Geometric deep learning for business classification · gheynst. "Convolutional neural networks on...
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Geometric deep learning for business classification Henry Kenlay, Xiaowen Dong, Stephen Roberts Introduction Graph signal processing • Processing of signals defined on vertex set of weighted and undirected graphs Geometric deep learning • Deep learning on data residing on graphs • Spectral approaches take inspiration from graph signal processing Business classification • Given attributes of a business and a graph topology relating businesses in Edinburgh, can we predict if the restaurant has closed down? • We use a graph convolutional neural network for classification hopefully exploiting pairwise relationships between businesses Graph signal processing and convolution on graphs Classification problem Graph CNN framework Filtering in the graph spectral domain • Eigenvalue decomposition is • Change of basis is also • Can approximate filtering using Chebyshev polynomials [1] • Further simplification by considering polynomials of order one [2] Results Results • Benchmarks were competitive • Review data gave slight advantage when using full set of features • Random graphs were detrimental to performance Feature selection • Using elastic net selected features improved performance for many approaches • The benefit is less significant in graph convolutional neural networks Thresholding edges • Thresholding the Jaccard graph improved performance despite disconnecting the graph and isolating 10.6% of the nodes Discussion Graph constructions • Social data • Statistical methods Longitudinal prediction • Can we predict business closure in the future? Other frameworks • Explore alternative graph convolutional neural network frameworks Theoretical approach to the importance of graph topology • Theoretical bound on the output given a small perturbation of the input graph topology O(n 3 ) O(n 3 ) Propagation rule: H (l+1) = ReLU ⇣ ˜ D - 1 2 ˜ A ˜ D - 1 2 H (l) W (l) ⌘ Degree matrix Adjacency matrix Parameter matrix Layer properties - CNN correspondence • Signal value on node i in layer K+1 is a function of the value at i as well as that at i’s 1-hop neighbours in layer K - localised filters • Weight sharing across nodes - filters independent of input size • Sparse multiplications give low computational cost Problem set up • Yelp dataset - multimodal data about businesses • Each restaurant is represented by a node in the graph with edges capturing relationship between businesses • Each restaurant has an associated feature vector • Label is if the restaurant is still operating (binary classification) • Elastic net model used to select subset of features Importance of graph topology • Edges provide a relational inductive bias • Using no edges is equivalent to a multilayer perceptron • We experiment with different ways to generate edge weights: • Randomly • Kernel-based methods (e.g., Gaussian RBF kernel) • Restaurant distance • Reviewer data - Jaccard index between two sets of reviewers (one for each restaurant) [1] Defferrard, Michaël, Xavier Bresson, and Pierre Vandergheynst. "Convolutional neural networks on graphs with fast localized spectral filtering." Advances in Neural Information Processing Systems. 2016. [2] Kipf, Thomas N., and Max Welling. "Semi-supervised classification with graph convolutional networks." arXiv preprint arXiv:1609.02907 (2016). Restaurant at node 9 is less expensive than average Restaurants at nodes 6 and 7 are related Restaurant at node 2 is more expensive than average Restaurants are nodes in the graph The scalar signal in this example is normalised price Contact: [email protected] -r 2 L = D - W Graph FT: Classical FT: Eigenfunction basis: Laplace operator: Graph Laplacian: Eigenvector basis: Classical convolution: Graph convolution: e i!x χ ` ˆ f (`) = hχ ` ,f i = N X i=1 χ ` (i) ⇤ f (i) f (i)= N X `=1 ˆ f (`)χ ` (i) ˆ f (!) = he i!x ,f i = Z (e i!x ) ⇤ f (x)dx f (x)= 1 2⇡ Z ˆ f (!)e i!x d! (f ⇤ h)(i)= N X `=1 ˆ f (`) ˆ h(`)χ ` (i) (f ⇤ h)(x)= Z f (⌧ )h(x - ⌧ )d⌧ = 1 2⇡ Z ˆ f (!) ˆ h(!)e i!x d!
Transcript of Geometric deep learning for business classification · gheynst. "Convolutional neural networks on...
Geometric deep learning for business classification Henry Kenlay, Xiaowen Dong, Stephen Roberts
IntroductionGraph signal processing • Processing of signals defined on vertex set of weighted and undirected graphsGeometric deep learning • Deep learning on data residing on graphs• Spectral approaches take inspiration from graph signal processingBusiness classification • Given attributes of a business and a graph topology relating businesses in
Edinburgh, can we predict if the restaurant has closed down?• We use a graph convolutional neural network for classification hopefully exploiting
pairwise relationships between businesses
Graph signal processing and convolution on graphs Classification problem
Graph CNN frameworkFiltering in the graph spectral domain • Eigenvalue decomposition is • Change of basis is also • Can approximate filtering using Chebyshev polynomials [1] • Further simplification by considering polynomials of order one [2]
ResultsResults • Benchmarks were competitive• Review data gave slight advantage when using full set of features• Random graphs were detrimental to performanceFeature selection • Using elastic net selected features improved performance for many
approaches• The benefit is less significant in graph convolutional neural networksThresholding edges • Thresholding the Jaccard graph improved performance despite
disconnecting the graph and isolating 10.6% of the nodes
DiscussionGraph constructions • Social data• Statistical methodsLongitudinal prediction • Can we predict business closure in the future?Other frameworks • Explore alternative graph convolutional neural network frameworksTheoretical approach to the importance of graph topology • Theoretical bound on the output given a small perturbation of the input
graph topology
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Propagation rule:
H(l+1) = ReLU
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Degree matrix Adjacency matrix Parameter matrix
Layer properties - CNN correspondence • Signal value on node i in layer K+1 is a function of the value at i
as well as that at i’s 1-hop neighbours in layer K - localised filters• Weight sharing across nodes - filters independent of input size• Sparse multiplications give low computational cost
Problem set up • Yelp dataset - multimodal data about businesses• Each restaurant is represented by a node in the graph with edges
capturing relationship between businesses• Each restaurant has an associated feature vector • Label is if the restaurant is still operating (binary classification)• Elastic net model used to select subset of features
Importance of graph topology • Edges provide a relational inductive bias• Using no edges is equivalent to a multilayer perceptron• We experiment with different ways to generate edge weights:• Randomly • Kernel-based methods (e.g., Gaussian RBF kernel)• Restaurant distance • Reviewer data - Jaccard index between two sets of reviewers
(one for each restaurant)[1] Defferrard, Michaël, Xavier Bresson, and Pierre Vandergheynst. "Convolutional neural networks on graphs with fast localized spectral filtering." Advances in Neural Information Processing Systems. 2016.[2] Kipf, Thomas N., and Max Welling. "Semi-supervised classification with graph convolutional networks." arXiv preprint arXiv:1609.02907 (2016).
Restaurant at node 9 is less expensive than average
Restaurants at nodes 6 and 7 are related
Restaurant at node 2 is more expensive than average
Restaurants are nodes in the graph
The scalar signal in this example is normalised price
Contact: [email protected]
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L = D �W<latexit sha1_base64="rj6lNmhYs1QdEDLcjyg3Vxp+iwk=">AAAB7HicbZA9SwNBEIbn/IzxK2ppsxgEG8OdCNoIQS0sLCJ4SSA5wt5mL1myt3fszgkh5DfYWChi6w+y89+4Sa7QxBcWHt6ZYWfeMJXCoOt+O0vLK6tr64WN4ubW9s5uaW+/bpJMM+6zRCa6GVLDpVDcR4GSN1PNaRxK3ggHN5N644lrIxL1iMOUBzHtKREJRtFa/v3V7WmjUyq7FXcqsgheDmXIVeuUvtrdhGUxV8gkNabluSkGI6pRMMnHxXZmeErZgPZ4y6KiMTfBaLrsmBxbp0uiRNunkEzd3xMjGhszjEPbGVPsm/naxPyv1sowugxGQqUZcsVmH0WZJJiQyeWkKzRnKIcWKNPC7kpYn2rK0OZTtCF48ycvQv2s4ll+OC9Xr/M4CnAIR3ACHlxAFe6gBj4wEPAMr/DmKOfFeXc+Zq1LTj5zAH/kfP4AxXWN/Q==</latexit><latexit sha1_base64="rj6lNmhYs1QdEDLcjyg3Vxp+iwk=">AAAB7HicbZA9SwNBEIbn/IzxK2ppsxgEG8OdCNoIQS0sLCJ4SSA5wt5mL1myt3fszgkh5DfYWChi6w+y89+4Sa7QxBcWHt6ZYWfeMJXCoOt+O0vLK6tr64WN4ubW9s5uaW+/bpJMM+6zRCa6GVLDpVDcR4GSN1PNaRxK3ggHN5N644lrIxL1iMOUBzHtKREJRtFa/v3V7WmjUyq7FXcqsgheDmXIVeuUvtrdhGUxV8gkNabluSkGI6pRMMnHxXZmeErZgPZ4y6KiMTfBaLrsmBxbp0uiRNunkEzd3xMjGhszjEPbGVPsm/naxPyv1sowugxGQqUZcsVmH0WZJJiQyeWkKzRnKIcWKNPC7kpYn2rK0OZTtCF48ycvQv2s4ll+OC9Xr/M4CnAIR3ACHlxAFe6gBj4wEPAMr/DmKOfFeXc+Zq1LTj5zAH/kfP4AxXWN/Q==</latexit><latexit sha1_base64="rj6lNmhYs1QdEDLcjyg3Vxp+iwk=">AAAB7HicbZA9SwNBEIbn/IzxK2ppsxgEG8OdCNoIQS0sLCJ4SSA5wt5mL1myt3fszgkh5DfYWChi6w+y89+4Sa7QxBcWHt6ZYWfeMJXCoOt+O0vLK6tr64WN4ubW9s5uaW+/bpJMM+6zRCa6GVLDpVDcR4GSN1PNaRxK3ggHN5N644lrIxL1iMOUBzHtKREJRtFa/v3V7WmjUyq7FXcqsgheDmXIVeuUvtrdhGUxV8gkNabluSkGI6pRMMnHxXZmeErZgPZ4y6KiMTfBaLrsmBxbp0uiRNunkEzd3xMjGhszjEPbGVPsm/naxPyv1sowugxGQqUZcsVmH0WZJJiQyeWkKzRnKIcWKNPC7kpYn2rK0OZTtCF48ycvQv2s4ll+OC9Xr/M4CnAIR3ACHlxAFe6gBj4wEPAMr/DmKOfFeXc+Zq1LTj5zAH/kfP4AxXWN/Q==</latexit><latexit sha1_base64="rj6lNmhYs1QdEDLcjyg3Vxp+iwk=">AAAB7HicbZA9SwNBEIbn/IzxK2ppsxgEG8OdCNoIQS0sLCJ4SSA5wt5mL1myt3fszgkh5DfYWChi6w+y89+4Sa7QxBcWHt6ZYWfeMJXCoOt+O0vLK6tr64WN4ubW9s5uaW+/bpJMM+6zRCa6GVLDpVDcR4GSN1PNaRxK3ggHN5N644lrIxL1iMOUBzHtKREJRtFa/v3V7WmjUyq7FXcqsgheDmXIVeuUvtrdhGUxV8gkNabluSkGI6pRMMnHxXZmeErZgPZ4y6KiMTfBaLrsmBxbp0uiRNunkEzd3xMjGhszjEPbGVPsm/naxPyv1sowugxGQqUZcsVmH0WZJJiQyeWkKzRnKIcWKNPC7kpYn2rK0OZTtCF48ycvQv2s4ll+OC9Xr/M4CnAIR3ACHlxAFe6gBj4wEPAMr/DmKOfFeXc+Zq1LTj5zAH/kfP4AxXWN/Q==</latexit>
Graph FT:Classical FT:
Eigenfunction basis:
Laplace operator: Graph Laplacian:
Eigenvector basis:
Classical convolution: Graph convolution:
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f(`) = h�`, fi =NX
i=1
�`(i)⇤f(i)
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f(i) =NX
`=1
f(`)�`(i)<latexit sha1_base64="humNT0UfpO6maUuWlkyaUKHb2jg=">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</latexit><latexit sha1_base64="humNT0UfpO6maUuWlkyaUKHb2jg=">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</latexit><latexit sha1_base64="humNT0UfpO6maUuWlkyaUKHb2jg=">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</latexit><latexit sha1_base64="humNT0UfpO6maUuWlkyaUKHb2jg=">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</latexit>
f(!) = hei!x, fi =Z(ei!x)⇤f(x)dx
<latexit sha1_base64="h8uWvFEDe2GkRQeLNRm2CNMf5Ss=">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</latexit><latexit sha1_base64="h8uWvFEDe2GkRQeLNRm2CNMf5Ss=">AAAC9HicbVFda9swFFW8ry77aLo97kUsDJJiip11NHsYlO6he+zG0hbiNMjytSMqy0aSOwfhf7K3sdf9k/2A/Y6+bjDF9kOa9oLg6Jyjq8s9Yc6Z0p73p+Pcu//g4aOtx90nT5893+7tvDhVWSEpTGjGM3keEgWcCZhopjmc5xJIGnI4Cy8/rvSzK5CKZeKrXuYwS0kiWMwo0Zaa95Ig5AWYYEE0jgdBlkJChhX+gANORMIBBxIiAxeGNRouq8rFsaUb2RqZ0Hhwlw8PL3Zt03KIo3Le63t7Xl34NvBb0Edtncx3OkUQZbRIQWjKiVJT38v1zBCpGeVQdYNCQU7oJUlgaqEgKaiZqTdS4TeWiXCcSXvseDW7/sKQVKllGlpnSvRCbWor8i5tWuh4PDNM5IUGQZuP4oJjneHVenHEJFDNlxYQKpmdFdMFkYRqG8KNX8pm1G4QQWzDq29mFUZlvhwfVcZzfX/f9cfvNyx2z61j5B+447fu6J11CPhGszQlIto1QdMl0FDqtbabrrrRmml1r7o2KH8zltvgdLTnW/x5v3941Ea2hV6h12iAfHSADtEndIImiKLf6Br9Rf+cK+e788P52VidTvvmJbpRzq//3Uvwlg==</latexit><latexit sha1_base64="h8uWvFEDe2GkRQeLNRm2CNMf5Ss=">AAAC9HicbVFda9swFFW8ry77aLo97kUsDJJiip11NHsYlO6he+zG0hbiNMjytSMqy0aSOwfhf7K3sdf9k/2A/Y6+bjDF9kOa9oLg6Jyjq8s9Yc6Z0p73p+Pcu//g4aOtx90nT5893+7tvDhVWSEpTGjGM3keEgWcCZhopjmc5xJIGnI4Cy8/rvSzK5CKZeKrXuYwS0kiWMwo0Zaa95Ig5AWYYEE0jgdBlkJChhX+gANORMIBBxIiAxeGNRouq8rFsaUb2RqZ0Hhwlw8PL3Zt03KIo3Le63t7Xl34NvBb0Edtncx3OkUQZbRIQWjKiVJT38v1zBCpGeVQdYNCQU7oJUlgaqEgKaiZqTdS4TeWiXCcSXvseDW7/sKQVKllGlpnSvRCbWor8i5tWuh4PDNM5IUGQZuP4oJjneHVenHEJFDNlxYQKpmdFdMFkYRqG8KNX8pm1G4QQWzDq29mFUZlvhwfVcZzfX/f9cfvNyx2z61j5B+447fu6J11CPhGszQlIto1QdMl0FDqtbabrrrRmml1r7o2KH8zltvgdLTnW/x5v3941Ea2hV6h12iAfHSADtEndIImiKLf6Br9Rf+cK+e788P52VidTvvmJbpRzq//3Uvwlg==</latexit><latexit sha1_base64="h8uWvFEDe2GkRQeLNRm2CNMf5Ss=">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</latexit>
f(x) =1
2⇡
Zf(!)ei!xd!
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(f ⇤ h)(i) =NX
`=1
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(f ⇤ h)(x) =Z
f(⌧)h(x� ⌧)d⌧
=1
2⇡
Zf(!)h(!)ei!xd!
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engs1973
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Henry Kenlay – Geometric deep learning for business classification
engs1973
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Henry Kenlay – Geometric deep learning for business classification
