LEAP: Local ECT-Based Learnable Positional Encodings for Graphs
Authors: Juan P Garcia Amboage
Ernst Röell
Patrick Schnider
Bastian Rieck
Venue: ICLR 2026 Poster (2026)
Keywords: Topology
Euler Characteristic Transform
Graph Neural Networks
Topological Data Analysis
TDA
Topological Deep Learning
Primary area: learning on graphs and other geometries & topologies
TL;DR: Local ECT-Based Learnable Positional Encodings for Graphs

Abstract: Graph neural networks (GNNs) largely rely on the message-passing paradigm, where nodes iteratively aggregate information from their neighbors. Yet, standard message passing neural networks (MPNNs) face well-documented theoretical and practical limitations. Graph positional encoding (PE) has emerged as a promising direction to address these limitations. The Euler Characteristic Transform (ECT) is an efficiently computable geometric–topological invariant that characterizes shapes and graphs. In this work, we combine the differentiable approximation of the ECT (DECT) and its local variant ($\ell$-ECT) to propose LEAP, a new end-to-end trainable local structural PE for graphs. We evaluate our approach on multiple real-world datasets as well as on a synthetic task designed to test its ability to extract topological features. Our results underline the potential of LEAP-based encodings as a powerful component for graph representation learning pipelines.

Published as a conference paper at ICLR 2026
LEAP: LOCAL ECT-BASED LEARNABLE POSITIONAL ENCODINGS FOR GRAPHS
Juan Amboage' Ernst Rö.11 Patrick Schnider BBastian Rieck 2,3
! 'Department of Computer Science, ETH Zurich, Switzerland
2 PADOS Lab, University of Fribourg, Switzerland
3 Institute Al for Health, Helmholtz Munich, Germany
*Technical University of Munich, Germany
6 'Department of Computer Science, University of Basel, Switzerland
ABSTRACT
Graph neural networks (GNNs) largely rely on the pessage-passing paradigm, where nodes iteratively aggregate information from their neighbors. Yet, standard message passing neural networks (MPNNs) face well-documented theoretical and practical limitations. Graph positional encoding (PE) has emerged as promising direction to address these limitations. The Euler Characteristic Transform (ECT) is invariant that characterizes shapes
efficiently computable geometric-topologer and graphs. In this work, we combine the differentiable approximation of the ECT (DECT) and its local variant (E-ECT) to propose LEAP, new end-to-end trainable local structural PE for graphs. We evaluate our approach on multiple real-world datasets well as synthetic task designed test its ability extract topological features. Our results underline the potential of (-ECT-based based encodings as a powerful component for graph representation learning pipelines.
INTRODUCTION
Graphs are the preferred modality in numerous scientific domains, permitting the study of dyadic relationships an efficient manner. Their broad applicability comes with several challenges that make them harder to process with standard deep learning architectures. Among these characteristies (i) a mixture of geometrical information (via node and edge fcatures) and topological information (via the edges), (ii) highly variable cardinalities even within the same dataset, and (iii) lack of canonical representation. The development of suitable models is thus crucial for advancing the field of graph representation learning. Contemporary research largely focuses on message passing neural networks (MPNNs), i.e., architectures that are based on local diffusion-like concepts. While powerful, they exhibit intrinsic limitations: For instance, MPNNs tend lose "signals" in graphs of high diameter (Di Giovanni et al., 2023; Rusch et al., 2023; Zhang et al., 2023), and may be incapable of efficiently leveraging substructure information (Chen et al., 2020).
As an alternative to MPNNs, inspired by the transformer architecture (Vaswani et al., 2017), recent work focuses on positional encodings (PEs) and structural encodings (SEs) of graphs, denoting functions that assign embeddings to nodes based on locality relational information, respec- tively (Dwivedi et al., 2023; Kreuzer et al., 2021; Rampášek et al., 2022). Most PEs/SEs are based on either geometrical aspects (like coordinates, curvature, distances) topological aspects (like Laplacians or random walks), which may limit their practical expressivity. To overcome this, we propose LEAP, topology. Being based on a local, learnable variant of the Euler Characteristic Transform (ECT), a geometrical-oopooogical invariant, LEAP is easy calculate and highly expressive.
and-to-end-trainbbl positional encoding that leverages both geometry and

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Our paper contains the following contributions:
1. We propose a new graph positional encoding based on local ECTs, which is highly flexible and permits end-to-end training, specifically geared to work with geometric graphs.
2. We observe that our method captures structural differences in graphs even in case the node features non-informative, thus also permitting solve learning tasks for non-attributed graphs. 3. We conduct extensive experiments on benchmark datasets that demonstrate that our method yields improved predictive power in comparison existing positional encodings when used in conjunction with graph neural networks.
BACKGROUND
Before introducing our learnalle positional encoding, we provide short self-contained summary of message-passing, positional encoding in the context of graphs, and the Euler Characteristic Transform.
2.1 MESSAGE PASSING
Graph Neural Networks (GNNs) are specifically designed to operate on graph-structured data. A large subclass of GNNs are Message Passing Neural Networks (Gilmer et al., 2017, MPNNs), MPNNs represent each node by a vector that iteratively updated by aggregating neighboring representations. Hence, the state of a node at step t, denoted h is computed as
h ( UPDATE (h!- AGGREGATE (nh u (eeN)))))
(1)
where both AGGREGATE and UPDATE are learnable functions and (w() denotes the neighbors of node c. Following von Rohrscheidt & Rieck (2025), we refer to graph G together with feature vectors for each of its nodes a featured graph and adopt the following notation.
Definition 1. A featured graph is pair (G,x), where G is (non-directed) graph, and x is a map that assigns each node u E V(G) a feature vector x(v) E Rd. We denote the set of nodes of by (G), and the set of edges by Eg)).
Despite their popularity, common MPNNs are limited by phenomena such as oversquashing (Di Gio- vanni et al., 2023), oversmoothing (Rusch et al., 2023; Zhang et al., 2023), or restricted expressive power (Chen et al., 2020; Xu et al., 2019). Multiple approaches have been proposed to address these challenges, for instance by (i) modifying graph connectivity via virtual nodes (Cai et al., 2023; Grötschla al., 2024), (ii) combining message passing with global attention (Rampášek et al., 2022), or (iii) imbuing model with topology-basee inductive biases (Horn et al., 2022; Verma et al., 2024).
2.2 GRAPH POSITIONAL ENCODINGS
Inspired by positional encodings transformers (Vaswani et al., 2017), graph positional encodings emerged as way to inject structural information into the node features. Architectures such as GPS (Rampášck et al., 2022) combine multiple PEs, enabling global-attention layers to incorporate graph structure. Graph PEs have also been shown to benefit standard MPNNs (Dwivedi et al., 2022; 2023; Ma et al., 2021; Verma et al., 2025). Rampášek et al. (2022) propose categorization of graph PEs into Positional Encodings and Structural Encodings, further subdivided into local, global, or relative variants. Two commonly-usee Encoding (Dwivedi al., 2022, RWPE) and the Laplacian Positional Encoding (Maskey et al., 2022, LaPE), which we will briefly introduce below. Both methods inspired several other non-learnable approaches (Grötschla et al., 2024: Lim et al., 2023; Maskey et al., 2022; Rampášek et al., 2022), as well as learnable ones (Lim et al., 2023, SignNet)..
graph positional encodings are the Random Walk Positional
Random Walk Positional Encoding (RWPE). For any node v E V(G), Dwivedi al. (2022) define the d-dimensional RWPE of v, denoted by p RWPF& as
RWPF, RWW... no: ney
(RW)............................ (RW) k
P
where RW == AG)) is the random walk matrix of the graph 9, A(G) denotes the adjacency matrix of G. and D()) denotes the degree matrix of G. Rampášek et al. (2022) categorize RWPE as local structural encoding.
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Laplacian Positional Encoding (LaPE). The normalized Laplacian matrix of G is given by L(G) = I (()))))) where I denotes the identity matrix. The LaPFs of the nodes in G are constructed from the eigendecomposition of L(g) = Q A Q. Given the eigenvalues sorted in ascending order X ,...... X(k),
(..,,,(() (K), Dwivedi with corresponding eigenvectors q ,..,,.,."""
et al. (2023) define the k-dimensional LaPE (p!-PE:) of node as
Pe
LaPEa = [a q ((+1)
( 11 RR,
where is the index of the first non-trivial eigenvector. Since LaPE employs the cigendecomposition of the full graph, it is considered to be a global positional encoding (Rampášek et al., 2022).
2.3 THE EULER CHARACTERISTIC TRANSFORM (ECT)
The Euler Characteristic Transform (ECT) originated as method to summarize simplicial complexes, i... higher-order domains (Turner al., 2014). Being an expressive and computationally favorable summary statistic, it has found many interesting applications in the biomedical sciences (Amézquita al., 2021), computer vision (Cisewski- Kehe et al., 2023; Jiang et al., 2020), statistical functional analysis (Crawford et al., 2020), and, more recently, in generative tasks (Röell & Rieck, 2025). For an extensive exposition of the applications and review of the literature, we refer the reader to Munch (2025) or Rieck (2025).
We will restrict our exposition to the case of graphs, consisting of vertices and edges. For graphs, the Euler characteristic is defined as the number of nodes minus the number of edges; it is topological invariant of the graph. When the Euler characteristic of two graphs is different, the graphs are not topologically equivalent, permitting us to distinguish them.' However, many graphs have the same Euler characteristic, its expressive power remains limited. By moving to a multi-scale variant of the Buler characteristic, we obtain the ECT, which combines geometrical and topological information to obtain a more expressive representation. Specifically, given a featured graph (G,z), we calculate the inner product of its attributes with a unit vector Sd-1, referred to direction, and consider the pre-image of the inner product to obtain a monotonically increasing sequence of subgraphs of G. Tracking the Euler characteristic along that sequence indexed by E E. yields the Euler Characteristic Curve (ECC) in the direction of f. The map that sends each direction vector to its corresponding ECC called the Euler Characteristic Transform (ECT). For graphs, is defined
ECT: gd
x XR +
(0,t)
1 a(v)), (t) VEV(C)
Σ
Σ 1[maxuce z(u)), oe) (t).
(4)
eEF(G)
Somewhat surprisingly, given sufficiently large finite number of directions, the ECT is injective on geometric graphs and geometric (simplicial) complexes (Curry et al., 2022: Ghrist et al., 2018), i.e., distinct inputs yield distinct ECTs. One limiting factor the applicability of the ECT in deep learning setting is the lack of differentiability with respect to the direction vectors and input coordinates. However, by approximating the indicator function of Equation (4) with a sigmoid function, we obtain the Differentiable Euler Characteristic Transform (Röell & Rieck, 2024, DECT), which may integrated into standard deep learning pipelines. This formulation of the ECT provides global summary of shape, but certain graph learning tasks benefit from a local perspective of the graph around a node of interest. As a static, i.c., non non-trainable, extension to the ECT, the local Euler Characteristic Transform (von Rohrscheidt & Rieck, 2025, (EECT), constitutes variant based on local neighborhoods with favorable properties for node classification. Given a featured graph (G,z) with z: V(g) + Rdd and vertex v, the local ECT of with respect to N is defined as
e-ECT | x; u| = ECT (v, 9), (v..).
(5)
where Nm (v. 9) denotes neighborhood of v, whose locality is controlled by the hyperparameter m. The following result by von Rohrscheidt & Rieck (2025) relates the é-ECT to MPNNs.
Theorem 1. Let (G,x) be a featured graph, and let {e-ECT, ( E-ECTs of all the vertices E V(G). Then {e-ECT, a,;; u}} provides all the (non-learnable) needed information to perform single message passing step on (G,z).
be the set of the 1-hop
Formally, homotopy-equivalee topological spaces have the same Fuler characteristic.
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2
3
Figure I: Steps for computing the LEAP PE using 1-hop neighborhoods. (1) The neighborhood of node in a featured graph is selected. (2) Normalization of the neighborhood features. (3) Computation of the differentiable ECT. (4) Projection of the matrix representation of the ECT to get the PE vector.
The required non-learnable information for a single message passing step refers to the fact that for a given vertex This result highlights the power of the 1-hop EEECT for graph representation learning. Moreover, Rohrscheidt & Rieck (2025) show that the EEECT sufficiently expressive perform subgraph counting, one of the limitations of traditional message passing architectures (Chen et al., 2020). This illustrates that ECT-based methods can be more powerful than traditional message passing neural networks in certain cases.
theoretically recover the features of the neighboring nodes from the E-ECT.
METHODS
This section introduces the Local ECT and Projection PE (LEAP), learnable local structural graph PE based on the (-BCT. As part of this encoding, we present strategies for projecting the ECT of shape into a lower-dimensional space.
3.1 E-ECT BASED POSITIONAL. ENCODING
Given featured graph (G, œ) with d-dimensional node features, which may be static (i.e., the original node features another PE), or learned (i.e.. hidden states at some step of MPNN), let [0, 1] be a finite set of thresholds and θ C gd-1 a finite set of directions. The k-dimensional LEAP PE of a node V()) constructed as follows:
1. Compute the m m-hop subgraph (v, G) around node U. (u, 9)), mean-center their feature set (a(u))..... ((,,,)) } and divide each element by the maximum norm in the centered set to ob- tain new features IF mapping between each node in the m-hop and its normalized feature vector.
2. Given the of nodes {ui.... ]n
((,))...; C $d-1, where f: (v,
denotes the
3. Compute the matrix M E [[|x|T]| whose (i,j) entry is the differentiable approximation of the ECT of (N (v. 9), (0,, ty) E x Lastly, a learnable projection final positional encoding of node v.
R|[|X|TI R maps M to vector PE(c) E IR* which the
Remark 1. LEAP is not static pre-processing step the graph. On the contrary, it can be integrated graph neural network architectures to be trained in an end-to-end fashion.
The previous remark highlights key difference between LEAP and graph PEs like LaPE and RWPE. This is also an important distinction from the prior use of the (-ECT, which was introduced as static, non-learnable extension of node features, with neighborhood connectivity being disregarded as the ECT was calculated node neighborhoods if they were point clouds rather than graphs (von Rohrscheidt Rieck, 2025). In addition, LEAP permits the set of directions θ to be randomly initialized and then either kept fixed optimized during training. LEAP also be applied to
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learned graph features and, since it integrates with any GCN, it is naturally applicable to both graph-level and node-level tasks. By contrast, the DECT is geared towards generating graph-level descriptors (Röell & Rieck, 2024).
Remark 2. Within the categorization of Rampášek et al. (2022), LEAP is a local structural encoding.
The locality of LEAP comes from computing each node's encoding only from its m-hop subgraph. Thus, locality controlled by the hop number m, which serves as a hyperparameter. By default, we suggest 1-hop neighborhoods, making our method as scalable as message passing, but we also describe two ways to control the locality of LEAP:
Use a larger hop number m. While straightforward, it should be noted that two nodes may differ in their vn-hop neighborhoods while becoming identical at (m + 1)-hops.2
Alternatively, we compute LEAP multiple times for each node with increasing m, then concatenate the results to obtain PE that captures how the m-hop neighborhoods evolve as m grows.
We also note that two nodes in graph that share identical m-hop neighborhoods receive the same LEAP PE, since the second step in the computation of the PE yields identical outputs. This aligns directly with the definition of local structural encoding given in Rampášek et al. (2022, Table 1). Moreover, consider a node whose normalized m-hop neighborhood features form a geometric graph embedding." If could access the ECT of that subgraph rather than approximation, then by the injectivity results of the ECT (Curry et al., 2022; Ghrist et al., 2018) we would have all the information required recover the neighborhood's structure."
3.2
ECT PROJECTION STRATEGIES
Since LEAP aims capture structural information, it should be invariant to scaling and rotations of neighborhood features. Step 2 above addresses normalization, but to minimize the effect of rotations, the projection in Step should be permutation invariant with respect the ECCs. However, this requirement is often ignored in practice (Röell & Rieck, 2024). In the remainder of this section, we present five projection strategies for LEAP, some of which explicitly enforce this invariance.
Linear projection: We "flatten" the E-ECT of each node into a vector v RD with D = |0] (T).
following Amézquita et al. (2021). We then apply a linear projection by multiplying v with learnable matrix W R This projection is not permutation invariant with respect to the ECC, and the number of learnable parameters with respect to [+] and [T] is O( (T]).
One-dimensional convolutions: We treat the 2-ECT of each node as a multichannel time series, where thresholds act as time steps and each ECC defines channel. Several ID convolutions are concatenated, and the resulting channels are averaged to produce a vector that is used as an input to an MLP. This projection is not permutation invariant with respect to the order of the directions, and the number of learnable parameters with respect to [ and [T] is ( + (T)).
DeepSets: We treat the ÉEECT of a node as a set of "Ti-dimensional vectors, corresponding to the ECCs along different directions in O processing this set using architecture inspired by DeepSets (Zaheer et al., 2017): Given the set of vectors corresponding to the ECCs we have PE MLP22Enefe MLP, (ECC,)). This projection strategy is permutation invariant wrt. the directions of the ECT, and its number of learnable parameters is independent of [e].
Attention: We treat the ÉEECT of a node as a set of ['T] "Tl-dimensional vectors, corresponding to the ECCs along the different directions in O and process this set by transformer encoder with a single attention head. To obtain the PE, we apply an MLP to the sum of the generated ECC representations. Due the use of self-attention without any positional encoding, the projection is permutation invariant, and the number of learnable parameters depends on [T] but not on [O]-
Attention with PE: As variant of the previous projection, instead of feeding the transformer encoder the set of ECCs directly, we concatenate each ECC, with the corresponding direction A θ before passing it to the encoder. This yields a permutation invariant projection strategy, while incorporating information about the directions along which the ECCs were computed.
Foor sufficiently large m. this strategy yields identical PEs for all nodes within the same connected component.
IIn general, there is no guarantce this will occur.
We design experiment test the ability of LEAP to capture topological features of graph, see Section 4.1.
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3.3 PROPERTIES
We first discuss the computational complexity of our method. Given an m-hop subgraph, Nn (v. () for each vertex 2, calculating the E-ECT has a total computational complexity of O Nm (v. 9)l. In the worst case, each subgraph is the complete graph on vertices, leading to an overall complexity of (n²). For sparse graphs whose m-hop neighborhood is of the order of m = (n). we obtain worst-case complexity of (nn)). Finally, assuming bounded degree, this reduces to a worst-case complexity of (n)) which is asymptotically equal to one step of message passing. Moreover, individual ÉEECT's can be computed in parallel. In terms of expressivity, von Rohrscheidt & Rieck (2025) provide the theoretical foundation for our work, stating that, given a sufficiently large number of directions, the injectivity of the E-ECT guarantees that it is more expressive than message passing. However, we consider the main contribution of our work to be the development of a novel local positional encoding and its empirical evaluation, in the spirit of Rampášek et al. (2022), leaving more in -e-hphhh theoretical analysis for future work.
4 EXPERIMENTS
We conduct experiments to evaluate different aspects of LEAP, investigating (i) its ability to capture structural properties independent of node features, (ii) its impact on the performance of different graph neural network architectures and the effect of learning the directions of the transform, (iii) its performance on a large-scale dataset with 202,579 graphs (Chen et al., 2019), (iv) its behavior when applied to learned node features in the H/V dataset (Wu et al., 2018), and (v) the effect of hyperparameters. Subsequently, L.EP-I. indicates that the directions for LEAP were randomly initialized and learned during training, while LEAP-F denotes that the directions remained fixed.
4.1 SYNTHETIC DATASET
We introduce a synthetic dataset of 40,000 graphs to test whether LEAP can capture structural differences independent of node features, thus proving that LEAP is indeed a structural encoding. Each graph has three nodes and contains either zero, one, two, or three edges, yielding classification task with four classes based on edge count. The node features are uniformly sampled from the unit disk D2 C R² to make the task purely structural. We use a standard GCN and GAT architecture as base models, and compare them to the same model enhanced with LEAP added as structural positional encoding. For the computation of the ECT used in LEAP, we use 16 directions with resolution of 16, summarizing each graph into 16 x 16 ECT. The models enhanced with LEAP achieve perfect accuracy of 100.0 + 0.0, demonstrating LEAP's ability to capture structural properties independent of the node features. By contrast, the GCN and GAT models alone exhibited lower accuracies (71.83 + 0.27 and 69.44 = 0.82, respectively), demonstrating their inability to capture relevant structural graph properties when informative node features are not available".
4.2 CLASSIFYING REAL-WORLD DATASETS
Table 1: Best approach (architecture, PE strategy. and projection strategy) and relative accuracy im- provement with respect to the worst performing baseline for TU classification datasets. In all cases the best result was achieved using our PE strategy.
We evaluate LEAP on several graph classifica- tion datasets from the TU benchmark (Morris et al., 2020). Our aim is to evaluate (i) the capac- ity of LEAP to enhance existing graph neural networks with structural information, (ii) com- pare LEAP with existing PEs, and (iii) inves- tigate in which architecture LEAP induces the largest increase in accuracy. Of particular inter- est is the evaluation the Alchemy -full (Chen et al., 2019) dataset, as the regression targets are rotation invariant with respect to the node features. For this dataset, we normalize the re- gression targets that all 12 tasks are on the
DATASET
BEST METHOD
WORST
BEST
GAIN (5)
LATTE--H
1.EARI ID
41.6
81.6
96.2
LUTTER-M
NeMP LEAP-L ID
57.8
88.5
53.1
LETTER-L.
NeMP+ 1LEAP- ID Cony
80.4
98.0
21.9
FINGURRRINT
NeMP 1.AP-1 Linear
48.8
55.7
14.1
LEEP--E-- Alln
77.7
BZR
1EAP- Linear
78.3
84.7
8.2
DHFR
GCN + 1.F.-P Atm wi PE
70.1
77.6
10.7
AAthhough LaPE and RWPE are computed independently of node features, we also performed this experiment using these PEs. Like I.EAP,, they achieved perfect accuracy.

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same scale. For the HIV dataset (Wu al., 2018), where nodes have categorical features, we exploit the end-to-end differentiability of the ECT by extending the architecture with a learnable embedding layer that maps these features into R where LEAP is computed.
Architectures. We fix four "backbone" architectures to which we add different positional en- codings, namely (i) GCN (Kipf& & Welling, 2017), (ii) GAT (Veličkovié et al., 2018), (iii) GIN (Xu et al., 2019), and (iv) NoMP ("no message passing"), ") a model that we introduce based on transformer encoder. Following Maggs et al. (2024), 32-dimensional hidden states for GCN and GAT. For the Alchemy dataset, we scale the architectures to 10 layers with 64-dimensional hidden states. The NoMP architecture projects node features into 16-dimensional latent space with linear layer. followed by a single self-attention layer that produces 16-dimensional state for each node. The final classification is performed by a feedforward layer. We chose hyperparameters to match the parameter count of GCN/GAT. By design, NoMP ignores graph structure unless given positional encodings, thus permitting us evaluate the ability of each PE to encode relevant structural properties. Finally, as positional encodings. we consider the following baselines: (i) No positional encoding, (ii) RWPE, which, like LEAP, a local structural PE under the categorization of Rampášek et al. (2022), making it particularly relevant baseline, and (iii) LaPE, a widely used graph PE that, unlike LEAP, captures global positional information.
use five message-passingg layers and
Experimental setup. All experiments use 5-fold cross-validation and are trained with the Adam optimizer for up to 100 epochs with early stopping enabled. As loss term, we use the cross entropy loss except for the Alchemy dataset, where we use the mean squared error loss. We use 10-dimensional PEs for all types and datasets. The only difference between the backbones with or without a PE is that the input dimension of the backbone increases by 10 when a PE is used. The Euler Characteristic Transform in LEAP is calculated with 16 directions and 16 thresholds. To simplify the setup, we keep all hyperparameters of LEAP's 's projection strategies fixed across all datasets. Despite this, describe below, we observe high predictive performance across a variety of datasets.
4.3 RESULTS
Table 2: Surprisingly, increasing the neighborhood size (Nm)) does not improve the efficacy of LEAP, showing that the 1-hop neighborhood is sufficient.
Table 4 reports the results for LEAP and the baselines in combination with different architec- tures across the various classification datasets. For every dataset-architecture combination, the two LEAP variants (F/L.) achieve the best and second- best performance. When combined with GCN and NoMP architectures, learning the di- rections of LEAP consistently improves perfor- mance in comparison to keeping them fixed. For GAT, learning the directions slightly reduces per- formance compared to the fixed variant of LEAP
METHON No.
LETTER-H
LETTER-M
LETTER-I.
81.29 4 1.91
88.00 1 1.89
96.27 i 0.84
LEAP-F
84.31 33.24
94.13 1.05
1,2
77.91 1.82
84.76 1.40
96.09 + 0.34
1
80.62 3.58
86.49 2.20
96.18 1.24
LEAP-L
2
72.76
85.11:1..9
93.38 a 0.84
1.2
2.83
85.96 1.42
95.64 1.51
in 3 of the 7 datasets. For all datasets, the overall best-performing architecture- combination always uses LEAP with learned directions.
Table 1 reports the relative increase of the perf-pernorning method compared the porst-pe method. We observe the largest gains from using LEAP on the Letter and Fingerprint datasets. For the datasets COX2, BZR, and DHFR, the advantage over the baselines is less pronounced, likely due to their smaller size, making it harder to benefit from richer features. We observe the largest improvement on DHFR, the largest dataset among the three. Table also shows that NoMP without positional encodings outperforms baseline GNNs on the Letter-High and Letter-Low datasets, highlighting the limitations of MPNNs, i.e., models without structural information may achieve better results.
We also evaluate LEAP on the Alchemy and HIV datasets from the TU Benchmark using the same setup described above. The Alchemy dataset is significantly larger compared to the other datasets allowing LEAP to extract more meaningful information from the data. For the Alchemy dataset, show the R² score and for the HIV dataset, report AUROC, due to large class imbalances. Figure shows the results for LEAP with the GCN backbone and the various projection methods, showing clear advantage for LEAP (with learned directions) over all baselines. For the HIV
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82.5
76.0
80.0
77.5
Test AUROC
74.0
72.5
70.0
70.0
67.5
RWPE LaPE
PE RWPE LaPE LEAP LEAP LEAP LEAP +LaPE +LaPE
-L
Alchemy
HIV
Figure 2: Results for different PE strategies on the Alchemy and H/V datasets reporting the R? and AUROC using a GCN. Colors rank the PEs from best, second best worst. LEAP with learnable direction significantly outperforms other methods on the Alchemy dataset while performing second best on the H/V dataset.
dataset, results exhibit large degree of variability, and this is the only case where neither of the Two LEAP variant (L/F) outperforms all baselines. Still, both variants surpass RWPE, which is in the same category of local structural encodings. LaPE yields the strongest performance for single PE, suggesting that global positional information. which cannot be captured by LEAP, may be particularly relevant in this case. For this reason, we also evaluate the use of LaPE and LEAP combined through concatenation. We observe that the combination of both PEs yields the best overall results. This showcases how these two PE strategies can successfully complement each other. Please refer to Figure S.I and Figure S.2 for the full results.
4.4 ABLATIONS
After having established that LEAP captures essential structural information to be used with multiple graph neural network architectures, we further aim to investigate the sensitivity of LEAP with respect to its various components. In particular hope to further understand how (i) the choice of projection method, (ii) the number of hops, and (iii) the embedding dimension impact the performance of LEAP.
Table Ablation for the embedding dimension of the projection. LEAP is stable wrt. the dimension, performing consistently well.
Projection strategies. We repeated all exper- iments using the five proposed LEAP projection strategies such that each projection strategy has approximately similar small parameter budgets, comprising 1K-5K parameters. Table S. re- ports the results; and we find single projec- tion consistently outperformed the others, show- ing the best projection be dependent the combination. However, a remarkable fact is that learnable directions did on average outperform the fixed set of directions, underpinning the benefits of learnale directions as compared to using them as static features.
EMB.
PE
LETTER-H
LETTER-N
LETTER-L
LaPE 66.31
94.71 : 1.29
76.76 =
RWPE
4
94.62 1 0.51
83.47 a 1.68
LEAP-F
81.1611.66
86.76 1.86
LEAP-L
3..33
95.56.1..33
0.99 87.38
fataset-architeoo
LaPE 94.22 3.54
82.09 1.64
RWPE
1..64
94.67 à: 0.96
85.82 + 0.76
LEAP-F
0.57
96.98 0.25
86.76 1.11
LEAP-L
1.440
96.44 0.59
88.36 a 0.64
LaPE
1.58
91.11 + 2.23
76,93 2.52
RWPE
79.24
94.67 4: 0.97
84.53 a 1.32
10
LEAP-F
2..03
96.68
86.59
Locality parameter sensitivity. We also study the effect of the locality parameter in LEAP by repeating the experiments on the Let- ter datasets, originally performed with 1-hop neighborhoods, using instead 2-hop neighbor- hoods and the concatenation of LEAP embeddings from 1- and 2-hop neighborhoods." For this
LEAP-L
80.68 2:4.2
80.68
96.99 4: 1.05
87.24
LaPE RWPE
9.924 2.49
77.64 2.11
1.336
95.38
86.49
20
LEAP-F
80.84 1.89
95.56 0.65
87.42 0.75
LEAP-L
1.59
96.40 A 0.79
86.89 à 1.56
In the concatenation setting, each emhedding is computed with half the target dimension so that the final representation matches the dimension of the other approaches.
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Table 4: Accuracy results for different PE strategies when using a variety of GNN architectures for multiple datasets from the TU Dataset benchmark. We evaluate the GCN, GAT and GIN architectures with and without LEAP. Additionally we evaluate LEAP on a non-message passing architecture (NoMP). Best results are green, second best are green, and worst are red. For every dataset, our approach achieves best and second best results.
MODEL
PE
COX2
BZR
DHFR
LETTER-H
LETTER-M
LETTER-L
FINGERPRINT
No PE
77.9
81.
71.6 1.4
41.64.1
6..5
80.4 1.0
48.8 1.4
RWPE
7.5
7.25
2.40
60.9
6.7
83.2
49.4
GCN
LaPE
7,94
8.2
7.
25.
2.6
8.22
48.1 1.8
LEAP-F 0.6
82,
7.21
7..2
12.6114
95.8
55.6 1.1
LEAP-L
1..4
1.6
7..6
1.52
83.6 1.3
96.0
5..1
PR
7.6
8.0
1.8
41.9
3.7
8.74
0.65
RWPE
1.40
1.1
7.,9
63.0
6.
9.88
0,8
GAT
LaPE
1.0
8.2
70.4
54.7
75.
89.6
48,9 1.0
LEAP-F LEAP-L
1.6
8..0
3..7
70,2
83.2
95,8
55.1
8..
83.7.29
7..5
73.5
8.64
9,92
5.7
PE
79.5 7.5
4.8
47.7
65.0
82.7
48.4
RWPE
1.6
79.3
7.90
2..4223
64.9
81.6
50.0 1.6
GIN
LaPE
7.5
1.7
61.00.11
55.0
75.223.5
84.4
4
LEAP-F 7.90
81.2
73.9
60.2
76.3
93.3
54.4 1.1
LEAP-L
1.66
81
76.2
62.7
7.76
94
55.3 1.4
0.8
7..1 2.6
1.
57.8
8.37
5.5
RWPE
1.3
8.79
1..
1..2
84.5
9..7
51.3
NoMP
LaPR
1.0
81.2
3.55
1.60
76.9
91.1
50.5 1.2
LEAP-F
7.6
83.2
7.3
813
88.0
9.
55.7 1.1
LEAP-L
7..6
84.7
75.7
81.6
88,5
98.0
56.3 1.4
ablation, we use the NoMP model so that the models can only access structural information through the PE, and we use atlention with PE as the LEAP projection strategy. Table 2 shows that the 1-hop neighborhood yields the best performance across all datasets, followed by the concatenated and 2-hop version, respectively.
Impact of PE dimension. To better understand the effect of increasing the embedding dimension for the projection strategies, we vary the size the embedding dimension on the Letter datasets. ran with PE dimension 10 for both LEAP and the baselines, and
The original experiment now repeat it with the embedding dimension set to {2, 5, 10, 20). respectively. As before, we fix the architecture to NoMP that models access structural information only through the PE, and use attention with PE as the projection strategy for LEAP. The results Table show that across all evaluated PE dimensions and datasets, LEAP outperforms both RWPE and LaPE.
DECT hyperparameters and comparison. We assessed LEAP's sensitivity the DECT hy- perparameters by varying the number directions in {2, 4, 8, 16, 32} and smoothing parameter in {2, 2, 8, 16, 32, 64, 128). LEAP remained robust, outperforming baselines across all settings; Figure S.5. Finally, Table S.2 summarizes the comparison of LEAP with DECT for graph classifica- tion tasks. LEAP outperforms two variants of the DECT (with different parameter budgets) most datasets, which further underscores the utility of learnable directions.
5 CONCLUSION AND FUTURE WORK
We presented LEAP, a new learnable local structural positional encoding for graphs based on the (-ECT, To the best of our knowledge, this is the first approach to integrate the (-ECT into deep learning architectures in end-to-end trainable fashion.
Our experiments show that LEAP consistently outperforms established baselines across multiple architectures and datasets, with learned directions further improving performance in most tasks, thereby highlighting the benefits of making this step trainable. Additionally, we introduced synthetic task in which our approach achieved perfect accuracy, demonstrating its ability to capture

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topological information independent of node features, which the evaluated MPNNS (GCN/GAT) alone failed to recover. Taken together, these results highlight the potential of &EECT encodings for topological deep learning (Papamarkou et al., 2024) and graph representation learning tasks. LEAP is particularly well-suited to provide local structural information to architectures that rely on global attention mechanisms, where graph structure not directly modeled and multiple PEs are combined to capture complementary notions of graph position.
Limitations. While LEAP provides a learnable way to capture local structural information has some limitations. First, it is not purely structural PE, as it requires node features to compute the ECTs, However, these features can be learned, and in the synthetic dataset, our approach succeeded even though the features were irrelevant to the prediction targets. Second, LEAP relies on a differentiable approximation the discretized ECT applied to normalized m-hop subgraphs, which are not necessarily geometrie, the theoretical guarantees of the exact ECT (e.g., injectivity) may not fully carry over; we expect this to be interesting for future work. Finally, unlike other graph PEs, such LaPE or RWPE, where the only hyperparameter is the embedding dimension, LEAP introduces several hyperparameters (among others, a smoothing parameter of the ECT approximation, the number of directions, and the number of discretization steps). In practice, however, we fixed these across datasets and architectures, nevertheless observing consistently strong performance. Our ablation studies further serve demonstrate the robustness these choices.
Future work. We envision several directions for future research. First, drawing on prior work (von Rohrscheidt & Rieck, 2025), we aim to formalize the theoretical expressivity of LEAP, noting that theoretical expressivity and empirical performance with positional encodings that capture complementary aspects of graph structure and embedding within more sophisticated architectures may further improve performance and expressivity. Another promising line work is to make the ECT step fully differentiable. Instead of discretizing along a fixed grid, treating thresholds as trainable parameters would allow the model to focus on informative regions and optimize their positions jointly with the other parameters. Using learned features, we also plan on assessing the performance of LEAP on non-atie aatrrbibbbted graph datasets, i.e., datasets that are fully structural. Finally, since the ECT can be applied to higher-orde datasets (Ballester et al., 2025) comprising, for instance, simplicial complexes or cell complexes (Hoppe et al., 2025), we believe that LEAP could be extended to this modality, thus serving as generalizable addition to the topological deep learning toolbox.
often not correlated. Combining LEAP
REPRODUCIBILITY STATEMENT
Our code can be accessed ht https: / comm/aaooos--a/LLLB All experiments used a fixed seed; the full configurations can be found in the exper iments folder.
ACKNOWLEDGMENTS
In loving memory of Teresa Puz Camps.
First and foremost, facing a turbulent decision-making process, want to thank the reviewers and the AC for believing in the merits of our work. We are particularly grateful for the support by reviewer f f41T, who continued to champion our work throughout the revision process. This work has received funding from the Swiss State Secretariat for Education, Research, and Innovation (SERI). B.R. wishes to dedicate this paper to his daughter Aurélie.
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Appendix (Supplementary Materials)
A Results for the Alchemy and HIV datasets
15
Validation metrics
16
Additional experiments
17
Computational Performance
19
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A RESULTS FOR THE ALCHEMY AND HIV DATASETS
0,80
0.82
0825
0,000
0800
6.775
C.75
Test
0.750
Test
C770
0.725
0.675
No
RWPE
LaPE
LEAP-F
LEAP-L
No PE RWPE
LaPE
LEAP-F
LEAP-L
GCN
GAT
Figure S.1: R2 results for different PE strategies on the Alchemy dataset using the GCN and GAT architectures. Best results in terms of mean R² are green, second best are green, and worst are red.
76.0
74.0
Test AUROC
74.0
Test AUROC
72.0
70.0
72.0
60.0
66.0
70.0
64.0
62.0
RWPE
LEAP-L LEAP-F LEAP-L +LaE +aPE
RWPE
LaPE LEAP-F LEAPL LEAP-F LEAP-L
+LaPE +aPPE
GCN
GAT
Figure S.2: AUROC results for different PE strategies on the HIV dataset using the GCN and GAT architectures, respectively. Best results in terms of mean AUROC are green, second best are green, and worst are red.
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B VALIDATION METRICS
GCN
Validation Accuracy
Validation Loss
0.5
0.9-
ECT
GAT
0.4
0/
0.6
8.
10
30
12
30
Epoch
Epoch
Validation
Validation Accuracy
Figure S.3: Validation loss and aceuracy per training epoch for the synthetic dataset for the baseline GCN, GAT, and LEAP. Our method achieves a perfect score in both metrics and convergence immediately. The shadows indicate standard deviation over 5 runs and the dashed line means that model training finished earlier because of early stopping.
0.7
0.6
0.5
0.4
0.4
0.3
LEAP
LEAP
RWPE
0.2
LaPE
LaPE
0.1
0.2
Letter High
Letter Medium
Figure S.4: Validation accuracy per training epoch for the Letter High (left) and Letter Medium
(right) datasets for different PE strategies using GCN architecture. Our method achieves the best results and converges faster. The shadows around the curves indicate the standard deviation over runs and the dashed line means that training ended due early stopping.
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C ADDITIONAL EXPERIMENTS
Table S.I: Accuracy results for all real-world datasets when varying the strategy of the LEAP PE with fixed and learnable directions for different models. The backbone architectures have around 4K parameters and we show the additional parameters each positional encoding introduces.
CON2
Estur
38 404
133w33
856 43+13
835
53
D0N
76-38
85--11
M1+11
Cl
7801 12
752222
798413
125
134425
031133
017435
134414
S40115
064112
713215
101222
13322
GALTh
201133
363:21
Are
$30.34
N9+11
711-18
03+11
G1eld
DAT
5:5-18
3---1
615-14
90-11
REASS
764258
16.2205
753:205
182213
739241
315140
66313
102233
525213
253114
OIN
155.06
00-11
M1+11
S1
Deppeex
11
767-05
740+13
M1-13
01+33
207463
113-17
636408
Lear
061236
100250
05.:00
112017
1ASES1
169235
1A3212
SALES
Alm
117241
NoMP
tt
4EAS5.
764184
758:003
19220
662143
113119
93100
131112
342114
8321
Deephex
-63-05
813+21
120+38
754-16
66166
9:1-E4
A0C
21
82:15
22:00
0:11
T1/3
7147
60.00
9923
7223
6047
0780
20
10-40
67
87%
17:15
16:93
00.4
M0-4
01:1
37
6142
04
81/11
00.0
55.00
M:27
5000
50.20
16.69
6747
9600
10:30
MC
Decal pi
Zeaaling 2t
Letter High
Letter Medium
Letter Low
X5:11
22:44
10.07
6076
62:3
02:11
(7:11
00
MI
0669
67:30
W1
00:C3
00:37
00.4
67-40
K:4
06.44
0r:11
50
10:44
6142
80
05.13
20:7
55.20
X:42
56,00
55.00
X:.6
06.44
422
67:9
10:0
07:2
10.09
06.0
60:00
1252
70.33
Shosking
Denaring pr
Letter High
Letter Medium
Letter Low
Figure S.5: We assess the sensitivity of LEAP with respect to the hyperparameters used in the ECT. Top row shows the effect of changing the hyperparameters for LEAP-F (fixed directions) and the bottom row for LEAP-L (learnable directions). LEAP consistently outperforms baselines across all settings and thus robust with respect to the hyperparameters.
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Table S.2: We provide comparison with DECT (Röell Rieck, 2024). DECT summarizes the graph with single global ECT and subsequently applies a convolutional neural network for the classification. We compare our method with two variants DECT, one with 4K parameters and one with 65K parameters. The parameter count in LEAP ranges from IK to 5K and therefore the comparison with DECT (4K) would be the most appropriate, although we outperform both variants on most datasets.
MODEL
COX2
BZR
DHFR
LETTER-H
LETTER-M
LETTER-L
DECT (4K)
7..4
8.2
6.0
63.8
76.2
91.5 + 2.1 96.8 + 1.2
DECT (65K)
74.6
8.13
7.69
8..4
86.3 2.0
LEAP-L(GCN)
79.4
1.6
1.52 7.86
13.611.3
96.0
(EAP-L(GAT)
80.1 2.2
83.7
7.15 7.85
95.2 8,.4
LEAP-L (NOMP)
8.77 0.8
75.7 2.7
81.6
2.5
9..0
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D COMPUTATIONAL PERFORMANCE
Table S.3: We report the average training time for the Letter High dataset. The average training time measures the total training time, including potential early stopping. For fair comparison between the various methods we also report the average training time per epoch. LEAP yields both high performance while remaining fast to train.
METHOD
Training TIME
AVERAGE EPOCH
LaPE
60.35
0.64
NoPE
44.60
0.51
RWPE 35.77
0,50
LEAP-L
32.51
0.93
LEAP-F
30.08
0.84
Table S.4: For the Roman Empire dataset (Platonov et al., 2023) the average training time per epoch reported well the initial preprocessing time. Since this dataset consists in single graph with 22662 nodes. 32927 edges, and dioeeimnsi features, we choose it to evaluate how the runtimes of the PEs scale for larger graphs. We report preprocessing time as for this graph size it is not negligible. Moreover, in this case, for the LEAP variants, the preprocessing consists in precomputing the node neighborhoods. LaPE requires the eigen decomposition of the full graph, leading to large initial preprocessing times. LEAP remains computationally efficient.
METHOD
Training TIME PER EPOCH
PREPROCESSING TIME
NoPE
0.09
0.01
LaPE
0.10
164.83
RWPE
0.10
0.18
LEAP-F
0.58
11.57
LEAP-L
1.13
11.57
19


Reviews:
[{"review_id": "SWjcVjXagV", "reviewer": "Reviewer_dZ9D", "summary": "This paper introduces LEAP, an end-to-end trainable local structural positional encoding that integrates geometric and topological information via the local Euler Characteristic Transform (ℓ-ECT). LEAP computes node-level embeddings by extracting m-hop neighborhoods, normalizing local features, applying a differentiable ℓ-ECT, and projecting the resulting matrix into a compact representation through learnable modules (e.g., linear layers, 1D convolutions, or DeepSets). Two variants are proposed: LEAP-F, which uses fixed ECT directions, and LEAP-L, which learns them during training. Extensive experiments on both synthetic and real-world datasets, using three GNN architectures, confirm that LEAP substantially enhances graph representation learning.", "questions": "Q1: On the HIV dataset, the locally encoded LEAP underperforms the globally informed LaPE. Could the authors elaborate on why global structural information is crucial for this task?\n\nQ2: The injectivity of the Euler Characteristic Transform underpins its discriminative power, yet LEAP employs a differentiable local approximation. Do the authors have theoretical or empirical evidence that this approximation preserves key topological information compared with the exact, global ECT?\n\nQ3: Is LEAP applicable to geometric graphs? If so, how does it leverage spatial coordinates or geometric relationships within the ℓ-ECT framework?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "S1: The paper proposes an end-to-end trainable framework for encoding positional information, eliminating the need for static or precomputed preprocessing.\n\nS2: The experimental setup is well-structured, evaluating three mainstream GNN architectures across multiple datasets, including both synthetic and real-world benchmarks. \n\nS3: Ablation studies demonstrate LEAP’s robustness to key hyperparameters and locality configurations. Moreover, the five projection strategies showcase LEAP’s adaptability to diverse model architectures and task requirements.", "weaknesses": "W1: Although the paper discusses LEAP’s robustness to hyperparameters (e.g., number of directions, thresholds), it omits computational efficiency analysis.\n\nW2: Since no single projection strategy yields the best results across all tasks, do the authors have heuristic or data-driven guidelines for choosing among them\n\nW3: Theoretical analysis and justification are insufficient, which is critical for this method-oriented paper.", "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": "GyEVtrGknM", "reviewer": "Reviewer_6i1u", "summary": "The paper proposes LEAP, a learnable positional encoding for graphs based on the local Euler Characteristic Transform ($l$-ECT), which integrates both geometric and topological information and supports end-to-end training. The authors validate LEAP on synthetic tasks and multiple real-world graph datasets, demonstrating its ability to capture structural information even when node features are non-informative.", "questions": "The ablation study (Table 2 ) shows that 1-hop neighborhoods perform better than 2-hop or a 1,2-hop combination. This seems counter-intuitive to the MPNN notion that larger receptive fields are better. Why is 1-hop sufficient?\n\nHow does LEAP perform on very large graphs?", "limitations": "", "rating": 4, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "(1) LEAP is the first work to integrate the local ECT into GNNs in an end-to-end trainable manner, combining geometric and topological insights.\n\n(2) The authors evaluate their method on synthetic data, small molecules, image graphs, and large-scale quantum chemistry datasets, showing consistent improvements. The synthetic task clearly demonstrates LEAP’s ability to classify graphs based solely on structural information, highlighting limitations of standard MPNNs.\n\n(3) The paper includes thorough ablation studies on various aspects (Projection strategies, Locality, PE dimension, and DECT hyperparameters) of LEAP, strengthening its conclusions.", "weaknesses": "(1) As the authors acknowledge in the limitations, LEAP is not a purely structural PE. It requires node features $x(v)$ to compute the ECT. While the synthetic experiment shows LEAP works even with uninformative features, it is unclear how, or if, LEAP would be applied to graphs without any node features (e.g., graphs represented only by an adjacency matrix).\n\n(2) LEAP introduces more hyperparameters (e.g., number of directions, smoothing, discretization steps) compared to LaPE or RWPE.\n\n(3) Although not analyzed in depth, computing ECT for m-hop subgraphs for every node could be expensive for large graphs.", "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": "jH8Q5RSkOG", "reviewer": "Reviewer_8cdq", "summary": "This paper proposes LEAP, a learnable graph positional encoding method based on a local Euler Characteristic Transform (ℓ-ECT). The key idea is to compute differentiable, direction-dependent topology-aware signatures localized around each node and incorporate them as positional embeddings in standard GNN architectures. By leveraging both geometric and topological structure, LEAP aims to address limitations of message passing GNNs that struggle to capture higher-order structural patterns. The authors evaluate the method on synthetic and real-world graph classification datasets, comparing to standard positional encodings such as RWPE and Laplacian-based LaPE. Results indicate that LEAP improves performance in many settings, particularly when used with architectures that are otherwise limited in structural awareness.", "questions": "1. Since LEAP is intrinsically local, have you considered augmenting it with a complementary global encoding? Could LEAP be combined with LaPE without redundancy?\n2. What is the computational overhead per graph and per layer when incorporating LEAP? How does it scale with graph size and number of directions?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The use of local, differentiable Euler characteristic transforms as graph positional encoding is an interesting new direction that meaningfully blends geometric and topological features.\n2. The paper clearly explains the limitations of pure message passing networks and why positional encoding is useful.\n3. The learnable direction variants of LEAP show thoughtful design toward practical applicability.", "weaknesses": "1. While LEAP is motivated as addressing expressivity limits of MPNNs, the paper does not provide formal analysis or guarantees regarding what structural distinctions LEAP can or cannot capture. The argument remains primarily empirical.\n2. While LEAP performs well in several TU datasets, its benefits diminish or disappear on datasets where global structure is more important (e.g., HIV), where LaPE performs best. This suggests that LEAP may be inherently limited in capturing global structural contexts, which should be more explicitly acknowledged or explored.", "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": "IruBPb6kBZ", "reviewer": "Reviewer_f41T", "summary": "Building upon the local Euler Characteristic Transform ($\\ell$-ECT), this paper introduces a new class of learnable *local strucutral encodings* (LSE) that can be integrated in any GNN architecture. To be precise, the main novelty is that such structural encodings are made learnable, since they were already adopted as static descriptors in another work, although very recent. Another novelty lies into a new \"projection\" layer in the proposed neural architecture, allowing one to reduce a matrix containing the differentiable approximations of $\\ell$-ECT, one per vertex, into a final vector: the positional encoding for that vertex. The approach is extremely versatile and the experimental section clearly shows that i) the proposed LSE allows standard GNNs to improve their performance in a variety of tasks and ii) learning the LSEs further increase the improvement and iii)  the choice of the hyper-parameters (e.g. number of directions, time grids) in not constraining, meaning that different choices still lead to good results, except for the hop's ray, in this case the smaller the better.", "questions": "Major remarks: \n\ni) It is unfortunate that the word \"projection\" is used in two different contexts: former concerning the definition of ECT, Eq. (4), latter concerning the $\\phi$ map at l.198. Moreover, this word plays a role in your acronym and it is quite clear that you refer to the second projection ($\\phi$) there but a few words about it would be appreciated. \n\nii) At line 225 you introduce the notion of *valid graph embedding*. It would be wise to recall the reader what it means.\n\niii) The experiments of the synthetic dataset are the only ones not being very convincing. I mean: what happens when you provide GCN/GAT with LaPE or RWPE? Isn't the prediction score equal to 1? Even if it were the case it would not be an issue, in my view, since in the rest of the section, by the way, you beat the state-of-the-art.\n\niv) Here's the weakest point of the paper in my view. There is a number of message passing architectures known to be more expressive w.r.t. GCNs. The first and maybe most important I can think of is graphs isomorphism networks (GINs, [1]). Including them in your Table 4 would turn your paper into a master  piece.  \n \nv) At l. 355 I read: \"An interesting observation from Table 4 ... \". I think you were too modest in your commentary! What I see in Table 4 is that on 5 columns over 7 the best accuracy score is obtained by a neural architecture **not** being a GNN, provided that it is equipped with LEAP-L. Did about Alchemy and HIV datasets? Did you test NoMP on them?\n\nMinor remarks: \n\ni) At line 239, you need a space between *to* and $|\\\\Theta |$  \nii) Line 248: *rojection* -> *projection*", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "This paper is really well written, clear and concise.  The experimental setup is solid and the results are impressive (more comments later), however I would say the contribution is just \"fair\" since local and global ECT descriptors already existed. However, I think that this paper has the potential to be accepted at ICLR, so I am **temporarily** ranking it with a 6, but I am ready to raise my note to 8 after the discussion, provided some key points are discussed/addressed.", "weaknesses": "A few points deserves to be discussed and/or presented differently. One major drawback, discussed below, and this is the only reason why I do not immediately rate this paper with a 8.", "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)