GLEE: Geometric Laplacian Eigenmap Embeddingleotrs.com/static/glee_poster.pdf1. GLEE replaces...
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Robustness to rewiring GLEE: Geometric Laplacian Eigenmap Embedding Leo Torres [email protected] Kevin Chan [email protected] Tina Eliassi-Rad [email protected] Work supported by NSF CNS-1314603, NSF IIS-1741197, Army Research Laboratory Cooperative Agreement W911NF-13-2-0045. Abstract ● Graph embedding builds a low-dimensional representation of a graph. ● Popular in the literature is the distance-minimization assumption: if two nodes are close (in the graph), their embeddings must be close (in embedding space). ● We dispose of the distance-minimization assumption. Instead, our new method Geometric Laplacian Eigenmap Embedding (GLEE) builds an embedding with geometric properties by leveraging the so-called simplex geometry of graphs. ● Benefits of GLEE: ○ Deterministic and interpretable. ○ Great performance, especially in the case of low clustering. ○ Robust to noise: it can recover graph structure in the presence of a high percentage of noisy edges. Conclusions and Future Work 1. GLEE replaces distance-minimization with the direct encoding of graph structure in the geometry of the embedding space. 2. GLEE performs best when the graph has low clustering coefficient, and performance increases as the embedding dimension increases. 3. What other geometric properties of embeddings can we utilize? Simplex Geometry and Embedding Experiments Graph reconstruction: a classification problem with extreme class imbalance Link Prediction: interpreting the geometry of GLEE 33-dimensional simplex ... ... ... Nodes 3 34 Example graph G Simplex of G GLEE of G triangle ... 4 tetrahedron 0 1 2 3 4 5 GLEE of node i i Problem: find optimal value of . GLEE (number of common neighbors, CN): GLEE-L3 (number of paths of length 3, L3): eigenvalues eigenvectors ...plus subsampling to deal with class imbalance. Three solutions: 1. Constant 2. Gaussian Mixtures 3. Density Estimation Dimension d Average clustering coefficient 32 128 512 0.04 0.56 0.14 Devriendt, Van Mieghem, The simplex geometry of graphs, Journal of Complex Networks, https://doi.org/10.1093/comnet/cny036 is the center of mass of neighbors of is the center of mass of neighbors of Graph reconstruction Link prediction
Transcript of GLEE: Geometric Laplacian Eigenmap Embeddingleotrs.com/static/glee_poster.pdf1. GLEE replaces...
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GLEE:Geometric Laplacian Eigenmap Embedding
Kevin [email protected]
Tina [email protected]
Work supported by NSF CNS-1314603, NSF IIS-1741197, Army Research Laboratory Cooperative Agreement W911NF-13-2-0045.
Abstract● Graph embedding builds a low-dimensional representation of a graph.
● Popular in the literature is the distance-minimization assumption: if two nodes are close (in the graph), their embeddings must be close (in embedding space).
● We dispose of the distance-minimization assumption. Instead, our new method Geometric Laplacian Eigenmap Embedding (GLEE) builds an embedding with geometric properties by leveraging the so-called simplex geometry of graphs.
● Benefits of GLEE:
○ Deterministic and interpretable.○ Great performance, especially in the case of low clustering.○ Robust to noise: it can recover graph structure in the presence of a
high percentage of noisy edges.
Conclusions and Future Work1. GLEE replaces distance-minimization with the direct encoding of
graph structure in the geometry of the embedding space.
2. GLEE performs best when the graph has low clustering coefficient, and performance increases as the embedding dimension increases.
3. What other geometric properties of embeddings can we utilize?
Simplex Geometry and Embedding
ExperimentsGraph reconstruction: a classification problem with extreme class imbalance
Link Prediction: interpreting the geometry of GLEE
33-dimensional simplex...
...
...
Nodes 3 34
Example graph G
Simplex of G
GLEEof G
triangle
...
4
tetrahedron
01
2
3
4
5
GLEE of node i
i
Problem: find optimal value of .
GLEE (number of common neighbors, CN):
GLEE-L3 (number of paths of length 3, L3):
eigenvalueseigenvectors
...plus subsampling to deal with class
imbalance.
Three solutions:
1. Constant
2. Gaussian Mixtures
3. Density Estimation
Dim
ensi
on d
Average clustering coefficient
32
128
512
0.04 0.560.14
Devriendt, Van Mieghem, The simplex geometry of graphs, Journal of Complex Networks, https://doi.org/10.1093/comnet/cny036
is the center of mass of neighbors of
is the center of mass of neighbors of
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