The skew spectrum of graphs - Semantic Scholar

The skew spectrum of graphs Risi Kondor

Gatsby Unit, UCL with

Karsten Borgwardt

University of Cambridge

Can just 49 features characterize a graph?

up to ~300 vertices

2 3 1

4 5

7



0 0  0  A= 1 0  0 0

0 0 1 1 0 0 0

0 1 0 1 0 0 0

1 1 1 0 0 1 0

0 0 0 0 0 1 0

0 0 0 1 1 0 1

 0 0  0  0  0  1 0

6

q(A) is a graph invariant if it is invariant to relabeling.

poly(n) time computable

complete set of invariants Graph isomorphism problem

efficiently computable set of invariant features Graph kernels, etc.

                           

0 1 0 0 1 0 1 1 0 1 0 1 0 1 1 0



                           

!→

                          

                           



v

π

      

       

=



    P (π) ⊗ P (π)   

[P (π)]i,j =

!

1 0

       

· v

if π(j) = i otherwise

! q = v v Our first invariant: 0

       

       



Ov

π



      

      

=



ρ1 (π)

      

ρ2 (π)



  ρ3 (π)    

· Ov

       

P (π) ⊗ P (π) = ρ1 (σ) ⊕ ρ2 (σ) ⊕ . . . ⊕ ρk (σ)

Now we have a whole bunch of invariants: q1 =

! v1 v1 ,

q2 =

! v2 v2 ,

. . . , qm =

! vk vk

       

f!(ρ) =

"

f (σ) ρ(σ)

σ∈Sn

ρ(σ2 σ1 ) = ρ(σ2 ) ρ(σ1 ) representation

f!(ρ) =

"

f (σ) ρ(σ)

σ∈Sn

• Ivertible • Unitary π −1 theorem • Translation f (σ) = f (π σ) • Convolution theorem • etc.

π ! f (ρ) = ρ(π) · f!(ρ)

•

f!(ρ) =

"

f (σ) ρ(σ)

σ∈Sn

Diaconis: Group representations in probability and statistics (1988) • Clausen, Maslen, Rockmore, Healy, ... : FFTs • Kondor, Howard and Jebara: Multi-object tracking with representations of the symmetric group (AISTATS, 2007) • Huang, Guestrin and Guibas: Efficient inference for distributions on permutations (NIPS, 2007)

The Fourier spectrum of graphs

f (σ) = [A]σ(n),σ(n−1) f!(ρ)

!→

! ! ! f (ρ) · f (ρ)

ρ(π) f!(ρ) invariant

Example ρ1 (σ) =

For σ = (1, 2, 3, 4, 5) ∈ S5 !

1



"

−0.5  0.866 ρ2 (σ) =   0 0



  ρ3 (σ) =   



   ρ4 (σ) =    

−0.289 −0.167 0.943 0

−0.204 −0.118 −0.0833 0.968

0.25 −0.433 −0.433 −0.25 −0.433 −0.25 0.75 −0.144 0 0.816

0.333 0.236 −0.471 0.0417 0.816 −0.0722 0 −0.484 0 0.839 0 0



−0.791 −0.456   −0.323  −0.25

0.433 −0.75 0.25 −0.433 0

0 0.217 0.125 −0.28 −0.161 0.913

−0.75 −0.433 0.144 0.0833 −0.471

0.913 0.161 −0.28 0.125 −0.217 0

0 0 −0.816 −0.471 −0.333

     

0 0 0.839 0 0.484 0 0.0722 0.816 0.0417 0.471 −0.236 0.333

       

Example f!(ρ1 ) = (6) 

-0.25  -0.323 f!(ρ2 ) =   0.913 0  1.33  0.471  f!(ρ3 ) =   0  0.816 0 

   ! f (ρ4 ) =    

-1.67 1.18 0 -0.913 0 -2.24

-0.323 -0.417 1.18 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0



0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0   0  0 

    

0 0 0 0 0 0

       



Ov

π

      

       

=



ρ1 (π)

      

ρ2 (π)



  ρ3 (π)    

· Ov

       

Now we have a whole bunch of invariants: q1 =

! v1 v1 ,

q2 =

! v2 v2 ,

. . . , qm =

! vk vk

       

Example f!(ρ1 ) = (6) 

-0.25  -0.323 f!(ρ2 ) =   0.913 0  1.33  0.471  f!(ρ3 ) =   0  0.816 0 

   ! f (ρ4 ) =    

-1.67 1.18 0 -0.913 0 -2.24

-0.323 -0.417 1.18 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0



0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0   0  0 

    

0 0 0 0 0 0

       

... but there is more ...

!

$ #! f"(ρ1 ) ⊗ f"(ρ2 ) Cρ1 ,ρ2 f"(ρ) ρ

non-commutative bispectrum [Kakarala ’92]

Skew spectrum q!ν (ρ) = r!ν (ρ) · f!(ρ) !

rν (σ) = f (σπ)f (σ)

Just 7 ν values!

[Kondor, 2007]

Sn

Bratelli diagram

http://www.gatsby.ucl.ac.uk/~risi/SnOB

49 graph invariants computable in

O(n ) 3

time

Number of instances/classes Max. number of nodes Reduced skew spectrum Random walk kernel Shortest path kernel

MUTAG 600/6 28 88.61 (0.21) 71.89 (0.66) 81.28 (0.45)

ENZYME 188/2 126 25.83 (0.34) 14.97 (0.28) 27.53 (0.29)

NCI1 4110/2 111 62.72 (0.05) 51.30 (0.23) 61.66 (0.10)

NCI109 4127/2 111 62.62 (0.03) 53.11 (0.11) 62.35 (0.13)

•

A general method for generating invariants

•

Only 49 graph invariants, but surprisingly powerful

•

Very fast to compute

•

Next: labeled graphs

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