Differentiable Lifting for Topological Neural Networks
Authors: Jorge Luiz Franco
Gabriel Duarte
Alexander V Nikitin
Moacir A Ponti
Diego Mesquita
Amauri H Souza
Venue: ICLR 2026 Poster (2026)
Keywords: Topological Deep Learning
Graph Neural Networks
graph classification
Primary area: learning on graphs and other geometries & topologies

Abstract: Topological neural networks (TNNs) enable leveraging higher-order structures on graphs (e.g., cycles and cliques) to boost the expressive power of message-passing neural networks. In turn, however, these structures are typically identified a priori through an unsupervised graph lifting operation. Notwithstanding, this choice is crucial and may have a drastic impact on a TNN's performance on downstream tasks. To circumvent this issue, we propose 
 ∂lift (DiffLift), a general framework for learning graph liftings to hypergraphs and cellular, simplicial, and combinatorial complexes in an end-to-end fashion. In particular, our approach leverages learned vertex-level latent representations to identify and parameterize distributions over candidate higher-order cells for inclusion. This results in a scalable model which can be readily integrated into any TNN. Our experiments show that  ∂lift outperforms existing lifting methods on multiple benchmarks for graph and node classification across different TNN architectures, with TNN+ ∂lift combinations surpassing standard GNN baselines. Notably, our approach leads to gains of up to 45% over static liftings, including both connectivity- and feature-based ones.

Published as a conference paper at ICLR 2026
DIFFERENTIABLE LIFTING FOR TOPOLOGICAL NEURAL NETWORKS
Jorge Luiz Franco University of São Paulo, Instituto Curvelo
Gabriel Duarte
Alexander Nikitin Aalto University
Federal Institute of Ceará
Moacir Ponti University of São Paulo
Diego Mesquita
Amauri H. Souza
Getulio Vargas Foundation, 26 Al
Federal Institute of Ceará, 26 Al
ABSTRACT
Topological neural networks (TNNs) enable leveraging high-order structures on graphs (e.g., cycles and cliques) to boost expressive power of pesssage-passing neural networks. In turn, however, these structures are typically identified priori through an unsupervised graph lifting operation. Notwithstanding, this choice is crucial and may have drastic impact TNN's performance downstream tasks. To circumvent this issue, we propose Ulift (DiffLift), a general framework for learning graph liftings hypergraphs and cellular- and simplicial complexes in an end-to-end fashion. In particular, our approach leverages learned vertex-level latent representations identify and parameterize distributions over candidate higher-order cells for inclusion. This results in scalable model which can be readily integrated into any TNN. Our experiments show that Ølift outperforms existing lifting methods on multiple benchmarks for graph and node classification across different TNN architectures. Notably, our approach leads to gains of up 45% over static liftings, including both connectivity- and feature-based ones.
INTRODUCTION
Topological neural networks (TNNs) (Papillon et al. 2023b Bodnar et al. 2021a Verma et al. 2024 have recently emerged as prominent class of models for learning on topological domains, such as hy- pergraphs and simplicial complexes, with many researchers arguing they represent the new frontier for relational learning (Papamarkou et al. 2024). Akin to graph neural networks (GNNs) (Scarselli et al.) 2009; Gilmer al.) 2017), typical TNNs employ message-passing layers where each element of the input (e.g., nodes or cells) updates its representation (features) based on those of its topological neigh- bors. Thus, these models generalize convolution- operations graphs higher-order relational objects. Importantly, the primary application of TNNs has been to enhance the capabilities of graph- based models, particularly in terms of expressivity (Bodnar al. 2 b). In this context, the input graphs must first be transformed to the domain on which TNN operates process known as lifting.
Lifting methods explore graph connec- tivity and features to create higher- order relational structures. For in-
Choice of domain dana-dependent
Choice lifting data-dependont
100
Simplicial Hypeyragh
E-hop
Aszuncy (3)
stance, clique lifting (Bodnar et al.)
60
20216) produces simplicial complex by leveraging cliques in the input graph while cycle lifting (Hajij et al. 2022)
40
60
Cor
Wisconsin
detects cycles to create cell complex. In general, there are many lifting pro-
Figure 1: [[eftt] Lifting to different domains can lead to disparate performances. Accuracies taken from the best TNNs in Telyarr nikov al. 2025). [Right] Performances of liftings to the same domain (hypergraph) vary greatly Values taken from Table 4
cedures for each topological domain c.f. Tab. 3 in Telyatnikov et al. (2025).
Not surprisingly, the optimal choice of topological domain and lifting procedure for each task is non- obvious, and its impact on performance is highly data-dependent. Figure I] compares TNNs on differ- domains, showing opposite behaviors depending on data, even within the same topological domain.

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Input graph
Node embeddings
(Hierarchical) sampler
Lifted cell complex
sampled hyper-odge
X
Graph neural
Topological
MLP
MLP
sampled rank-1 cell
sempled rank-2 call
Figure 2: Overview of Olift. For a given input graph, we first compute node embeddings using GNNs. Then, these embeddings select cells/hyperedges. Cell-level embeddings run through MLPs responsible for returning acceptance probabilities. For hierarchical domains (e.g., cell complexes), cells are generated in increasing dimensionality, From the accepted cells, we form relational object that is sent to an off-the-shelf TNN for graph/node-level predictions. The model is trained end-to-end.
Strikingly, despite the high impact of the lifting operation on TNNs, most lifting methods are not supervised and thus not informed by the task hand (Hajij al. 2022] Telyatnikov al., (025)). which may lead to suboptimal architectures. To date, differentiable lifting has only been explored the context of cell complexes (Battiloro al. 2024).
This work proposes Ülift (DiffLift) general, differentiable lifting framework applicable to various domains, including hypergraphs and simplicial/cell complexes. Our method uses a probabilistic approach to sample candidate cells of adaptive sizes. Specifically, parameterize distributions over cells using node embeddings derived from arbitrary graph models (e.g., GNNs or Graph Transformers (Rampášek et al. 2022)). For each candidate cell, we compute its embedding and use multilayer perceptron (MLP) to estimate the probability accepting or rejecting the cell that is, determining whether it should be included in the output structure. Figure 2 provides schematic overview of Òlift.
To model the typical hierarchical structure of topological objects, we propose an iterative sampling procedure, where cells are generated in increasing order of dimensionality: samples of dimension i are used to inform the sampling of dimension (1 + 1) -eellss. Notably, our approach generalizes across multiple topological domains and be seamlessly integrated into standard TNN pipelines.
We evaluate olift on 12 datasets spanning graph and node classification tasks using four different TNN models. Our results show that Alfft consistently outperforms unsupervised lifting methods in nearly all graph-level classification benchmarks - achieving superior performance in 22 out of 24 experiments, often by substantial margin. These gains are robust across all TNN architectures. For example, when using CW Networks (Bodnar et al. 202 Olift yields performance gains of up to 45%. For node classification, Ulift achieves competitive performance relative to static lifting methods and outperforms DCM (Battiloro et al., 2024) (a differentiable lifting baseline) overall. Additionally, we analyze the sensitivity of Ali to the choice of its GNN component, highlighting that while this choice often impacts the overall performance, our design is robust and produces strong empirical results even when adopting simple GNNs (c.g., graph isomorphism networks (Xu et al. 2019
2 BACKGROUND
This section overviews the main types of relational structures and respective neighborhood notions, message-passing networks for relational data, and graph lifting methods. In the following, we assume readers are familiar with basic notions in topology: see (Munkres 2000) for reference.
set of vertices (or nodes) and E is a set of unordered vertex pairs, i.c., edges. The set of neighbors of a node v in G denoted by A (v) = {u V {u, u} E E}. Hypergraphs generalize graphs by allowing edges connect multiple nodes. Formally, hypergraph (V, K), where K C 2 ( and its elements are called hyperedges.
Graphs and hypergraphs. We denote an undirected graph as tuple = (V, F.) where
nonempty set V is pair
Simplicial complexes topological spaces comprised of simple mathematical objects called simplices (points (0-simplices), line segments (I-simplices), triangles (2-simplices), and their higher-dimennsonal analogues). In particular, abstract simplicial complex (ASC) over vertex
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Graph
Simplicial Complex
Cell Complex
Hypergraph
Combinatorial Complex
Figure 3: Examples of topological domains.
set V is set K of subsets of V (the simplices) such that, for every C a, we have that E K. Thus, we can define ASCs as a family of subsets K C 2 of V that is closed under taking subsets. The dimension of a simplex is equal to its cardinality minus 1, and the dimension of an ASC is the maximal dimension of its simplices, We say T is on the boundary of simplex denoted by the boundary relation of K. We note that undirected graphs correspond to I-dimensional ASCs.
K and every non-emply
a, iff r C and there is no ó such that C C i.e., < defines
Cell complexes. A regular cell complex (Hansen and Ghrist 2019) is topological space X with partition (Xa hoePx of subspaces X, X called cells such that
1. For each E X, there is open neighborhood of that intersects finitely many cells;
2. For all o, T Px, X, nx, * 0 only if X S X., where X, denotes the closure of X, (the intersection of all closed sets containing X,);
3. Every cell X, is homeomorphic to I2 De for some D3 which we call X, 's dimension;
4. For all Px. there is homeomorphism é of closed ball in RD Xa such that the restriction of è to the interior of the ball is a homeomorphism onto X₂.
Importantly, the conditions (2) and (4) impose poset structure S a = X, C Xa which fully characterizes the topology of the underlying cell complex X. This topological information can be and there is cell ó such that a < 8 < т, where denotes the strict version of the partial order < above. We note that the class of cell complexes subsumes simplicial complexes. For more details on cell complexes, we refer to No/Hatcher (2002); Bodnar et al. (2021a).
described by the boundary relation between two cells: a < t ilf
Combinatorial complexes. combinatorial complex (CC) (Hajij et al. 2022 is a tuple (V, K, rk) where V is finite set, K C 2 \6 comprises a set of cells, and rk K - Zzo is ranking function s.t.
1. For all v E V. (V K; 2. For all o, K.a Go = <r((o'))
The idea of CCs is generalize hierarchical structures (e.g.. simplicial complexes) by imposing mild relationships between cells via ranking functions - CCs only require the order- preservrse property condition (2) while being flexible accommodate non-hierarchical structures such
hypergraphs. Figure 3 depicts the most popular relational structures in topological deep learning.
Neighborhood structures. We can exploit boundary relations (or rank functions) to specify local neighbors for each cell. In particular, [Bodnar et al. (2021b) introduce four neighborhood structures:
Boundary and co-boundary: (n (T: o and Nc(o) = (T: < <T), respectively st st 8, x 8) and (1) 36 st 8 x T, 5 o
Upper/lower adjacency:Ny((r)==
Analogs of these neighborhoods can also obtained via ranking functions (Hajij et al. 2022 p.
Features signals. In this work, consider relational structures equipped with features. Let K be a set cells or hyperedges relational domain. Its attributed counterpart is tuple (K, x) where K Rd assigns a feature vector 2(ar) each cell Hereafter, we denote the features of a by Co-
Topological neural networks (TNNs). Most TNNs use message-ppssingg mechanisms to obtain cell- level representations (Papillon et al. 2023b). In particular, let be finite sequence of neighborhood structures, No(a, = Nela) Nc(c. and Na r) NB(aa) (WB(T). In its general form,
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starting from hg Io for all message-passingg TNN Bodnar al. 2021a recursively computes
(or,th, Mg kg)) : Ni()), NB(a,T) if = =4 m (oe,lhy Ny. hg) : Ni(o), Nc(a,T) if MM
(1)
(he,s : : (N(o)) otherwise.
1 40
No X
Agge (mi,o)
(2)
where by is the embedding of layer l, & and Agge are inter- and intra-neighborhood aggregation functions, respectively, and is an update function (e.g., MLP).
Graph lifting. A graph lifting is map lift G - T from the space of attributed graphs, G, to target domain, T, such that G F G' = lift(G) En lift(G"), where दy denotes the isomorphism relation in domain T. One of the most widely used methods for lifting graphs to cell complexes is cycle lifting (Bodnar et al. (021a). This a static (non-learnable) approach that constructs 2-cells by identifying basic cycles (elements of cycle basis) chordless cycles (Bodnar al. 202 Ia in input graphs. Specifically, the vertices involved in a basic cycle are grouped to form a 2-cell in the a minimal set of cycles such that any other cycle in G' can be expressed as a modulo-2 sum of cycles from this set.
resulting cell complex. cycle basis of graph
3 DIFFERENTIABLE LIFTING
In this section, we introduce Qlift (read DiffLift)), a general framework for learning graph lifting functions. /Section provides an iterative description of our method, allowing for learning structures of increasingly higher order when the target domain is hierarchically structured. [Section and and Section 3.3 instantiate Alift. for hopp-to--hypergraphh and graph-to-cell-comple. liftings, respectively. Moreover, we formulate our approach for simplicial- and combinatorial complexes in (Apeendix A
3.1 GENERAL FORMULATION
Lifting consists of determining which higher-order cells should be added to an input graph G, satisfying the constraints of the target domain. To do so, propose the following recipe.
Ulift general recipe for differentiable graph liftings
Input: Attributed graph = E,z,, target domain T, and maximum dimension Dmax-
Compute node embeddings. Use an arbitrary GNN to compute a vector representa- tion (embedding) 2u for each node V. This GNN component can be either pre-trained model or learned end-to-end. Set the current domain dimension to D = 1.
Step 1:
Step 2: Elicit candidate cells. Given the node embeddings {ae]vev, define set of candidate cells C C 2 of dimension D. For each cell C E C. compute an embedding 2C DDec 2n. where + is an arbitrary permutation-invarian aggregation function. Note that the exact procedure for defining candidate cells depends on the target domain T, as candidates must respect possible hicrarchical constraints.
Step 3: Accept/reject candidate cells. Apply neural network ó (e.g., MLP) that defines an acceptance probability ((zc: for each candidate cell C. Finally, draw a sample yc from Bernoulli distribution with parameter ((2c) indicating whether cell C is accepted not. The resulting domain is then given by V U U {C E yc = with yo ~ Ber(o(=c))
Step 4: Termination check. If D = Dmax. halt; otherwise, D + D + and return to Step 2.
Importantly, Olift is learned in an end-to-end fashion, using the straight-through estimator (Bengio et al.) 2013) to propagate gradients through samples at Step 3. For hypergraphs, we assume hyperedges have dimension one, causing Ulift stop once it reaches Step 4. We note that Steps 2 and are the only domain-dependen ingredients of our algorithm. Next, we explain how these
steps can be adapted to specific domains.
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Learning I-dim celks
Leurning 2-dim cells
ly
Ri
Find busic cycles
Set Punetion
DocppSer
Tu
Figure 4: Two iterations of Olift for cell complexes. the first iteration, leverage each node u's GNN embedding Su to delineate candidate 1-dim cells. Specifically, we consider cell-equivalent of edges linking to each of its ki NNs in embedding space, where ky a random variable parameter- ized by Zu. We use set function over the embedding of nodes within each cell to compute their accep- tance probabilities.
the second iteration onwards, we use cycle lifting in our augmented cell com- plex to elicit candidate cells, whose acceptance probabilities are computed similarly to the first step.
3.2
LIFTING
+ [Step 2 For notational convenience, suppose we wish learn up to one hyperedge per node. For each node u, we define candidate hyperedge C(v) using the k. nearest neighbors of in the embedding space:
C(v) = (S [S] Ky and w & S = dist(Zw: 20)
max dist zo)3,
(3)
wES
where - denotes a dissimilarity metric. Here, consider the Euclidean distance.
To allow for adaptive hyperedge sizes, we sample ki according to probability distribution param- eterized by (a function of) v's emhedding 2,- More specifically, we define the (kmax kiin + 1)- dimensional probability vector ne X exp MLP(Zz)) and draw k2 ~ (TT.)) where kmin and kmax lower- and upper-bounds on kg.
+ [Step 3] We define the probability of acceptance (i.e., of b, 1) for C(v) as a function of the (multiset of embeddings of) nodes C(v). More specifically, we define by as
by Ber (V( E C(v))))
(4)
where V is learned and maps from multisets (i.e., V is
of elements in F to (0, 1).
Feature liftiing Each accepted hyperedge C(v) receives a feature vector IC(e) computed as a multiset operation over (I : u E C(v) B.- Specifically, we employ a scaled sum projection:
TC(()
Σ
Tur
Vo such that by 1.
Too wEC(v)
3.3 GRAPH-TO-CELLI-CMMPLEE LIFTING
For computational reasons, we split the lifting procedure for cell complexes into two cases. We provide an overview of our proposed graph-to-cell-comple lifting in Figure
Case D = 1: Learning edges
+ [Step ] Similarly to Step 2 of rraph-to-hypergee lifting, for each node v, we sample neighborhood size k, and define a set C(v) C V \ (v) containing the nodes associated with the ki nearest neighbors
in the embedding space, excluding e itself.
+ [Step 3] Next, we construct candidate edges (1-cells) by considering each pair (v, v') and define their probability of acceptance (i.e., of by = as function of the embeddings of and v'. Specifically, we set by,ar ~ Ber (f( 2y B)), where V is order-invariant function.
At end of this iteration, the obtained cell complex is given by:
Kd = V E.
V, C(v) with by = 1).
Regarding feature lifting, apply scaled sum projection, identically the hypergraph case.

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Case D 2 2: Learning D-cells
To select candidate cells of arbitrary dimension, we need the notion of n-cycles of a cell complex. Let (K) denote the n-chains of the cell complex K equipped with 2/222 space structure. Also let 0,, (K( + 1n-1(K) be the boundary linear map on K. Then, the n-cycles of K are given by Zn (K) = k.r()n). We provide further details in the supplementary material.
+ [Stepp 2] Let K be the cell complex at the end of iteration D -ー 1. We select a basis for of cycles Zp- (K in KP-1 serve as candidate cells. Recall a basis for cycles is minimal collection of cycles such that any cycle can be written as modulo-2 sum of cycles in the basis. We note that we can also employ chordless eycles define the 2-cells, in (Bodnar al. 2021a).
+ [Step 3] Let C be the set of candidate (D - 1) cyycles from Step 2. We define the probability of accepting C' E C (i.e., setting be 1) using DeepSet model (Zaheer et al. 2017) over the multiset of embeddings 2 Bree of all nodes in C. The output complex at this iteration is then
KD D-1 UC EC E C bc =1.
For simplicity, again, the features of D-cells are obtained via sum projection lifting.
Remark 1 Despite the generality of Alif,, in the experiments we only consider 2-dimensional cell complexes (Dmax = 2) and use the algorithm in (Paton) 1969) (available at the toolbox Net- workX (Hagberg al. 2008)) to identify basic 1-cycles in graphs. This is mainly due to the fact that current implementations TNNS for cell complexes only support 2-dimensional objects- for instance, TopoBench (Telyatnkovv et al. 2025
Remark 2 We can obtain deterministic version of Olift using a probability threshold, i.e., we simply set be = 1 (.)( > 7 with. e.g., = 0.5. for all candidate cells C.
Experiments regarding the deterministic version can be found in the Appendix E.2
4 RELATED WORKS
Topological deep learning. Traditional graph deep learning methods are limited to modeling only
pairwise interactions, making them unsuitable for capturing higher-order dependencies involving
multiple nodes (Hajij et al. 2022 Papillon et al. 2023a). To address this limitation, a variety of
deep TopologicaTTerrning learning methods have been developed for hypergraphs (Bai et al., 202T Yadati
et al. 2019), simplicial complexes (Hajij al., 2202: Goh et al.) 2022 Maggs et al. [2024]]] Yang
ct al. 2022). and cell complexes (Haji) et al.; [022] (222)) the works that are based on the topological
signal processing field (Barbarossa and Sardellitti 2020 [chhau et al. 2021 Roddenberry al.
2022, Sardellitti et al.] (202). Paapllonn et al. (2025) also use GNNs to enhance TDI., where the lifted
topological domain is transformed into augmented Hasse graphs. These methods have demonstrated their effectiveness across several practical applications, including action recognition (Wang et al.
2022
Hao et al. 2021), bioinformatics
(Liu et al. 2022) and neuroscience (Wang et al. 2023).
Liftings to topological domains. Most relational datasets and benchmarks defined on discrete structures such as graphs. To apply topological deep learning methods to these datasets, a trans- formation process known as lifting is required, which maps discrete data into topological domains (Telyatnikov et al. 2025 Hajij et al. 2022 Bernárdez et al., 2024). This lifting process can be either predetined e.g., based on structural features like node proximity or the presence of cycles or learned
directly from the data (Battiloro et al. 2024 Kazi et al. 2022). Graph structure learning methods
QQian et al. 2024; Kazi et al. 2022 Franceschi
2019 Topping at 2022 Sun al.) 2023
30233/her Chen
ct al. 12200 Jin et al. 20200) are closely related to the graph lifting literature and can be interpreted as
instances of graph lifung graph domain. Our approach represents the most general form of learnable
lifting proposed so far and empirically outperforms the aforementioned methods in many henchmarks.
Statie liftings. To the best of our knowledge, Bodnar et al. (2021b a were the first to combine static liftings and high-order message passing, focusing on simplicial- and cell complexes. These static liftings embed graph into a topological domain by, e.g., aggregating each node's n-hop neighborhood or by tracing its cycles. The repertoire of statie liftings was later broadened by

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the ICML TDL challenge (Bernárdez et al. 2024), which added methods based on kNN, Voronoi decompositions, and random walks. Our work proposes a more flexible, data-driven approach to defining liftings, which offers benefits across range of tasks.
5 EXPERIMENTS
In this section, we evaluate Ølift on two complementary tasks: graph classification and node classifi- cation. We compare it against broadly used lifting schemes for both hypergraphs and cell complexes. We also report results across different TNNs within each of these domains. We run experiments using PyTorch (Paszke al.) 2017) and Py PyTorch Geometric (Fey and Lenssen 2019); our code is anonymously available at https: / thub. com/ Homm/Jorgeuiiae www.i//aaannnnniiiiiiiie
5.1 GRAPH CLASSIFICATION
Datasets, We evaluate model performance on six widely used graph- benchmark datasets for
molecular property prediction: NCIL, NC1109, MUTAG, MOLHIV, PROTEINS, and ZINC (Kersting
et al. 2016; Dwivedi et al. 2023 Hu et al., 2020). These datasets are standard benchmarks in the
literature for assessing the effectiveness of graph-based models (Dwivedi et al. [023] Telyatnikov
ct al. 2025). All tasks are binary classification problems, with the exception of ZINC, which is regression task. We provide more details regarding datasets in the Appendix.
Baselines. We compare θlift with four existing graph lifting methods: cycle lifting, k-hop lifting, n--narest-neighhor (k-NN) lifting. and kernel lifting. Among these, cycle lifting is the most widely adopted strategy for graph-to-cet wiw.-too-elll-lpeese liftings and has become the de facto standard in most of TNNs operating on cell complexes (Telyatnikov et al., (2255). Similarly, k-hop lifting is the predomi- nant approach for constructing hypergraphs from graphs and is often the sole method considered in re- cent benchmarks such as (Telyatnikov et al. 2025). We also consider k-NN lifting as it shares similar- ities with our approach due to the use of K-NN. Finally, we consider kernel lifting, one of the most suc- cessful approaches in the ICML TDL challenge (Bernárdez et al. (2224). Notably, our choice of base- lines covers lifting methods based on connectivity (cycle and k--hop liftings), features (k-NN lifting), and both connectivity and features (kernel lifting). We provide formulations for the baseline liftings in Appendix B We consider the following TNNs: CWN, CIN (Bodnar et al. 2b22 and CXN (Hajij et al. 2020) for cell complexes, and UniGCNII and UniGIN (Huang and Yang 2021 for hypergraphs.
Evaluation setup. For ZINC and MOLHIV, we the publicly available train/valltest data splits; for the remaining datasets, we use a random 80/10/10% split. We optimize the hyper-parameters of the lifting methods and take the optimal hyperparameter values from (Telyatnikov al. 2025) whenever available; otherwise, we select optimal values based on the optimal results using cycle or k-hop lifting. We provide further details on the choice of hyperparameters and model selection in the supplementary material. We compute the mean and standard deviation of the performance metrics (MAE J. for ZINC, AUC 1 for MOLHIV, and accuracy t for all other datasets) over three independent runs.
Results. Table shows that Alift is the best- rest-performing lifting method in over 90% of the TNN/dataset combinations, both for cell complexes and hypergraphs. Notably, Alift resulted an improvement in average accuracy of up to s 45% compared to static liftings using the same TNN. For CWN, CIN, and UnIGCNII, our method outperforms static liftings on all datasets. On NCI109 and ZINC, Olift is con- sistently better than the static liftings across all TNN backbones. We also observe that TNNs perform better than GNNs in this setup with Olift. Overall, these results validate the effectiveness of Ulift.
Impact of GNN choice performance. We aim to assess how sensitive our approach is to the choice of GNN. We report results using GIN (Xu et al. 2019) and GPS (Rampášek et al. 2022).
Table 2 indicates that choosing GNNs that are able to generate richer and more informative latent node representations leads to better results in Aifft. In particular, GPS performs better than GIN in most datasets. A possible explanation for this observation is the greater expressivity of GPS, which benefits from the incorporation of positional encodings. Notably, on cell complexes and ZINC dataset, GPS allows reducing the MAE from 0.46 to 0.
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Table Graph classification: Olift static liftings. We denote the best-performing model for each dataset/TNN in bold. For any fixed TNN and dataset, Olift is better than static liftings in 90% of cases, offering a performance improvement up to 45%.
Domain
TNN
Lifting
NCI1
NC1109 f MOLHIVI
MUTAG
Protcinst
ZINCI
GCN
74.45=1005
76.46.1.03
14.99
64.91 1:496
1..18
0.64.20.4
Graph
GIN
76.89
76.90.0.80
70.76.2.46
80.70 48
32.50.2.1
0.59...000
Cycle Alift
76.93=1.18
76.7
1.334
70.15:23.0
66.67
69.05
0.46:0.01
CWN
19.81
80.55
10.30
75.37
85.96 14996
70.54.3.
0.17 10.00
Cycle
72.02=1.09
75.01
=0062
69.1
39.17::1..00
61.40=2.44
70.83
0.79=20.22
CXN
Cellular
Olitt
82.08 +1.50
82.57 =:0440
74.83 :..9
84.21 =4.30
69.94=2.10
a0.7
Cycle Olift
75.91=1.11
76. 1:00
68.46
66.96=1.44
67.86=0..8
0.42=0001
CIN
79.59
81.06
72.37 1:1.6
88.72 44.30
72.43 2::11
a0.01
k-hop
72.70=0.62
72.01=1..5
61.40=2.44 48
72192
0.66=e0.02
UniGCN2
k-NN
71.78
68. 60.50.33
57.73...
L.6.84
64.91
48
13.41.10.42
1.10.0.01
kernel
=0.80
72.66 =0.64
57.07
63.16=88.0
73.21:0.78
0.79
Alift
77.45 1.88
75.30
69.32
89.47 14.30
73.51
0.56
0.56100000
Hypergraph
k-hop
65.50=1.00
66.97
63,49
64.91 1/2
71.43.0.77
1.15
k-NN
72.83
1..14
52.34
59.65
72.62.1155
1.1020.02
UniGIN
kernel
60.50
66.59 1..49
49.60
57.89
14.30
66.67
1..80
1.45.0.02
Ulift.
64.88 1.00
79.74=0.25
72.04 =0.88
66.67 :0.56
73.81 1:55
0.92:0.05
Table 2: Effect of GNN backbone on the performance of Alift. The results suggest that the expressive power of backbone GNNs have a direct impact in 0lift's 's performance. Except for MOLHIV and NCII, GPS leads to better performance than GIN overall.
TNN
GNN
NCI1
NC1109 1
MOLHIV 1
MUTAG1
Proteins?
ZINC/
GPS
79.81
80.55
15.32
87.72 22.48
70.54
0.17 10.000000
CWN
GIN
81.59 10.80
18.69
75.37
82.16
71.13:2.77
0.46:00.00
GPS
79.97
$2.57=0.40
65.58
84.21 1:330
69.94
0.17
CXN
GIN
81.35 #:229
79.98:1.0
72.25
77.19
67.86=2.19
0.43:t0.01
GPS
78.67 1:
74.50=1.16
68 08.22 22.38
89.47 1:30
73.81 #0.42
0.56
UniGCN2
GIN
75.38
74.98
11.12
68.73
64.91 ::6.56
73.51 20.84
0.63:10.01
GPS
66.42
79.74.0.20
6333213311
66.67
16656
72.32
1000:
UnIGIN
GIN
64.40=0.11 41
78.53
68.86::3.05
70.18 6.556
72.02
1.12:10.26
5.2 NODE CLASSIFICATION
Datasets. For node classification, we evaluate Ølift on four datasets: Cora, Citeseer (Yang et al., 2016), Texas, and Wisconsin (Rozemberczki et al. (02T). Within these datasets, two are knowingly homophilic (Cora and Citeseer) and two are heterophilic datasets (Texas and Wisconsin). Dataset statistics can be found in Amppendix D
Baselines. We also compare our method (for cell domains) against the learnable approach in (Battiloro et al. (024), called Differentiable Cell Complex Module (DCM), which was originally evaluated on node classification tasks. To do so, we consider Ulift combined with CWN and TopoTune (Papillon et al., 2025). We also include results of them with cycle lifting. We consider the same hypergraph TNN baselines as in /Section 5.1
Evaluation setup. For all datasets, we use random train/val/test data split with 60/20/20% split. Similarly to the experiments for graph classification, we optimize the hyper- hyper-parameters of the lifting methods and take the optimal TNN hyperparameters from Telyatnikov et al. (2025) when available. Otherwise, we choose them to maximize the validation accuracy using k-hopp lifting. For more details, please refer to the supplementary material. We report the average accuracy and standard deviation over three independent runs.
Results. Table commpres 0lift against DCM and cycle lifting. Notably, Olift is the besf-perg method in all datasets except for Wisconsin, in which it achieves the second-best performance. It is
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Published as a conference paper at ICLR 2026
also worth mentioning that Olift (with either CWN or TopoTune) outperforms DCM for all datasets, sometimes by a large margin - c.f., Texas and Cora.
Table 3: Comparison of DCM and Olift on node classification. Olift achieves the highest average performance across all datasets (rhs) and significantly outperforms DCM on heterophilic datasets.
TNN
Lifting
Cora
Citeseer
Texas
Wisconsin
Avg
GCN
85.64 = 0.51
70.43 + 0.71
58.91 0.76
49.14 0.66
66.03
GAT
86.17 = 0.33
73.82 = 0.45
58.38 + 1.05
49.41 = 0.95
66.95
GIN
85.50 + 0.54
72.20 + 0.60
59.10 0.80
48.50 + 0.70
66.33
DCM
80.73 + 0.33
77.90 0.80
56.76 + 6.62
73.86 + 1.85
72.31
CWN
Cycle
74.80 0.08
75.83 0.90
63.06
80.39 4.24
73.52
CWN
@Lif
80.17 + 1.59
72.83 + 2.15
50.1813.:0
77.78 3.70
77.74
Tone
Cycle
69.03 + 0.88
72.90 + 0.85
71.56 1.27
70.16 + 1.85
70.91
TopoTune
ALif
86.82 + 0.75
78 23 1.08
72.97 0.00
65.36 2.45
75.84
Table 4 reports results of lifting methods for hypergraph neural networks. Compared to k-hop. our approach is better on heterophilic datasets but worse on homophilic ones for UniGCN2. Additionally, Table shows that Ølift leads to better average accuracy than other static liftings.
Table 4: Comparison of Olift and static lifting baselines for hypergraphs node classification. Our
method outperforms all static liftings on average across the selected node classification datasets.
TNN
Lifting
Cora
Citeseer
Texas
Wisconsin
Avg
k-hop
86.03 + 0.63
78.40 = 0.36
66.67 + 6.74
69.28 + 5.62
75.09
k-NN
74.00 + 0.65
18.17 + 0.05
70.27 + 3.82
79.74 + 2.45
60.54
UniGCN2
kernel
29.93 + 1.33
18.07 0.21
57.66 7.75
57.52 + 10.42
40.79
ØLif
81.93 + 1.11
78.03 + 0.91
69.37 2.55
73.20 + 5.62
75.63
k-hop
78.73 + 0.66
7
9.19
58.82 + 5.77
69.44
k-NN
62.00 + 1.08
19.33
0.48
65.77
73.20 + 4.03
55.07
UniGIN
kernel
40.93 + 2.52
18.53
+ 0.61
58.56 + 7.09
51.63 + 4.03
42.41
öLif
84.23 + 0.53
77.97 + 0.45
63.96
63.40 + 4.03
72.39
CONCLUSION
Topological neural networks (TNNs) are receiving increasing attention in the graph machine learning community. Yet, their effectiveness depends crucially on the choice of graph lifting procedure. Despite its central role, lifting has remained largely unsupervised and task-agnostic, which can lead to the construction of suboptimal topological representations for downstream learning.
To address this limitation, we introduced Ölift, a general-purpose, purpose, differentiable lifting framework that is compatible with multiple topological domains. Across a broad set of benchmarks and TNN architectures, Olift consistently outperformed traditional unsupervised lifting methods, demonstrating the benefit of making the lifting process learnable and inforinfrdddd
Limitations. For hypergraph domains, Olift can create candidate hyper-edges and decide whether keep them in an embarrassingly parallel fashion rendering Ølift especially compute-efficte efficieen for this domain. However, for cell complexes, we need to compute cycle basis to elicit candidate cells, which may come cubic cost with respect to the number of nodes in the input graph. this case, may reduce the number of candidate cells by, for instance, regularizing the k., variables or shifting their distribution towards zero. Nonetheless, devising more efficient algorithms for candidate identification in hierarchical domains is clear direction of improvement for future works.
Future work. Our method can be extended to other topological domains, such as point clouds, making it applicable to 3D mapping tasks. Additionally, future work could focus addressing the computational challenges of DiffLift and scaling it to handle larger graphs. Another promising direction is to explore differentiable lifting in dynamic temporal graphs, where topological

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Published as a conference paper at ICLR 2026
structures evolve over time. Moreover, integrating Olift with pretraining strategies could yield generalizable topologiaal priors across tasks.
We also believe that formally analyzing the impact of enriching topologiaal structures with learnable liftings on mitigating oversmoothing and oversquashing in TNNs an interesting research direction. For instance, in scenarios where long and narrow paths connect dense substructures, static liftings are limited to adding cycles within each local region, leaving information between communities to traverse the original bottleneck. In contrast, Olift learn to introduce 2-cells that effectively create shortcuts across such bridges, reducing the effective distance between distant nodes and improving information flow. While a full theoretical treatment is left for future work, this adaptive ability provides a plausible mechanism for mitigating oversquashing.
Ethics Statement. We do not identify any immediate, direct societal harms from the technical contributions presented in this work. Our method operates on standard, non-sensitive benchmarks and does not require or expose personally identifiable information.
Reproducibility Statement. The repository containing the code is available at https:
Joan:/Jorgeuiiaa wwwpeeeeewirrrearaaareeeeeeee dit.rranccccfffffiiinng We provide further de- tails on used datasets and implementation details (e.g., parameter selection) in Appendices D and
ACKNOWLEDGMENTS
We acknowledge the support by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) (8888..76399/2225-000 Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ) SSHL-26000330200442222 Nacional de Desenvolvimento Científico Tecnológico (CNPq) (404336/2023-0, 305692/2025. 9, 315158/2023-9, 163868/2023-9, 408974/2025-7, 312068/2025-5), We also acknowledge the
SEI-2600003/200094222555 and the Conselho
computational resources provided by the Aalto Science-IT Project.
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A ADDITIONAL BACKGROUND AND FORMULATIONS
A.1 CHAINS, BOUNDARY OPERATORS, AND CYCLES
Here, we introduce some basic notions in algebraic topology. For simplicity, our exposition considers abstract simplicial complexes (ASCs) equipped with coefficients in the finite field 2,227 = (0, 1).
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The space of n-chains is the vector space of all formal sums of n-dimensional simplices of an ASC K. Formally, let n 2 O and (n) = {o E dim((r = nj be the n-skeleton of K. The n-chains of is the set (K) whose elements take the form
Σ 6
(8)
oe (n)
where for all (m)) 6o E 22/22.
(n) Let E E and Eo) e be two n-chains. The sum two chains c') and
the product of a chain by a scalar (Ac) are respectively defined by
c'd= -((r t2)o
(9)
= [Acooo
(10)
where and products modulo-2.
We define the boundary of a n-simplex o, denoted by d, as the sum of its constituents (n 1)- simplices, i.e.,
0n
Σ
T
(11)
TCo:71110111
This boundary extends linearly to chain spaces. In particular, the boundary operator O, is a linear map 0n C. - (n-1(K) defined by
c = O KKK(m)
Σ
60
Σ
GoOno.
(12)
Finally, we can define n-cyycles. For 2 0, the n-cycles of K is the set (n) given by the kernel of On, that is
Zn(K)= fcE (x)) dnc 0}.
(13)
A.2 Olift FOR SIMPLICIAL COMPLEXES
Note that for when D - i.e. when we must decide which edges to add Steps and 3 of Olftt for cell-complexes naturally result in simplicial complex. To fully specify Ølift for simplicial complexes, we are left with defining these steps when D 1.
Case D > 2: Learning D-simplices
+ [Step 2 When creating simplices of dimension D, we must ensure they respect the hierarchical structure of simplicial complexes. Let Kt be the cell complex at the end of iteration e < D To identify a preliminary set of candidates C', we run static D-clique lifting on K For D > 2, it is possible that lower-order clique within some C E C" does not belong KD-1, Therefore, must filter out these elements, defining refined set of candidates:
= (C e C'IS e P-1 for all SCC.
(14)
+ [Step 3] Similarly to this respective step for cell complexes, we define the probability of accept- ing C E C (i.e., setting be = 1) applying DeepSet over the embeddings f(z Jeecs subsequently sampling the Bernoulli variables be Cec. The output complex at this iteration is then
KD = K D-1 D c: by = 1}.
(15)
We define the features for D-simplices using projection lifting.
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A.3 Olift FOR COMBINATORIAL COMPLEXES
There are multiple ways to combine cell complexes with hypergraphs to obtain valid combinatorial complexes (CC). Here, we would like preserve the property that hyperedges exchange messages with nodes via boundary (or lower incidences) neighborhoods. Thus, we propose first running Đlift to either cell or simplicial complexes
where ranking functions are given by cell/simplex dimensions. Let K be the resulting complex. Then, we employ (in parallel) Ölift to a hypergraph H, where edges/hyperedges have rank 1. To ensure valid combinatorial complex, we prune the sampled hyperedges to include only those that are not supersets of any cell of rank greater than in K. Formally, the resulting CC given by {h E H :o E K s.t. o Chhuk.
B TOPOLOGICAL LIFTINGS
Clique lifting. The set of cliques in graph G is given by CI(G) fc V(G) : [u, v} E }. i.e., each element of C((G) is complete subgraph of G. The k-cliques of G are the elements of CI(G)) of size k. for k >> 1, and we denote them ClA (G). Formally, the k-clique lifting operation is given by
lifteelque, (G) = V(G) 1 CL(G).
(16)
Note that the inclusion of all cliques of size smaller than ensures the function returns a valid abstract simplicial complex.
Cycle lifting. The idea of cycle lifting is to identify basic cycles in the input graph and use the tuple of vertices in cycle as 2-rank cell of the output complex.
Let us consider modulo-2 sum operations for vertices and edges. Also, let O be the edge boundary map for graph G. i.e., 0 (fu, 0)) {u} {v) for any edge (u, E(G). The cycles of G are L(G) = (7 C E(G) Eeer (e) = 0}.
A basis for cycles of G is minimal collection of cycles such that any cycle in G can be written as a sum of cycles in the basis - i.e., the smallest set B C L such that VI E L,3B' S B with
1 EvEB b. The cycle lifting map is
lifteyale (G) = V(G) U U(G) ( (V(b) be B(G)):
(17)
where (b)) denotes the set of vertices the cycle and (V(6) b e B(G) is the set of 2-dim cells.
DCM. Battiloro et al. 2024) proposed a novel layer composed of several modules, with the Differentiable Cell Complex Module (DCM) being central latent topology inference. The DCM first samples the 1-skeleton of the latent cell complex using the a-Differentiable Graph Module grrgonssreeprsennnnn
(a-DGM). then selects sampled graph using the Polygon Inference Module (PIM). For a detailed description of a-DGM and PIM, we refer the reader to Section of Battiloro et al. 2004)).
higher-order interactions-fomete by cycles in the
k-hop lifting. The k-hop neighborhood of node E V(G) is defined as
Nu(u] = {u E V(G) dist(u, v) S k):
(18)
where dist(u, v) is the shortest-path distance in the graph G, measured by the number of edges in the path.
To construct the h-hop hypergraph from G, hyperedge is formed for each node v E VVG) based on its k-hop neighborhood:
lift (G) = [N&[v] : V(G)).
One can note that when k 1, k-hop is equal to neighborhood lifting. The parameter & controls the extent of the neighborhoods included as hyperedges, with larger & values progressively incorporating nodes farther away in terms of shortest-pan path distance.
k-NN lifting. K-NN lifting constructs hyperedges by identifying the k nearest neighbors based on their node features (feature space). For every node, a separate hyperedge formed that includes the node itself and its & closest neighbors.
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Kernel lifting. Kernel lifting is procedure that constructs hyperedges based similarity measures derived from kernels over graph nodes. These kernels can be defined in three ways: (i) over the graph structure itself. (ii) over the node features, or (iii) composition that jointly incorporates both graph and feature information. For a given reference node v, the method computes similarities between e and all other nodes vz using kernel function. A hyperedge is then formed by selecting fixed fraction (typically 0.5) of the nodes that are most similar to u according to the chosen kernel. This process is repeated for each node construct a set of hyperedges. The kernels can be defined in several forms: over nodes Kg (u. D²), features K2 (x, (x,,,,,, or over nodes and features C(K (x, x= u, e e²)), where C a valid composition function. Kernels over features are calculated as standard RBF or exponential kernels (Duvenaud] (014)), whereas kernels over graphs can be calculated as heat or Matérn kernels (Schölkopf and Smola 2002 [Brovitskiy et al. 2021: Nikitin et al. 2022).
C IMPLEMENTATION DETAILS
C.I MODELS
Our implementation relies mainly on the Pytorch (Paszke et al., 2017) and Pytorch Geometric (Fey and Lenssen 2019) libraries. For TNN models and static lifting we used TopoX (Hajij et al. 2024) and Topo (Telyatnikov et al. 2025).
Regarding the base TNNs, we use the hyperparameters (including learning rate, optimizer, batch size, width, depth, and so on) reported in TopoBench for CWN, CXN, and UniGCNII on NCI1, NC1109, MOLHIV, MUTAG, Proteins, ZINC, Cora and Citeseer (Telyatnikov et al. (025). Since TopoBench does not report optimal hyperparameters for UniGIN, we use the same used for UnIGCNII. While for
CIN, we used part of the hyperparameters reported in (Bodnar al., 202 1b) and for the TopoTTune,
since they do not report the hyperparameters, we used a grid search similar to the one available in their repo.
We note that MOLHIV, Texas and Wisconsin datasets were not present in TopoBench. We use two TNN layers for MOLHIV and one for Texas and Wisconsin with respective learning rates 10-2, 5 x 10-3 and 5 x 10-3, For these datasets, we fix the embedding size in 64 for all layers. And for Texas and Wisconsin we used weight decay 5 x 10-6
We are left with the task of optimizing the hyperparameters for the lifting operations (Olif, k-NN, kernel). For Alift, we consider using both GPS and GIN as backbone GNNs, with embedding dimensions in {32, 64, 128). network depth in {2, 3}, and kmax {3, 5, 7, 9, 113. For k-NN lifting we choose in {3, 5, 7, 9]. For kernel lifting, we consider equally-spaced temperature values within 0.1 and 9.6, with 0.5 increments.
All models were trained for 200 epochs and with early stopping after 50 epochs without improvement on validation accuracy. We run three independent trials for computing mean and standard deviation of the performance metrics. We select the optimal hyperparameters based on validation accuracy.
C.2 HARDWARE
For all experiments, we use a cluster with Nvidia V100 GPUs details regarding the compute infrastructure are omitted for anonymity.
D DATASETS
Graph-level tasks. The datasets NCTI, NCT109, PROTEINS, and MUTAG are part of the TUDatascts (Kersting et al.] (016)) dataset collection broadly used for benchmarking GNNs. We also use ZINC-12K and MOLHIV (Hu et al., 2020), popular benchmarks for molecular property prediction. Statistics for each dataset given Table
Node-level tasks. For node classification, we use four popular benchmarks: Cora, Citeseer (Sen et al., 2008) Shhhhur et al.) 2018), Texas, and Wisconsin (Rozemberezki al.) 2021). Cora and Citeseer are citation networks where nodes represent papers and edges denote citation between them. Node features are given by bag-of-word vectors and node labels comprise the academic topies of
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Table 5: Statistics of datasets for graph-level tasks.
Dataset
graphs
cllasses
Avg 1/ nodes
Avg /edgees
Train%
Val%
Test%
NCII
4110
2
29.87
32.30
80
10
10
NC1109
4127
29.68
32.13
80
10
10
MUTAG
188
2
17.93
19.79
80
10
10
PROTEINS
1113
2
39.06
72.82
80
10
10
MOLHIV
41127
2
25.5
27.5
Public Split Public Split
ZINC
12000
23.16
49.83
the underlying articles. Texas and Wisconsin are datasets of webpages from university departments. Nodes represent webpages and edges are hyperlinks between them.
For citation networks, we use same data split as in (Chen et al. 2018), and for the remaining ones we use the split in (Pei et al. 2020h. These are the standard and most used splits. Tablle 6 provides more details about the datasets.
Table 6: Statistics of datasets for node classification.
Dataset
Nooees
#Edges
# Features
#Classes
#Train
#Val
#Test
Cora
2708
5429
1, 1,333
7
1, 208
500
1. 0.000
Citeseer
3327
4732
3. 703
6
1, 827
500
1, 000
Texas
183
309
1703
5
87
59
37
Wisconsin
251
499
1703
5
120
80
51
E ADDITIONAL RESULTS
E. RUNTIME AND COMPLEXITY COST
Wall-clock time. Table reports per-epoch training and test times (seconds) for the different lifting methods and target neural networks (TNNs) used in our experiments. Overall, Olift incurs only modest runtime overhead compared to static liftings while offering the flexibility of task-adaptive topology.
Memory usage. Table Srreports GPU and RAM consumption (GB) for the different architectures and lifting schemes. Reported values mean + standard deviation over runs. Overall, Olift demonstrates moderate memory requirements across both cellular and hypergraph domains, with GPU usage scaling proportionally to the complexity of the lifted topology.
Complexity analysis. Hypergraph Ølift runs in O(N2D+N' 22 log + DNkmax)) where the N term comes from computing all pairwise node distances in D-dimensional embedding space, the N2 log kmax term for selecting each node's top-k neighbors, and the Nhmma D term for aggregating across the sampled hyperedges. By contrast, static kernel-based lifting requires O(N D + N*) time, while a k-hop static lifting scales as O(N kimxxd) with d the average node degree. Cellular Olift runs in O(N2 D + (E + N among lifted edges or candidate cells and CED covers the embedding aggregation over C' candidate cells of average size A static cycle-basis basis lifting costs O(N(N E) + C?)) whose dominant component is the cycle-basis computation. In both variants, the modest additional runtime is justified by consistent accuracy gains of approximately 5-10 percentage points compared to static liftings.
kxmxa) ) CLD), where (E N Nmaxx )2 captures pairwise adjacency
Choice of Hyperparameters. Although ölift includes additional learnable components such as the GNN used for neighborhood scoring, the networks computing acceptance probabilities and adaptive neighborhood sizes, and the maximum neighbor budget kimax the resulting hyperparameter overhead remains modest. Our search space comprised fewer than 30 configurations overall (Appendix C), and we found that tuning the adaptivity introduced by these components leads to consistent performance gains while adding only minor computational and memory costs, as reflected in Tables and
somewhat stable across datasets. Crucially,
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CXN Olift Published as a conference paper at ICLR 2026
Table 7: Per-epoch training and test times (seconds) for the different lifting methods. Reported values are mean standard deviation over runs.
Dataset/Phase
CWN Cycle
CWN Olift CXN Cycle
Cellular Domain
NCII Train
47.41 2.59
66.77
NCII Test
1.42 0.05
4.33 0.62
1.05
2.67 0.06
NCI109 Train
52.11 2.58
07.05
53.50 0.97
NCT109 Test
1.67 0.02
5.19
1.56
2.15
MUTAG Train
2.55 -: 0.12
3.229
1.37
3.60
MUTAG Test
0.09 0.00
0.20
0.05
0.02
Proteins Train
18.07
20.12
15.18 1.222
Proteins Test
0.65 = 0.00
0.82 0.10
0.46
0.72
ZINC Train
58.24 - 3.10
118.45 + 6.20
16.18
82.30 + 4.50
ZINC Test
2.15 0.08
5.92 + 0.35
1.68 + 0.06
3.84 0.18
Dataset/Phase
UniGCNII k-hop
UniGCNII Olift
UniGIN k-hop
UniGIN Olift
Hypergraph Domain
NCTI Train
42.26 = 0.46
69.34 2.55
37.46 0.31
61.32 + 2.83
NCII Test
1,23 1 0.01
1.04
0.00
2.03 1 0.06
NCI109 Train
31.66 2.26
3.22
69.61 2.33
NC1109 Test
0.89 0.05
2.72
0.002
2.30
MUTAG Train
1.94 1.94=0.. 0.14
5.12
2.04
3.16
MUTAG Test
0.06 = 0.00
0.14
0.00
0.09
Proteins Train
11.93 0.89
12.45
11.50 1..1
Proteins Test
0.36 + 0.00
0.60
0.50 0.03
ZINC Train
51.80 1.50
7
45.30
66.40
ZINC Test
1.58 0.04
2.68 0.12
1.28 + 0.03
2.85
Table 8:
Memory usage (GB) for the different lifting methods and architectures. Reported values are standard deviation
mean
Memory Type
TNN
MUTAG
NCI1
NCI109
PROTEINS
ZINC
GPU Memory (GB)
UniGCNII k-hop
0.02:0..0
0.02:10.00
0.02:0.00
0.03
0.02:1..0
UniGCNII ôlift
0.02
0.02:00.00
0.02:10.00
0.05
0.02:1..00
UniGIN k-hop
0.0220.00
0.0220.00
0.02=40.00
0.02=0.00
0.0240.00
UniGIN Ølift
0.02
0.02=0.00
0.02++0.00
0.04
0.0240.00
CWN Cycle
0.02=0.00
0.02#00.0
0.02-+a.00
0.03=40.00
0.02++0.00
CWN Mift
0.02+0.00
0.02#10.00
0.02++0.00
0.07
0.02++0.00
CXN Cycle
0.02
0.02
0.0244.00
0.03
0.024..0
CXN Olift
0.02#0.00
0.02
0.02++0.00
0.04
0.02++0.00
RAM (GB)
UniGCNII k-hop
1.2220.02
1.3120.01
1.31
1.24
1.34
UniGCNII Olift
1.4520.02
1.46=0.01
1.4540.00
1.4620.01
1.4740.01
UniGIN k-hop
1.21
1.31.0.01
1.31:0.01
1.2420.00
1.34
UniGIN Olift
1.4220.01
1.45.0.01
1.45:10.22
1.42.0.00
1.47:0.01
CWN Cycle
1.2020.01
1.4620.02
1.1544.00
1.352a.01
1.7140.01
CWN Ulift
1.38
1.42=0.01
1.43+a.01
1.394a.01
1.42+o.0
CXN Cycle
1.21
1.47
1.46:1.01
1.37
1.70.0.00
CXN Olift
1.40
1.40
1.44:0.00
1.43
1.45
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Published as a conference paper at ICLR 2026
E.2 DETERMINISTIC VERSION
As we mentioned in the main text, we can derive a deterministic version of Olift by thresholding probabilities. Table compares Olift with its deterministic variant trained using threshold 0.5. Note that (random) ilift outperforms its deterministic counterpart in most cases. Nonetheless, while we fixed the threshold 0.5 here we leave open the possibility that tuning it hyper- parameter could yield performance improvements.
Table 9: Comparison between Olift (random) and its deterministic variant (thresholding at 0.5).
Domain
TNN
Alift
NCI1
NC1109
ZINC (MAE)
PROTEINS
MUTAG
Cellular
CWN
deterministic
80.62 + 0.75
77.64
0.30
0.30
1.33 4: 5.33
70.54 0.00
82.46 2.48
Cellular
CWN
random
79.81 +: 0.40
80.55
0.50
0.17 + 0.00
70.54
85.96
Hypergraph
UniGCNII
deterministic
71.53 1: 1.92
69.76
6.03
0.70
0.42
75.44
2.48 2.48
Hypergraph
UniGCNII
random
77.45 +: 1.88
1.10 75.30
0,56 1 0.03
73.51
89.47
4.30
E.3 FURTHER COMPARISON AGAINST DCM
DCM versus Uift. DCM Battiloro et al. 2024 introduces learnable lifting approach specifically for cell complexes through a two-step procedure: first, the a-Differentiable Graph Module (a-DGM) learns the 1-skeleton (edges) using o-entmax sampling to generate sparse, non-regular graphs; then, the Polygon Inference Module (PIM) samples polygons from induced cycles of the learned graph. DCM uses a-entmax for both edge and polygon sampling, and trains end-to-end using auxiliary reward-based losses (Eqs. 11 and 13 in their paper) that encourage edges/polygons involved in correct predictions. The method is evaluated exclusively on node classification tasks (homophilic and heterophilic datasets) and is limited to 2--22dimm 2-dimensional cell complexes.
Some key differences between DCM and @iif include:
1. Domain generality: our framework applies to hypergraphs, simplicial complexes, and combina- torial complexes, not just cell complexes. Extending DCM to hypergraphs non-trivial because it relies on cycle-based based candidate generation, which is inherently tied to the graph structure and does not naturally translate to hyperedge formation;
2. Sampling mechanism and training: use Bernoulli sampling with straight-through estimators rather than a-entmax, and critically, we do not require auxiliary reward-based losses our framework is trained purely with the task loss, making it simpler and better suited for end-to- end learning. Additionally, our hypergraph variant is embarrassingly parallel whereas DCM's two-step procedure sequential;
3. Adaptive cell sizes: we learn distributions over ki (neighborhood sizes) allowing adaptive hyperedge/cell cardinalities, while DCM's polygon sizes are constrained by the induced cycles in the learned graph;
4. Evaluation scope: assess both node and graph classification across 12 datasets with multiple TNN architectures, demonstrating broader applicability.
Experiments point clouds. We compared Ulift against DCM point-cloud node classification
(graphs with no edges) to provide a direct comparison. Table shows results across four datasets. Alift combined with different TNNs achieves competitive or superior performance: UniGCNII+Olift achieves 84.97% on Wisconsin (vs. 71.24% for DCM), and UniGIN+Olift achieves 83.78% on Texas (vs. 62.16% for DCM), demonstrating substantial gains on heterophilic datasets. On the homophilic dataset Cora, DCM achieves 73.07% compared to 71.47% for the best Ulift variant, showing comparable performance. As expected, the performance in heterophilic datasets are higher in this setting compared to homophilic when referring Table 3 Table 4) These results demonstrate that Alift's framework provides competitive performance while offering greater flexibility across topological domains and TNN architectures.
E.4 COMPARISON AGAINST PROBABILISTIC REWIRING
IPR-MPNN versus Wift. IPRNMPNN Qian et al. 2024) introduces probabilistic graph rewiring by connecting original graph nodes small set of virtual nodes in end-to-end differentiable
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Published as a conference paper at ICLR 2026
Table 10: Point cloud performance comparison of Ölift across TNNS versus DCM.
TNN + alift
Citeseer
Texas
Wisconsin
Cora
UniGCNII
74.73 10.12
$1.9811.2
84.97
70.83 10.37
CWN
51.80
71.1711.27
75.8220.99
55.90112.45
CXN
62.03
81.08:00.00
78.43=0.00
57.43
UniGIN
72.73
83.78
83.66-0.92
1..47
DCM
74.40.0.45
62.16
71.24
73.07
manner. The method uses an upstream MPNN to compute priors for assigning each original node to & virtual nodes (from m total virtual nodes, where m n²,, sampling assignment matrices via differentiable k-subset sampling. A downstream MPNN then operates on the augmented graph with message passing among: (1) original nodes virtual nodes, (2) among virtual nodes (forming a complete subgraph), and (3) among original nodes.
There are relevant distinctions between Ölift and IPR-MPNN: (i) Explicir vs. implicit structure: IPR-MPNN implicitly routes long-range information through virtual nodes, while ölift explicitly constructs higher-order cells (hyperedges, simplices, polygons) that directly encode multi-way interactions; (ii) Domuin flexibility: IPR-MPNN operates within the graph domain augmented with virtual nodes, whereas Ulift learns liftings to diverse topological domains (hypergraphs, simplicial complexes, cell complexes, combinatorial complexes); (iii) Sampling strategy: both use differentiable k-subset sampling, but IPR-MPNN samples which higher-order cells include in the lifted structure; (iv) Computational approach: IPR-MPNN requires managing virtual node features and specialized pesssage-passing between hierarchies, while Ølitt integrates directly with standard TNN architectures designed for each topological domain.
tovii-tto-irtualle -ode assignments while ölift samples
Results on graph classification. Table 1] compares IPR-MPNN to Ulift combined with various TNNs five molecular datasets. To ensure fair comparison, we evaluated IPR-MPNN using our experimental setup, with the data splits described in Appendix D and a hyperparameter grid similar the one in Appendix Cj with embedding dimensions (32, 64, 128) and network depths in {2, 3). All other hyperparameters remained fixed, taken from the configurations in the official IPR-MPNN repository. For instance, on ZINC we do not use edge features.
Ulift achieves substantial improvements multiple benchmarks: CXN+ôlift reaches 82.08% on NCI1 and 82.57% on NCI109 compared to IPR-MPNN's 77 744 and 77.08%. On ZINC, both CWN+0lift and CXN+ôlift achieve 0.17 MAE versus IPR-MPNN's 0.39, representing a 56% error reduction. UniGCNI++Alift obtains the highest MUTAG accuracy at 89.47%. While IPR-MPNN demonstrates the ad advantages of learnable graph augmentation through virtual nodes, our results show that explicitly learning higher-order topological structures through Qlift can provide complementary benefits, particularly when combined with TNNs designed to exploit these structures.
Table 11: Graph classification performance comparison between IPR-MPNN and Ölift combined with various TNNs
Method
MUTAG?
NCI1
NC11091
PROTEINS
ZINC
IPR-MPNN
70.18 12.48
77.44
77.08 0030
73.73 1..93
1..9
CWN Cycle
66.67:12..0
76.93
76.71=1.35
69.05 12.95
0.46
CWN alift
85.96
79.81:0.40
80.55 0.50
1..54
0.17
0.17::0.00
CXN + Cycle
61.40#2.45
72.02+1.69
75.01
=0.62
70.83 1.52
0.79
CXN + Alift
84.21 14.30
82.08 1.50
82.57 -0.40
69.94-2..10
0.17.10.01
UniGCNII + k-hop
61.40.2..4
72.70
72.01 1155
72.92 :::
0.66.10.02
UniGCNII Alift
89.47 L4.300
77.45.1..8
75.30
73.51 0.884
0.56
UniGIN k-hop
64.91 12.48
65.50 1..9
66.97
17.25
71.43 10.73
1.15
UniGIN Olift
16656
64.88 1.00
79.74.0.23
73.81 1..52
L..05
20


Reviews:
[{"review_id": "ftqF9p7Wcl", "reviewer": "Reviewer_AwbL", "summary": "This paper introduces DiffLift, a differentiable framework for end-to-end learnable graph lifting to higher-order topological domains, including hypergraphs and cell/simplicial complexes. Instead of relying on static, task-agnostic liftings (e.g., cycles, k-hop neighborhoods), DiffLift learns probabilistic distributions over candidate higher-order cells using node embeddings from a backbone GNN. The method integrates with multiple TNN architectures and supports hierarchical sampling for cellular complexes. Experiments across 12 benchmarks show improvements over static liftings.", "questions": "1. How does Difflift behave on very large graphs (e.g., OGB-LSC scale), where cycle-basis computation and candidate sampling may be expensive?\n\n2. Could the authors elaborate on how sensitive Difflift is to stochastic sampling noise, the backbone GNN used for embedding generation, and the choice of aggregation function? Did you evaluate alternative backbone models or pooling operators, and do you observe increased variance or instability when weaker embeddings or different set functions are used? Are variance-reduction techniques (e.g., Gumbel-softmax, annealing, deterministic thresholds) helpful?\n\n3. Does Difflift mitigate known issues in deep TNNs, such as over-smoothing or over-squashing, more effectively than static lifting?\n\n4. Can the authors report runtime and memory overhead relative to common static lifting pipelines, particularly in high-dimensional molecular tasks?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper proposes a general formulation for differentiable lifting applicable across multiple domains (hypergraphs, cell complexes, simplicial complexes), which enables task-aware structure learning.\n\n2. Experimental validation across multiple datasets and TNN backbones shows performance gains.\n\n3. The proposed framework is robust enough to be integrated with existing TNN frameworks.", "weaknesses": "1. Computational bottlenecks remain for cell complexes, especially in cycle-basis computation (cubic in nodes), limiting scalability to larger graphs.\n\n2. The gains on node classification are less uniform, particularly for homophilic datasets with hypergraph models.\n\n3. The proposed method is Dependent on NN-based k-selection, and probability estimators may introduce additional hyperparameter complexity.\n\n4. There are no explicit theoretical guarantees on representation expressiveness or stability under stochastic lifting.", "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": "GRiF24AWt7", "reviewer": "Reviewer_1MFQ", "summary": "The paper proposes a differentiable lifting to create an end-to-end learnable\npipeline for the lifting of graphs into higher order structures to enhance\ntheir expressivity. Their pipeline consists of an initial embedding layer that\nmaps a combinatorial complex (specialized to graphs in the experiments) into a\npoint cloud. A cell sampler subsequently samples rank-k cells from this point\ncloud and learns whether or not the k-cell should be included in the final\nlifted cell complex. Hitherto the standard procedure for this lifting has been\ndeterministic, based on either cliques or cycles, for instance. The author\nargue that the optimal lifting depends heavily on the dataset, motivating the\nneed for a *learned* lifting in order to mitigate this need for  pre-defined\nchoices. An experimental suite shows that this method works rather well on a\nwide range of standard benchmarks for both node classification as well graph\nclassification.", "questions": "- Is there a particular reason the method is restricted to graphs? I could\nimagine that, since the idea of TNNs is to operate on higher dimensional\ncomplexes as well, higher dimensional simplicial data also to be interesting. \n- Have you considered applying the method to point clouds as well? \n- One limitation is that to construct the cell complexes, on has to loop over\nthe full power set, which get quite intense. It is understandable that one\nwould only consider 1- and 2-cells, but even that would become rather\nprohibitive. E.g. A \"tiny\" point cloud or mesh of an object quickly contains up\nto 2-4k points and this is where your method already could become quite\nexpensive since even constructing the 1-cells (graph) for a single point clouds\nwould already have to consider $2048 choose 2 = 2,096,128$ or $4096 choose 2 =\n8,386,560$ possibilities and for 2-cells one would have to consider billions of\npossibilities. Some remarks could shed some light into these considerations.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "Overall the paper is well-written and the idea is interesting. In particular\neliminating the choice of lifting provides a more general approach for TNNs to\nwork on a wider set of graphs. This is an important fact, as multiple works\nhave already shown that various graph datasets have very different properties\nand that it is not always clear what type of networks operates best on each\nparticular dataset. Hence overcoming this challenge is a good contribution to\nthe TDL community. The code to reproduce the experiments are provided and is\nwell-documented, which is great to see.", "weaknesses": "While the reviewer views the experiments section good it is a little surprising\nthat the methods only compare to other TNN methods, whereas other comparison\npartners could be interesting as well, such as DiffPool or other methods\nproposed by the TDA community and graph learning community in general. As this\nmethod seems to have a higher complexity that other methods, it is in the\nopinion important to discuss what may persuade a practitioner to use this\nmethod over say a simpler method if it performs more or less equal.", "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": "qJC3Vg75fL", "reviewer": "Reviewer_gFdZ", "summary": "This paper addresses the problem of learning liftings, i.e., functions that map graphs to higher-order domains for use in Topological Neural Networks (TNNs). While most existing approaches rely on handcrafted, \"static\" lifting procedures, this work explores the possibility of learning liftings directly from data in an end-to-end manner. The authors introduce DiffLift, a general framework that can learn to lift attributed graphs to various higher-order topological structures.\n\nThe method runs steps:\n(i) Generating node embeddings using a standard GNN.\n(ii) Constructing candidate higher-order cells or hyperedges.\n(iii) Assigning scores to each candidate via a learnable module.\n(iv) Sampling candidates using a Bernoulli distribution based on the scores above, using Straight Through Estimators to ensure differentiability.\n\nTwo main instantiations of steps (i,ii) are described:\n- Graph to Hypergraph lifting: Candidate hyperedges are formed via k-NN search on node embeddings. Their cardinalities are sampled, and scores are computed using a permutation-invariant multiset function.\n- Graph to Cell complex lifting: Candidates are derived from cycles in a chosen cycle basis, and scored using a DeepSets-based model.\n\nExperiments are conducted on both graph-level and node-level prediction tasks. When compared with static lifting approaches, DiffLift generally achieves significant improvements. However, the benefits are less consistent on node-wise classification in heterophilic datasets. An additional experiment shows that the choice of GNN backbone (e.g., GPS vs. GIN) in the first stage can noticeably affect performance.", "questions": "[Q1] What is the difference between using a DeepSet model in the case of Hypergraphs and a Set Function in the case of Cell Complexes in Figure 4?\n\n[Q2] Line 314 – Hajij et al, characterised various lifting together, but other previous uncited works introduced them first?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "[S1] The paper is easy to understand and the narrative quite straightforward.\n\n[S2] The experiments are comprehensive, covering a broad set of use-cases.", "weaknesses": "[W1] The relationship between this work and the DCM method is not clearly articulated. The authors cite it, but the concrete technical advantages over DCM are to be discussed. What limitations of DCM are addressed? How? If DiffLift claims more generality across topological domains, the authors should explain why DCM cannot be extended similarly and highlight key methodological differences.\n\n[W2] The proposed learnable lifting mechanism resembles differentiable pooling approaches as well as methods that *learn* to wire virtual nodes in graphs. This paper, e.g., is quite relevant (https://proceedings.neurips.cc/paper_files/paper/2024/file/31df6a082046111e605abfec26ef5ccc-Paper-Conference.pdf). The authors should position their method relative to these lines of work and provide a minimum experimental comparison, in particular, w.r.t. pure hypergraph lifting, where no cycle-like inductive bias is leveraged.\n\n[W3] The requirement to restrict candidate cells to cycles within a chosen basis seems somewhat arbitrary and potentially limiting. If only a subset of those cycles is ultimately selected, it is unclear why a cycle basis is necessary in the first place. Previous works also consider lifting based on any induced or non-induced cycles (see e.g. Bodnar et al., 2021). Is this primarily a computational convenience? A clearer rationale or explanation is needed.\n\n[W4] On graph property prediction benchmarks, the TNN-based baselines very often do not outperform standard GNNs, which raises concerns. Is this expected? It is unclear whether the experimental setup or hyperparameter tuning might be limiting performance.\n\n[W5] For node classification tasks, TNN models are not compared against vanilla GNN baselines. Some lifting methods (e.g., “kernel” on CiteSeer) perform suspiciously poorly without explanation.", "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)