Learning from Interpretations: A Rooted Kernel for Ordered Hypergraphs. Roni Khardon Gabriel Wachman Tufts University Department of Computer Science
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1 Learning from Interpretations: A Rooted Kernel for Ordered Hypergraphs Roni Khardon Gabriel Wachman Tufts University Department of Computer Science
2 Motivation Chemicals For example a water molecule: O Relational Data siblings(a, B, C) father(d,a) father(d,b) father(d,c) Can be represented by this hypergraph: H H A B D siblings() father() C
3 Agenda Introduction A Hypergraph Kernel Experiments and Results Future Work Questions
4 Kernel Definition A kernel takes objects and calculates their inner product in some feature space. For example: H O H Φ(G1)= Graph G1 has H has O has C has Hg H C H Φ(G2)= Graph G1 has H has O has C has Hg Therefore K(G1,G2) = <Φ(G1),Φ(G2)> = 1... but K(.,.) performs this calculation without explicitly representing the actual feature vectors.
5 Kernel Definition A kernel takes objects and calculates their inner product in some feature space. For example: H O H Φ(G1)= Graph G1 has H-C-H has H-C-O has H-O-H has H-C-Hg H C H Φ(G2)= Graph G1 has H-C-H has H-C-O has H-O-H has H-C-Hg Therefore K(G1,G2) = <Φ(G1),Φ(G2)> = 1... but K(.,.) performs this calculation without explicitly representing the actual feature vectors.
6 Previous Work Propositionalization [DeKuKa2003], [LaDz94] ILP Systems with Explicit Relational Hypotheses Progol [Mu1995], Tilde[BlDe1997] Graph Kernels Labeled Walk Variants [GaFlWr2004],[KaTsIn2003]
7 Walk Types A walk designates a set of connected edges in the hypergraph and the position of the vertices that connect them. Example: v1 v2 v3 v4 v5 l(e1)=p l(e2)=q v0 va vb v6 v7 v8 v9 l(e3)=r A walk in this graph goes from e1->e2 via v5, then e2->e3 via V8. The walk type is p 5 1 q 4 3 r p(v1,v2,v3,v4,v5) Ʌ q(v5,v6,v7,v8,v9) Ʌ r(v0,va,v8,vb)
8 Graph Kernel and Variants Theorem: K n p 1, p 2 = w of length n count G 1, p 1,w count G 2, p 2,w w is a walk type, count(g,p,w) gives number of walks of type w in G starting at p Variants: K ' n G 1,G 2 = p 1 E 1 n K D n = i=1 p 2 E 2 K n p 1, p 2 i K ' i G 1,G 2 K D K ' D n G 1,G 2 = n G 1,G 2 K D n G 1,G 1 K D n G 2,G 2
9 Edge Kernel Input G 1 = V 1,E 1,G 2 = V 2, E 2 p 1 E 1, p 2 E 2 p 1 i =the i 'th vertex of edge p 1 K 1 p 1,p 2 K 1 p 1,p 2 K n p 1, p 2 = 1 iff label p 1 =label p 2 = 0 o/w k = K 1 p 1,p 2 a=1 max arity b=1 p' 1 : p 1 a =p' 1 b K n 1 p' 1,p' 2 p' 2 : p a b 2 =p' 2 vb v1 v2 v3 v4 v5 e1 va e3 e2 v6 v7 v8 v9
10 Edge Kernel Input G 1 = V 1,E 1,G 2 = V 2,E 2 p 1 E 1,p 2 E 2 p 1 i =the i ' th vertex of edge p 1 K 1 p 1, p 2 = 1 iff label p 1 =label p 2 K 1 p 1, p 2 = 0 o/w K n p 1, p 2 k = K 1 p 1, p 2 a=1 max arity b=1 p' 1 : p 1 a =p' 1 b p' 2 : p 2 a =p' 2 b K n 1 p' 1, p' 2 a = 5 b = 1 p 1 vb v1 v2 v3 v4 v5 va e2 e1 v6 v7 v8 v9 possible p' 1
11 Example G1 v1 v2 v3 v4 v5 label(e2) = r v6 v7 v8 v9 label(e1) = q Walk type q 5 1 r matches in G1 only v1 v2 v3 v4 v5 label(e1) = q G2 v6 v7 v8 v9 label(e2) = r Walk type q 4 1 r matches in G2 only
12 A Hypergraph Kernel Handles multi-arity data directly Bounded length walks Easily implemented DP algorithm
13 A Hypergraph Kernel Handles multi-arity data directly Fixed length walks Easily implemented DP algorithm Different Feature space:
14 Experiments NCTRER 232 examples, 60% majority class Predictive Toxicology 4 label sets, all ~350 example, 60% majority class NCI-HIV 41,606 examples, 99% majority class Mutagenesis 188 chemical structures, 60% majority class High-arity (>2) edges
15 Edge Encoding [Gärtner05] l(v4) = C l(v1) = B e1 l(v3) = C l(v5) = D l(v2) = A l(v6) = A l 1 (e1) = bond(v1,v3) (also B(v1),A(v2),etc.) l 2 (e1) = bondbtoc(v1,v3) l 3 (e1) = bondb ACtoC ABCD(v1,v3)
16 Edge Encoding l(v4) = C l(v1) = B e1 l(v3) = C l(v5) = D l(v2) = A l(v6) = A l 1 (e1) = bond(v1,v3) (also B(v1),A(v2),etc.) l 2 (e1) = bondbtoc(v1,v3) l 3 (e1) = bondb ACtoC ABCD(v1,v3)
17 Edge Encoding l(v4) = C l(v1) = B e1 l(v3) = C l(v5) = D l(v2) = A l(v6) = A l 1 (e1) = bond(v1,v3) (also B(v1),A(v2),etc.) l 2 (e1) = bondbtoc(v1,v3) l 3 (e1) = bondb ACtoC ABCD(v1,v3)
18 Edge Encoding l(v4) = C l(v1) = B e1 l(v3) = C l(v5) = D l(v2) = A l(v6) = A l 1 (e1) = bond(v1,v3) (also B(v1),A(v2),etc.) l 2 (e1) = bondbtoc(v1,v3) l 3 (e1) = bondb ACtoC ABCD(v1,v3)
19 Edge Encoding l(v4) = C l(v1) = B e1 l(v3) = C l(v5) = D l(v2) = A l(v6) = A l 1 (e1) = bond(v1,v3) (also B(v1),A(v2),etc.) l 2 (e1) = bondbtoc(v1,v3) l 3 (e1) = bondb ACtoC ABCD(v1,v3)
20 Edge Encoding l(v4) = C l(v1) = B e1 l(v3) = C l(v5) = D l(v2) = A l(v6) = A l 1 (e1) = bond(v1,v3) (also B(v1),A(v2),etc.) l 2 (e1) = bondbtoc(v1,v3) l 3 (e1) = bondb ACtoC ABCD(v1,v3)
21 Edge Encoding l(v4) = C l(v1) = B e1 l(v3) = C l(v5) = D l(v2) = A l(v6) = A l 1 (e1) = bond(v1,v3) (also B(v1),A(v2),etc.) l 2 (e1) = bondbtoc(v1,v3) l 3 (e1) = bondb ACtoC ABCD(v1,v3)
22 Edge Encoding l(v4) = C l(v1) = B e1 l(v3) = C l(v5) = D l(v2) = A l(v6) = A l 1 (e1) = bond(v1,v3) (also B(v1),A(v2),etc.) l 2 (e1) = bondbtoc(v1,v3) l 3 (e1) = bondb ACtoC ABCD(v1,v3)
23 Edge Encoding l(v4) = C l(v1) = B e1 l(v3) = C l(v5) = D l(v2) = A l(v6) = A l 1 (e1) = bond(v1,v3) (also B(v1),A(v2),etc.) l 2 (e1) = bondbtoc(v1,v3) l 3 (e1) = bondb ACtoC ABCD(v1,v3)
24 Experimental Setup Perceptron with Margin [KrMe1986] Encoding versus walk length Discounting/incrementing paths High-arity versus binary edges Comparison to other methods
25 Results on NCTRER Encoding 3 vastly improves results Length Encoding 1 Encoding 2 Encoding / / / / / / / / / / / / / / / / / / / / / / / / / / / [LaPaDe06] /- 0.09
26 Results on NCTRER Encoding 3 vastly improves results Length Encoding 1 Encoding 2 Encoding / / / / / / / / / / / / / / / / / / / / / / / / / / / [LaPaDe06] /- 0.09
27 Results Competitive or better than existing methods on all datasets Dataset Length HG [FWSZ05] [KaTsIn03] PTC(FM) / / / PTC(FR) / / / PTC(MM) / / / PTC(MR) / / / (Accuracy) Dataset Length HG [HoGaWr04] [GaFlWr03] NCI-HIV / / / (Area under ROC curve)
28 Results Discounting/incrementing gives little or no improvement Best length walk is dataset dependent
29 Results Hyperedges in Mutagenesis Length AB AB+H AB+LC AB+H+LC / / / / / / / / / / / / / / / / / / / / / / / / Encoding 1 Length AB AB+H AB+LC AB+H+LC / / / / / / / / / / / / / / / / / / / / / / / / Encoding 3
30 Results Hyperedges in Mutagenesis Length AB AB+H AB+LC AB+H+LC / / / / / / / / / / / / / / / / / / / / / / / / Encoding 1 Length AB AB+H AB+LC AB+H+LC / / / / / / / / / / / / / / / / / / / / / / / / Encoding 3
31 Results Hyperedges in Mutagenesis Length AB AB+H AB+LC AB+H+LC / / / / / / / / / / / / / / / / / / / / / / / / Encoding 1 Length AB AB+H AB+LC AB+H+LC / / / / / / / / / / / / / / / / / / / / / / / / Encoding 3
32 Results Hyperedges in Mutagenesis Length AB AB+H AB+LC AB+H+LC / / / / / / / / / / / / / / / / / / / / / / / / Encoding 1 Length AB AB+H AB+LC AB+H+LC / / / / / / / / / / / / / / / / / / / / / / / / Encoding 3
33 Results Hyperedges in Mutagenesis Length AB AB+H AB+LC AB+H+LC / / / / / / / / / / / / / / / / / / / / / / / / Encoding 1 Length AB AB+H AB+LC AB+H+LC / / / / / / / / / / / / / / / / / / / / / / / / Encoding 3
34 Hypergraph kernel Summary Competitive with ILP methods as well as graph kernels Uses different feature space Ability to use hyperedges improves performance Data encoding critical Open questions Extend to infinite path Use on other multi-arity data Capture encoding 3 performance Effect of encoding 3 on ILP solvers Kernels for more complex walk types
35 Thank You Gabriel Wachman Roni Khardon
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