1. Discrete Kernels

2. Kernels for Sequences ACGGTTCAA ATATCGCGGGAA
Similarity between sequences of different lengths
ACGGTTCAA
ATATCGCGGGAA

3. Count Kernel Inner product between symbol counts
Extension: Spectrum kernels (Leslie et al., 2002)
Counts the number of k-mers (k-grams) efficiently

4. Motivations for graph analysis
Existing methods assume ” tables”
Structured data beyond this framework
→ New methods for analysis …
Osaka
Female 31 ××
0002
Tokyo
Male 40 ○○
0001
Address
Sex
Age
Name
Serial Num

5. Graph Structures in Biology
DNA Sequence
RNA
Compounds A C G H C CG U UA H C C O H C C
フェノール H C H H

6. Graph Kernels Going to define the kernel function
(Kashima, Tsuda, Inokuchi, ICML 2003)
Going to define the kernel function
Both vertex and edges are labeled

7. Label path Sequence of vertex and edge labels
Generated by random walking
Uniform initial, transition, terminal probabilities

8. Path-probability vector

9. Kernel definition Kernels for paths
Take expectation over all possible paths!
Marginalized kernels for graphs

10. Classification of Protein 3D structures
Graphs for protein 3D structures
Node: Secondary structure elements
Edge: Distance of two elements
Calculate the similarity by graph kernels
Borgwardt et al. “Protein function prediction via graph kernels”, ISMB2005

11. Biological Networks Protein-protein physical interaction
Metabolic networks
Gene regulatory networks
Network induced from sequence similarity
Thousands of nodes (genes/proteins)
100000s of edges (interactions)

12. Physical Interaction Network
Undirected graphs of proteins
Edge exists if two proteins physically interact
Docking (Key – Keyhole)
Interacting proteins tend to have the same biological function

13. Metabolic Network

14. Diffusion kernels (Kondor and Lafferty, 2002)
Function prediction by SVM using a network
Kernels are needed !
Define closeness of two nodes
Has to be positive definite
How Close?

15. Definition of Diffusion Kernel
A: Adjacency matrix, 　
D: Diagonal matrix of Degrees
L = D-A: Graph Laplacian Matrix
Diffusion kernel matrix
：Diffusion paramater
Matrix exponential, not elementwise exponential

16. Computation of Matrix Exponential
Definition
Eigen-decomposition

17. Adjacency Matrix and Degree Matrix

18. Graph Laplacian Matrix L

19. Actual Values of Diffusion Kernels
Closeness from the
“central node”