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Dynamic Edge-Conditioned Filters in Convolutional Neural Networks on Graphs

Martin SimonovskyNikos Komodakis

article2017CVPR1,389 citations

Introduces edge-conditioned convolutions that dynamically generate filter weights from edge attributes, allowing neural networks to generalize across arbitrary graph structures and excel at 3D point cloud classification.

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Deep learning models, particularly convolutional neural networks, have achieved outstanding success on structured grid data such as images and audio. However, many critical problems in fields like 3D spatial perception, biochemistry, and network analysis rely on irregular, non-Euclidean data represented as graphs or 3D point clouds. Traditional graph neural networks often struggle because they smooth out directional and relational information across neighboring nodes and fail to process graphs of varying sizes and structures dynamically.

The main objective of the article is to develop and evaluate Edge-Conditioned Convolution (ECC), a novel spatial convolution operation for neural networks that dynamically generates filter weights conditioned on specific edge attributes. The authors demonstrate the versatility and effectiveness of this approach on 3D point cloud classification and general graph classification benchmarks.

The evaluated approach uses a small multi-layer network to dynamically compute custom filter weights for each connection based on edge labels, such as spatial coordinate offsets in point clouds or bond types in chemical compounds. This operation operates directly in the spatial domain, allowing the network to process diverse graphs of varying sizes. The authors built deep classification architectures incorporating hierarchical graph coarsening (pooling) and evaluated their method on standard benchmarks, including the Sydney Urban Objects LiDAR dataset, ModelNet 3D shapes, and five biochemical graph classification datasets.

The evaluation produced several key findings. First, on real-world 3D point clouds from the Sydney dataset, the method achieved a new state-of-the-art weighted F1 score of 78.4% (and 79.5% with identity connections), outperforming existing volumetric voxel-based approaches like VoxNet (73.0%) and ORION (77.8%). Second, on chemical graph classification datasets with edge labels (NCI1, NCI109, and MUTAG), the method achieved top-tier performance (e.g., 83.8% on NCI1), outperforming other deep learning methods and matching leading graph kernel techniques. Third, incorporating rich edge labels proved vital; removing edge labels caused performance on Sydney to drop sharply from 78.4% to 38.9%. Finally, the approach proved highly robust to missing data and spatial sparsity, maintaining a 99.14% test accuracy on MNIST even when 80.9% of background pixels were discarded.

These findings show that treating 3D point clouds directly as sparse graphs avoids the high memory costs, resolution loss, and discretization artifacts associated with standard 3D grid voxelization. The results confirm that edge information preserves critical structural and geometric relationships that standard graph aggregations lose. Organizations working with 3D spatial data, autonomous systems, or molecular modeling can achieve higher fidelity and competitive performance by applying edge-conditioned graph convolutions rather than forcing data into uniform grids.

Moving forward, practitioners should consider direct graph representations for 3D sensing and relational data pipelines. For datasets lacking edge labels, appending node degree features can yield substantial accuracy gains (up to 5 percentage points). Future research and development should focus on applying this architecture to surface meshes and implementing randomized clustering techniques to reduce memory usage on large graphs with continuous edge attributes.

The conclusions are well-supported by thorough comparative experiments, though some limitations apply. The method incurs higher memory consumption when processing very large graphs with continuous labels. Additionally, on graphs lacking inherent edge labels (such as certain protein datasets), performance remains slightly below specialized graph kernels. Overall, confidence is high that the proposed edge-conditioned formulation offers a robust, highly effective approach for learning on irregular and spatial graph data.

École des Ponts ParisTechbipartite graphsclassificationconvolutional neural networksdynamic edge-conditioned filtersedge-conditioned convolutionconvolutional neural networks on graphs
arXiv: 1704.02901mys007/ecc
Cover for Dynamic Edge-Conditioned Filters in Convolutional Neural Networks on Graphs

Abstract

A number of problems can be formulated as prediction on graph-structured data. In this work, we generalize the convolution operator from regular grids to arbitrary graphs while avoiding the spectral domain, which allows us to handle graphs of varying size and connectivity. To move beyond a simple diffusion, filter weights are conditioned on the specific edge labels in the neighborhood of a vertex. Together with the proper choice of graph coarsening, we explore constructing deep neural networks for graph classification. In particular, we demonstrate the generality of our formulation in point cloud classification, where we set the new state of the art, and on a graph classification dataset, where we outperform other deep learning approaches. The source code is available at this https URL

Topics

bipartite graphsclassificationconvolutional neural networksdeep learningpoint cloudsource codesdirected graphsweighted graphclassification models

Keywords

dynamic edge-conditioned filtersedge-conditioned convolutionconvolutional neural networks on graphsdynamic filter networksgraph Laplaciangraph kernelsgraph coarseninggraph signalsWeisfeiler-Lehman graph kernelslayer normalization

Table of Contents

  • 1 Introduction
  • 2 Related Work
  • 3 Method
  • 3.1 Edge-Conditioned Convolution
  • 3.2 Relationship to Existing Formulations
  • 3.3 Deep Networks with ECC
  • 3.4 Application in Point Clouds
  • 3.5 Application in General Graphs
  • 4 Experiments
  • 4.1 Sydney Urban Objects
  • 4.2 ModelNet
  • 4.3 Graph Classification
  • 4.4 MNIST
  • 5 Conclusion
  • References
  • A Overview
  • B Details on Graph Classification Benchmark
  • C Robustness to Noise
  • D Edge Labels for Point Clouds
  • E Identity Connections
  • F Vertex Degrees in Edge Labels
  • G Vertex Degrees in Normalization

Knowls

  1. Knowl 1 — Edge-Conditioned Convolution

    model/method

    Edge-Conditioned Convolution (ECC) is a spatial graph convolution operation that dynamically generates localized filter weights conditioned on continuous or discrete edge labels.

    Let G=(V,E)G = (V, E) be a directed or undirected graph where VV is a set of nn vertices and E⊆V×VE \subseteq V \times V is a set of mm edges. Let l∈{1,…,lmax⁡}l \in \{1, \dots, l_{\max}\} denote the layer index of a feed-forward neural network. Each vertex i∈Vi \in V has a feature vector Xl(i)∈RdlX^l(i) \in \mathbb{R}^{d_l} (with X0(i)X^0(i) being the initial input features), and each directed edge (j,i)∈E(j, i) \in E is associated with an edge label or attribute vector L(j,i)∈RsL(j, i) \in \mathbb{R}^s. The neighborhood N(i)={j∈V∣(j,i)∈E}∪{i}N(i) = \{j \in V \mid (j, i) \in E\} \cup \{i\} contains all predecessor vertices adjacent to ii, including a self-loop (i,i)(i, i).

    The filtered vertex feature Xl(i)X^l(i) at layer ll is defined as: Xl(i)=1∣N(i)∣∑j∈N(i)Fl(L(j,i);wl)Xl−1(j)+bl=1∣N(i)∣∑j∈N(i)ΘjilXl−1(j)+blX^l(i) = \frac{1}{|N(i)|} \sum_{j \in N(i)} F^l(L(j, i); w^l) X^{l-1}(j) + b^l = \frac{1}{|N(i)|} \sum_{j \in N(i)} \Theta_{ji}^l X^{l-1}(j) + b^l where:

    • Fl:Rs→Rdl×dl−1F^l : \mathbb{R}^s \to \mathbb{R}^{d_l \times d_{l-1}} is a parameterized filter-generating network (such as a multi-layer perceptron) with learnable model weights wlw^l.
    • Θjil=Fl(L(j,i);wl)∈Rdl×dl−1\Theta_{ji}^l = F^l(L(j, i); w^l) \in \mathbb{R}^{d_l \times d_{l-1}} is the dynamic weight matrix computed on the fly for edge (j,i)(j, i).
    • bl∈Rdlb^l \in \mathbb{R}^{d_l} is a learnable bias vector.
    • ∣N(i)∣|N(i)| is the degree of vertex ii, providing neighborhood size normalization.

    Computing XlX^l across all vertices in GG requires at most mm evaluations of FlF^l (or ss evaluations if edge labels are discrete with s<ms < m) and m+nm + n (for directed graphs) or 2m+n2m + n (for undirected graphs) matrix-vector multiplications.

  2. Knowl 2 — Equivalence of Edge-Conditioned Convolution to Standard Grid Convolution

    theoretical result

    Standard discrete convolution on regular grids is a special case of Edge-Conditioned Convolution (ECC).

    Consider a 1D grid represented as an ordered set of vertices VV forming a path graph. For a discrete convolution kernel with spatial support size ss centered at each vertex, an edge set EE is formed by connecting each vertex ii to its ss nearest spatial neighbors jj (including itself) via a directed edge (j,i)(j, i). Each edge is assigned an edge attribute vector L(j,i)∈{0,1}sL(j, i) \in \{0, 1\}^s consisting of the one-hot encoding of the discrete spatial offset δ=j−i\delta = j - i.

    Setting the filter-generating network Fl:Rs→Rdl×dl−1F^l : \mathbb{R}^s \to \mathbb{R}^{d_l \times d_{l-1}} to be a single-layer perceptron without bias, parameterized by weight matrix wl∈R(dl×dl−1)×sw^l \in \mathbb{R}^{(d_l \times d_{l-1}) \times s}, evaluating Fl(L(j,i);wl)F^l(L(j, i); w^l) yields wl(δ)w^l(\delta), where wl(δ)w^l(\delta) is the reshaped column of wlw^l corresponding to offset δ\delta.

    Ignoring the boundary normalization factor 1/∣N(i)∣1/|N(i)|, the ECC operation reduces to: Xl(i)=∑j∈N(i)ΘjilXl−1(j)=∑δwl(δ)Xl−1(i−δ)X^l(i) = \sum_{j \in N(i)} \Theta_{ji}^l X^{l-1}(j) = \sum_{\delta} w^l(\delta) X^{l-1}(i - \delta) This matches standard discrete convolution on regular grids with identical parameter count and computational complexity.

  3. Knowl 3 — Point Cloud Graph Representation and Hierarchical Coarsening

    model/method

    Point clouds are modeled directly as graphs embedded in Euclidean space with multiscale hierarchical coarsening pyramids to enable deep Edge-Conditioned Convolutions (ECC) without voxelization.

    Graph Construction: Given a 3D point cloud P={pi}i=1nP = \{p_i\}_{i=1}^n with point coordinates pi∈R3p_i \in \mathbb{R}^3 and optional per-point features XP(pi)X_P(p_i) (such as laser return intensity or RGB color):

    1. A vertex i∈Vi \in V is created for each point pi∈Pp_i \in P with initial signal X0(i)=XP(pi)X^0(i) = X_P(p_i) (or 0 if unfeatured).
    2. Directed edges (j,i)∈E(j, i) \in E connect vertex ii to all points pjp_j within a fixed metric Euclidean radius ρ\rho: ∥pj−pi∥≤ρ\|p_j - p_i\| \le \rho.
    3. The spatial offset vector δ=pj−pi\delta = p_j - p_i is mapped to a 6D continuous edge attribute vector combining Cartesian and spherical coordinates: L(j,i)=(δx,δy,δz,∥δ∥,arccos⁡δz∥δ∥,arctan⁡δyδx)L(j, i) = \left(\delta_x, \delta_y, \delta_z, \|\delta\|, \arccos\frac{\delta_z}{\|\delta\|}, \arctan\frac{\delta_y}{\delta_x}\right)

    Hierarchical Coarsening: A pyramid of downsampled point clouds {P(0),P(1),…,P(hmax⁡)}\{P^{(0)}, P^{(1)}, \dots, P^{(h_{\max})}\} is computed using the VoxelGrid algorithm with voxel resolution parameters r(0)<r(1)<⋯<r(hmax⁡)r^{(0)} < r^{(1)} < \dots < r^{(h_{\max})}. Points falling within each voxel are replaced by their geometric centroid to maintain sub-voxel accuracy. At each level hh, a graph G(h)G^{(h)} is constructed with radius ρ(h)\rho^{(h)}. The pooling assignment M(h):V(h−1)→V(h)M^{(h)} : V^{(h-1)} \to V^{(h)} maps each point in P(h−1)P^{(h-1)} to its nearest spatial centroid in P(h)P^{(h)}, over which max-pooling aggregates vertex features.

  4. Knowl 4 — Graph Coarsening and Multiresolution Hierarchy for General Graphs

    model/method

    For general graph classification datasets lacking spatial Euclidean coordinates, multiscale coarsening pyramids are generated using spectral reduction.

    The multiresolution pyramid {G(0),G(1),…,G(hmax⁡)}\{G^{(0)}, G^{(1)}, \dots, G^{(h_{\max})}\} is constructed through iterative downsampling and reduction steps:

    1. Spectral Downsampling: The vertex set V(h−1)V^{(h-1)} is partitioned into two components according to the sign of the eigenvector corresponding to the largest eigenvalue of the graph Laplacian, dropping approximately half of the vertices.
    2. Kron Reduction: Kron reduction computes the coarsened edge connectivity E(h)E^{(h)} and assigns continuous scalar edge labels L(h)L^{(h)} between the remaining vertices.
    3. Randomized Spectral Sparsification: Effective resistance-based spectral sparsification thins out dense edge sets. Because sparsification is stochastic, generating multiple different coarsened pyramids per graph at training time acts as structural data augmentation.

    In the deep architecture, vertex features are pooled across reduction maps M(h):V(h−1)→V(h)M^{(h)} : V^{(h-1)} \to V^{(h)} using max-pooling, followed by global average or max pooling at the coarsest graph resolution hmax⁡h_{\max} to obtain a graph-level embedding.

  5. Knowl 5 — Point Cloud Classification Performance on Sydney Urban Objects

    empirical result

    Edge-Conditioned Convolution (ECC) was evaluated on the Sydney Urban Objects dataset, consisting of 588 outdoor LiDAR scans across 14 imbalanced categories. Performance is evaluated using mean F1 score weighted by class frequency over four standard splits.

    The evaluated network architecture is C(16)-C(32)-MP(0.25,0.5)-C(32)-C(32)-MP(0.75,1.5)-C(64)-MP(1.5,1.5)-GAP-FC(64)-D(0.2)-FC(14)C(16)\text{-}C(32)\text{-}MP(0.25, 0.5)\text{-}C(32)\text{-}C(32)\text{-}MP(0.75, 1.5)\text{-}C(64)\text{-}MP(1.5, 1.5)\text{-}GAP\text{-}FC(64)\text{-}D(0.2)\text{-}FC(14), where C(c)C(c) denotes an ECC layer with cc channels, batch normalization, and ReLU; MP(r,ρ)MP(r, \rho) denotes max-pooling down to grid resolution rr meters with neighborhood radius ρ\rho meters; GAPGAP is global average pooling; FC(c)FC(c) is a fully connected layer with cc channels; and D(p)D(p) denotes dropout with rate pp.

    Model Mean F1
    Triangle+SVM 67.1
    GFH+SVM 71.0
    VoxNet 73.0
    ORION 77.8
    ECC (2ρ2\rho) 74.4
    ECC (1.5ρ1.5\rho) 76.9
    ECC (ρ\rho) 78.4
    ECC (1.5ρ1.5\rho with residual identity connection) 79.5

    ECC operating directly on irregular point graphs achieves a state-of-the-art weighted F1 score of 78.4% (and 79.5% with identity skip connections), outperforming dense volumetric 3D CNNs (VoxNet at 73.0% and ORION at 77.8%). Increasing neighborhood search radius ρ\rho without identity connections degrades performance by over-smoothing central vertex representations.

  6. Knowl 6 — 3D Object Classification Performance on ModelNet

    empirical result

    Edge-Conditioned Convolution (ECC) was evaluated on ModelNet10 and ModelNet40 benchmarks using synthetic point clouds of 1,000 points sampled uniformly from mesh faces.

    Model ModelNet10 ModelNet40
    Class Acc. (Inst. Acc.) Class Acc. (Inst. Acc.)
    3DShapeNets 83.5% (—) 77.3% (—)
    MVCNN — 90.1% (—)
    VoxNet 92.0% (—) 83.0% (—)
    ORION 93.8% (—) —
    SubvolumeSup — 86.0% (89.2%)
    ECC 89.3% (90.0%) 82.4% (87.0%)
    ECC (12-orientation voting) 90.0% (90.8%) 83.2% (87.4%)

    ECC operates directly on raw sparse point graphs without voxel grid rasters, obtaining competitive accuracies of 90.8% instance accuracy on ModelNet10 and 87.4% on ModelNet40 when voting over 12 orientations.

  7. Knowl 7 — Graph Classification Benchmark Performance and Impact of Edge Labels

    empirical result

    Edge-Conditioned Convolution (ECC) was evaluated on five standard graph classification benchmarks: NCI1, NCI109, MUTAG (edge-labeled chemical compounds), ENZYMES, and D&D (unlabeled protein graphs). Results report 10-fold cross-validation mean classification accuracy.

    Model NCI1 NCI109 MUTAG ENZYMES DD
    DCNN 62.61% 62.86% 66.98% 18.10% —
    PSCN 78.59% — 92.63% — 77.12%
    Deep WL 80.31% 80.32% 87.44% 53.43% —
    structure2vec 83.72% 82.16% 88.28% 61.10% 82.22%
    WL graph kernel 84.55% 84.49% 83.78% 59.05% 79.78%
    ECC (no edge labels) 76.82% 75.03% 76.11% 45.67% 72.54%
    ECC 83.80% 81.87% 89.44% 50.00% 73.65%
    ECC (5-run voting) 83.63% 82.04% 88.33% 53.50% 73.68%
    ECC (5-run score avg.) 83.80% 82.14% 88.33% 52.67% 74.10%

    On edge-labeled graphs (NCI1, NCI109, MUTAG), ECC matches the Weisfeiler-Lehman (WL) graph kernel and outperforms prior deep learning methods. Edge conditioning is crucial: removing edge labels (ECC no edge labels) degrades accuracy by 6.98% on NCI1, 6.84% on NCI109, and 13.33% on MUTAG.

  8. Knowl 8 — Validation of ECC on Dense, Sparse, and One-Hot Encoded MNIST

    empirical result

    The properties of Edge-Conditioned Convolution (ECC) were validated on the MNIST handwritten digit classification dataset (28×2828 \times 28 pixels) formatted as 2D point cloud graphs:

    Model Train accuracy Test accuracy
    ECC 99.12% 99.14%
    ECC (sparse input) 99.36% 99.14%
    ECC (one-hot) 99.53% 99.37%

    Key empirical findings:

    1. Grid Matching: Full-grid ECC achieves 99.14% test accuracy, matching spectral graph CNN benchmarks.
    2. Invariance to Irregular Sparsity: In the sparse input configuration, all background pixels (X0(i)=0X^0(i) = 0, accounting for 80.9% of all points) are removed, leaving an irregular graph with sample-varying topology. Test accuracy remains identical at 99.14%, confirming stability under varying graph structures.
    3. One-Hot Equivalence: Emulating exact grid convolution via single-layer filter generators and one-hot offset encoding reaches 99.37% test accuracy.
    4. Continuous Filter Visualization: Sampling the dynamically generated filter weight matrix Θ1=F1(L(j,i);w1)\Theta^1 = F^1(L(j, i); w^1) across spatial coordinate offsets (δx,δy)(\delta_x, \delta_y) reveals smooth, structured spatial filter patterns analogous to standard CNN first-layer filters.

  9. Knowl 9 — Residual and Degree-Conditioned Extensions of Edge-Conditioned Convolution

    model/method

    Several architectural variants extend the baseline Edge-Conditioned Convolution (ECC) formulation:

    1. ECC-ResNet (Identity Skip Connections): To facilitate residual learning and alleviate over-smoothing: Xl(i)=1∣N(i)∣∑j∈N(i)ΘjilXl−1(j)+bl+id(Xl−1(i))X^l(i) = \frac{1}{|N(i)|} \sum_{j \in N(i)} \Theta_{ji}^l X^{l-1}(j) + b^l + \text{id}(X^{l-1}(i)) where id(⋅)\text{id}(\cdot) is the identity mapping if dl=dl−1d_l = d_{l-1} and a linear projection otherwise.

    2. Vertex Degree Augmented Edge Labels: Edge labels are augmented with vertex degree terms: Ldeg(j)L_{\text{deg}}(j) and Ldeg(i)L_{\text{deg}}(i) for edge (j,i)(j, i), tested with variants 1/deg(i)1/\sqrt{\text{deg}(i)}, 1/deg(i)1/\text{deg}(i), deg(i)\sqrt{\text{deg}(i)}, and deg(i)\text{deg}(i), where deg(i)=∣N(i)∣\text{deg}(i) = |N(i)|. Supplying deg(i)\sqrt{\text{deg}(i)} improves accuracy on datasets without native edge labels, raising ENZYMES from 50.00% to 55.00% and D&D from 73.65% to 75.79%.

    3. ECC-Z (Learned Degree Normalization): A factor-generating MLP Zl:R→RZ^l : \mathbb{R} \to \mathbb{R} dynamically computes degree-dependent normalization factors: Xl(i)=Zl(∣N(i)∣;wl)∣N(i)∣∑j∈N(i)ΘjilXl−1(j)+blX^l(i) = \frac{Z^l(|N(i)|; w^l)}{|N(i)|} \sum_{j \in N(i)} \Theta_{ji}^l X^{l-1}(j) + b^l

Coverage note — None; all core theoretical results, point cloud and general graph processing pipelines, experimental benchmark results, and architectural extensions from the paper and its appendix are covered.

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Citation

MLA
Simonovsky, M., and N. Komodakis. “Dynamic Edge-Conditioned Filters in Convolutional Neural Networks on Graphs”. arXiv, 2017, http://arxiv.org/abs/1704.02901v3.
APA
Simonovsky, M., & Komodakis, N. (2017). Dynamic Edge-Conditioned Filters in Convolutional Neural Networks on Graphs. arXiv. http://arxiv.org/abs/1704.02901v3
Chicago
Simonovsky, M., and N. Komodakis. 2017. “Dynamic Edge-Conditioned Filters in Convolutional Neural Networks on Graphs”. arXiv. http://arxiv.org/abs/1704.02901v3.
Harvard
Simonovsky, M. and Komodakis, N. (2017) “Dynamic Edge-Conditioned Filters in Convolutional Neural Networks on Graphs”, arXiv [Preprint]. Available at: http://arxiv.org/abs/1704.02901v3.
Vancouver
1. Simonovsky M, Komodakis N (2017) Dynamic Edge-Conditioned Filters in Convolutional Neural Networks on Graphs. arXiv

BibTeX

@article{simonovsky2017dynamic,
  title = {Dynamic Edge-Conditioned Filters in Convolutional Neural Networks on Graphs},
  author = {Simonovsky, Martin and Komodakis, Nikos},
  year = {2017},
  journal = {arXiv},
  url = {http://arxiv.org/abs/1704.02901v3},
  eprint = {1704.02901}
}
Metadata:arXiv

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License: IEEE