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Semi-supervised learning of edge flows

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Slides from my talk at the Higher-order Network Models Satellite at NetSci.
May 28, 2019.
Burlington, VT

Published in: Data & Analytics
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Semi-supervised learning of edge flows

  1. 1. 1 Joint work with Junteng Jia (Cornell), Michael T.Schaub (MIT),& Santiago Segarra (Rice) Graph-based semi-supervised learning for edge flows Austin R. Benson · Cornell University NetSci HONS May 28, 2019 Slides. bit.ly/arb-HONS-19
  2. 2. 2http://www.vidiani.com/road-map-of-manhattan/ Traffic flow sensors
  3. 3. Two major questions in semi-supervised learning. 3 1. Interpolation. Given the measurements, how do I interpolate to locations where I don’t know have data. 2. Active learning. Where are the best locations to make my measurements, knowing step 1?
  4. 4. Background. Classical graph-based semi-supervised learning interpolates from labels on a few vertices. 4 Key idea. My label is similar to the labels of my connections. minimize labels x X (i,j)2E (xi xj)2 subject to x matches given labels<latexit 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  5. 5. In the higher-order case of edge flows,we have a different type of objective. 5 Key idea (“divergence-free”). Net flow into a node should be similar to net flow out of a node.
  6. 6. An edge flow represents net flow along an edge. 6 • As an alternating function: F(i, j) = -F(j, i) • For the linear algebra, first orient each edge i → j if i < j. Then vector f gives flows on these oriented edges. • If fi,j > 0, if net flow aligns with orientation • If fi,j < 0, net flow is opposite of orientation. 1 2 3 4 5 6 7 f1,3 > 0 f5,7 < 0
  7. 7. In the higher-order case of edge flows,we have a different type of objective. 7 Key idea (“divergence-free”). Net flow into a node should be similar to net flow out of a node. minimize flows f X i 2 4 X j>i,(i,j)2E fij X k<i,(k,i)2E fki 3 5 2 subject to f matches labels<latexit 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  8. 8. There is a close relationship between node-based SSL and edge-based SSLobjective functions. 8 X (i,j)2E (xi xj)2 = xT Lx = xT BBT x = kBT xk2 2 <latexit 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X i 2 4 X j>i,(i,j)2E fij X k<i,(k,i)2E fki 3 5 2 = fT BT Bf = kBfk2 <latexit 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Bk,(i,j) = 8 >< >: 1 k = i, i < j 1 k = j, i < j 0 otherwise<latexit 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One catch. • Having labels in node case gives unique answer. • Having labels in edge case is under-constrained.
  9. 9. We add regularization to get a nice sparse linear least squares problem. 9 minimize flows f kBfk2 2 + kfk2 2 subject to f matches labels<latexit 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• We use iterative solvers LSQR or LSMR to compute the solution efficiently.
  10. 10. Key Idea My label is similar to the labels of my connections. Objective Net in flow = net out flow at all nodes. minimize vertex values x kBT xk2 2 subject to x matches labels<latexit 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minimize edge flows f kBfk2 2 + kfk2 2 subject to f matches labels<latexit 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  11. 11. 11 0.0 0.2 0.4 0.6 0.8 1.0 correlation Anaheim Barcelona 0.0 0.2 0.4 0.6 0.8 1.0 ratio labeled (| L |/| |) 0.0 0.2 0.4 0.6 0.8 1.0 correlation 0.0 0.2 0.4 0.6 0.8 1.0 ratio labeled (| L |/| |) Winnipeg Chicago ZeroFill LineGraph FlowSSL
  12. 12. Why do we do so poorly on Chicago? 12 ftruth = y z, y 2 R, z 2 C<latexit 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Our divergence-free assumption says that ftruth ⇡ z.<latexit 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Does this actually hold in our data?
  13. 13. 13 0.0 0.2 0.4 0.6 0.8 1.0 correlation Anaheim Barcelona 0.0 0.2 0.4 0.6 0.8 1.0 ratio labeled (| L |/| |) 0.0 0.2 0.4 0.6 0.8 1.0 correlation 0.0 0.2 0.4 0.6 0.8 1.0 ratio labeled (| L |/| |) Winnipeg Chicago ZeroFill LineGraph FlowSSL kyk2 kzk2 ⇡ 0.2. <latexit 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kyk2 kzk2 ⇡ 0.5. <latexit 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