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Distance Metric Learning using Graph Convolutional Networks: Application to Functional Brain Networks
Sofia Ira Ktena, Sarah Parisot, Enzo Ferrante, Martin Rajchl, Matthew Lee, Ben Glocker, Daniel Rueckert
TL;DR
Comparing functional brain graphs requires an application-sensitive similarity metric, but irregular graph structure complicates standard convolutional approaches. The paper learns a graph metric with a siamese spectral GCN and tailored global loss, obtaining improved matching-versus-non-matching separation on ABIDE data, especially within acquisition sites.
Problem
Functional brain-network comparisons require accurate graph similarity metrics, while existing approaches have limitations in capturing structure, retaining information, computational cost, or setting edit costs.
Method
The method uses a siamese graph convolutional neural network with spectral polynomial filters and a global loss that separates matching from non-matching graph embeddings.
Results
The learned metric improves separation between matching and non-matching ABIDE graphs, with AUC increases of up to 0.44 for same-site pairs.
Takeaways & Limitations
The proposed metric shows promising performance for comparing functional brain connectivity graphs and is described as flexible enough for other graph-comparison problems.
Takeaways & Limitations
The relatively simple architecture could be improved with more sophisticated networks, while site-independent representations and comparisons between differently structured graphs remain extension directions.
Abstract
from arXiv · showhide
Evaluating similarity between graphs is of major importance in several computer vision and pattern recognition problems, where graph representations are often used to model objects or interactions between elements. The choice of a distance or similarity metric is, however, not trivial and can be highly dependent on the application at hand. In this work, we propose a novel metric learning method to evaluate distance between graphs that leverages the power of convolutional neural networks, while exploiting concepts from spectral graph theory to allow these operations on irregular graphs. We demonstrate the potential of our method in the field of connectomics, where neuronal pathways or functional connections between brain regions are commonly modelled as graphs. In this problem, the definition of an appropriate graph similarity function is critical to unveil patterns of disruptions associated with certain brain disorders. Experimental results on the ABIDE dataset show that our method can learn a graph similarity metric tailored for a clinical application, improving the performance of a simple k-nn classifier by 11.9% compared to a traditional distance metric.
1 Introduction
Accurate graph similarity metrics are needed to compare functional brain networks and support analyses of neurodevelopmental disorders. The paper proposes learning such a metric for irregular graphs using graph convolutions and applies it to ABIDE functional connectivity data.
- Accurate similarity or distance metrics are required to compare brain networks for statistical and machine learning analyses of neurodevelopmental disorders.
- Existing graph-comparison methods include graph kernels, graph embedding, motif counting, and graph edit distance, each with limitations involving global structure, information loss, computation, or predefined edit costs.
- Graph convolutions offer a route to applying convolutional neural networks to irregular brain graphs through filtering in the graph spectral domain.
- The proposed method learns a similarity metric between irregular graphs with known node correspondences using a siamese graph convolutional neural network and a global loss function.
- The method is demonstrated on functional connectivity graphs from 871 ABIDE subjects collected across international sites with different protocols.
2 Methodology
The method applies spectral graph convolutions within a siamese network to learn similarity between brain graphs with known node correspondences. It combines localized Chebyshev filters, shared convolutional branches, global similarity loss, and graph structures based on anatomical k-NN relationships.
- 2.1 Spectral Graph Filtering and Convolutions: Spectral graph theory defines graph convolutions by filtering signals in the graph spectral domain using the graph Laplacian.The normalized Laplacian is decomposed into eigenvectors and eigenvalues, which represent graph-frequency components.
- 2.1 Spectral Graph Filtering and Convolutions: Chebyshev polynomial approximations make the filters K-localised in space while reducing computational complexity.The filtering operation uses a truncated expansion whose polynomials are recursively defined.
- 2.1 Spectral Graph Filtering and Convolutions: Each GCN output feature map is formed from trainable Chebyshev coefficients applied to input feature maps.The network contains Fin × Fout vectors of coefficients θi,j ∈ R^K.
- 2.2 Loss Function and Network Architecture: The siamese architecture processes two input graphs through identical weight-sharing convolutional branches, then predicts similarity with an inner product and sigmoid fully connected output layer.The fully connected layer integrates global information from preceding localized graph features.
- 2.2 Loss Function and Network Architecture: The global loss increases mean similarity for matching graphs, decreases it for non-matching graphs, and reduces variance within both pair types.The margin m separates matching and non-matching similarity means, λ balances mean and variance terms, and L2 regularization is added to fully connected weights.
- 2.3 Dataset and Functional Brain Graphs: The graph structure is shared across subjects and defined anatomically as a k-NN graph, while node signals encode each ROI’s Pearson correlation connectivity profile.This shared structure is required because spectral graph convolution parameters depend on the graph Laplacian.
3 Results
On the ABIDE test set, the learned metric improved separation between matching and non-matching brain graphs and improved k-nn classification over PCA-based Euclidean distance. Performance gains were strongest for pairs from the same acquisition site, while cross-site heterogeneity limited overall performance.
- Metric separation: The learned metric significantly improved separation between matching and non-matching pairs for the total test set and most individual sites.The comparison used distances learned by the proposed GCN model versus Euclidean distances after PCA.
- Subject classification: 11.9% improvement was achieved in k-nn classification on the total test set, with gains up to 40% for individual sites.The classifier used k = 3 and the estimated graph distances.
- Matching classification: Higher AUC values were obtained for all five biggest sites, with increases up to 0.44 for site 18.ROC curves evaluated matching versus non-matching graph classification using the proposed metric and Euclidean distance.
- Site effects: The overall improvement across all sites was limited relative to same-site comparisons, which the authors attribute to heterogeneity across acquisition sites.The test-set evaluation included pairs from different sites in the “all sites” condition.
4 Discussion
The proposed siamese GCN metric produced promising results for functional brain connectivity graphs, especially for same-site comparisons, while remaining broadly applicable to graph-comparison problems. The authors identify architectural simplicity as an opportunity for further improvement and propose more sophisticated, site-independent extensions.
- Discussion: The method produced promising results across sites, with significant performance increases for pairs acquired at the same site.The approach was applied to functional brain connectivity graphs from the ABIDE database.
- Discussion: The method is described as flexible and general enough for graph-comparison problems beyond brain networks, including shape analysis.This broader applicability is presented as a supported scope of the proposed metric-learning approach.
- Future extensions: The relatively simple network architecture leaves room to improve performance through more sophisticated network designs.The authors specifically suggest autoencoders and adversarial training for lower-dimensional, site-independent connectivity representations.
- Future extensions: Generalisable spatial-domain GCNs could enable training similarity metrics for graphs with different structures.This is proposed as an extension beyond the current setting of irregular graphs with known node correspondences.