with principled spectral properties and a unifying view that recovers prior (graph/hypergraph/magnetic) operators as special cases. The theoretical results give a well-posed basis for diffusion/convolution with per-incidence maps and directional phases.\n- **Empirical competitiveness within the sheaf/hypergraph family.** The proposed models outperform or match prior sheaf/hypergraph baselines on a range of benchmarks.", "weaknesses": "### Weaknesses\n\n- **High-level intuition is hard to understand.** For a reader unfamiliar with sheaf neural networks, I struggled to understand some of the motivation and would appreciate if the paper expanded on the intuition for the approach. One example of where I think expanding on intuition would help is why sheaves mitigate oversmoothing, oversquashing, and heterophily. This would help the reader appreciate the goal to extend them to directed hypergraph case. Another is if the paper could expand at a high level why the theoretical results are important. Why is it necessary that DSHN satisfies all the proven properties? Though each of these results are sound and clearly presented, I struggled to understand why each of these are desirable to have.  \n- **Empirical setup and baselines.** I am not convinced by the decision to process datasets such as Cora, Chameleon, and Squirrel as directed hypergraphs. What does this star-expansion gain over modeling the dataset as an undirected graph (how the datasets are usually processed)? Additionally, if the authors are aware of or could evaluate on datasets with native directed-hypergraph representations (where one-to-many interactions are intrinsic), this would strengthen the empirical evaluation. As it stands now, the datasets evaluated on are usually processed as undirected graphs and I'm unsure what the issue is with these standard ways of processing them. Finally, I would appreciate if the authors could justify when to use DSHN over standard MPNNs and MPNNs designed for heterophily. Currently, there is no discussion of or empirical comparison against against strong heterophilous GNNs (e.g., H2GCN, GCNII). I think these should be discussed and ideally evaluated against since much effort has been dedicated to these types of GNNs for homophilous and heterophilous datasets.", "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": "ifaefIFsol", "reviewer": "Reviewer_YUAy", "summary": "The paper addresses the absence of a directed Sheaf Hypergraph Laplacian in the literature by proposing a novel definition, together with the introduction of corresponding Directed Hypergraph Cellular Sheaves and a Directional Sheaf Hypergraph Network. The authors motivate their approach by noting that existing work fails to properly address node classification in directed hypergraphs, particularly under heterophilic conditions. They evaluate the proposed model on hypergraph datasets with varying levels of homophily and further assess its effectiveness in a synthetic experimental setting.", "questions": "**General concerns**\n- Section 3.3 provides solid theoretical results, but two elements are missing: (1) a discussion of the practical meaning of these properties, how they relate to the model’s behavior when used to define a convolutional architecture, and (if possible) a connection to the experimental results; and (2) a comparison to SHN, as mentioned above, to explicitly support your claims.\n- Could you be more specific regarding the novelty of the objects you use in your definitions? For example, it is not clear to me whether the object $\\mathcal{S}^{(q)}$ represents a novel contribution or not.\n- What is the computational and runtime overhead associated to your model with respect to SHN and other non-sheaf-based hypergraph networks?\n- In the experimental section, there is no introduction or citation of some of the most relevant baselines (e.g., GeDi-HNN, DHGNN).\n\n**Structure of the paper**\n- Looking at the overall structure, I would argue that too little space is dedicated to properly introducing the background and previous methods that are central to the work (e.g., SHN), while a large portion focuses on detailed derivations of various sheaf objects. I suggest moving some of these explicit equations to the Appendix and keeping more intuitive explanations in the main text.\n\n**Notation**\n- I recommend to define more thoroughly the notation used in Section 2, for example specify how $| . |$ is used in different contexts, and unify the notations $\\delta_e$ and $\\delta(e)$.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- The work addresses a problem that was not thoroughly investigated in the literature before, providing a theoretically interesting solution.\n- The proposed method is well justified theoretically, with several proofs regarding the properties of the newly defined Laplacian.\n- The motivations, goal and method are overall clear. \n- The study on the effects of the parameter $q$ in Fig. 2 provides a nice and easily understandable justification for its introduction.", "weaknesses": "A major concern is the lack of an introduction to the Sheaf Hypergraph Network (SHN) [1] method, along with a comparison to your proposed method, in the main text. Strong claims are made in the introduction, stating that SHN does not actually possess some of the properties claimed by its authors, and that the proposed approach addresses this issue. However, in the main text there is no direct reference - even at an intuitive level - to these claims or to how the new method compares to SHN.\n\nAdditionally, the introduction of a sheaf structure generally adds computational and runtime overhead compared to a simple prediction model, so its inclusion should be well justified in two respects: (1) whether the additional cost is outweighed by the gain in prediction accuracy, and (2) whether there is a real practical need for a sheaf-based model. Regarding (1), a discussion of the computational cost, ideally comparing the model with non-sheaf baselines, is missing. Regarding (2), I refer to the absence of experiments on natively directed hypergraph datasets, since the considered ones are artificially derived from existing directed graph benchmarks. I understand that this limitation may be difficult to address experimentally due to time constraints, but it would still be helpful to provide some discussion about the availability of such datasets and whether they have been used in similar experimental settings, to clarify the data landscape and the actual necessity for directional hypergraph methods.\n\n[1] Duta et al., Sheaf hypergaph networks, 2023", "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)