Poster Session 6E Tonight!

The Kendall and Mallows Kernels for Permutations

Yunlong Jiao & Jean-Philippe Vert

MINES ParisTech

ICML Lille, July 8, 2015

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Introduction

Recommender system,e.g. CollaborativeFiltering.

Learn from converted rankings,e.g., gene expression dataanalysis for leukemiaclassification [Tan et al., 2005].

– Data: n × p matrix (p � n).

– Rule: if SPTAN1 ≥ CD33then ALL; else AML.

– Accuracy: 93.80% (LOOCV).

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Introduction

Recommender system,e.g. CollaborativeFiltering.

Learn from converted rankings,e.g., gene expression dataanalysis for leukemiaclassification [Tan et al., 2005].

– Data: n × p matrix (p � n).

– Rule: if SPTAN1 ≥ CD33then ALL; else AML.

– Accuracy: 93.80% (LOOCV).

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Outline

1 Data type

– Rankings and permutations.

2 Methods

– Computationally efficient kernels for total rankings, partialrankings and rankings converted from quantitative vectors.

3 Experiments

– High-dimensional classification in biomedical applications.

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Total Rankings and Permutations

A total ranking is a strict ordering of n items {x1, x2, . . . , xn} ,

xi1 � xi2 � · · · � xin .

A permutation is a rearrangement of n indices,

σ : {1, 2, . . . , n} → {1, 2, . . . , n} such that σ(i) 6= σ(j) for i 6= j .

A total ranking is equivalently represented by a permutationif σ maps item index to item rank, e.g.,

x2 � x4 � x3 � x1

⇐⇒ σ =

(2 4 3 14 3 2 1

)— index— rank

⇐⇒ σ(1) = 1, σ(2) = 4, σ(3) = 2, σ(4) = 3 .

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Total Rankings and Permutations

A total ranking is a strict ordering of n items {x1, x2, . . . , xn} ,

xi1 � xi2 � · · · � xin .

A permutation is a rearrangement of n indices,

σ : {1, 2, . . . , n} → {1, 2, . . . , n} such that σ(i) 6= σ(j) for i 6= j .

A total ranking is equivalently represented by a permutationif σ maps item index to item rank, e.g.,

x2 � x4 � x3 � x1

⇐⇒ σ =

(2 4 3 14 3 2 1

)— index— rank

⇐⇒ σ(1) = 1, σ(2) = 4, σ(3) = 2, σ(4) = 3 .

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Kendall tau Distance for Permutations

Kendall tau distance [Kendall, 1938]counts the number of discordant pairsbetween permutations, i.e.,

nd(σ, σ′) =∑i<j

1σ(i)<σ(j)1σ′(i)>σ′(j)

+ 1σ(i)>σ(j)1σ′(i)<σ′(j) .

The number of concordant pairsbetween permutations is

nc(σ, σ′) =

(n

2

)− nd(σ, σ′) .

E.g.,

index e 1 2 3 4

rank σ 2 3 4 1rank σ′ 3 1 4 2

nd(σ, σ′) = 1 + 1 + 0 = 2

nc(σ, σ′) =4(4− 1)

2−2 = 4

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Kendall tau Distance for Permutations

Kendall tau distance [Kendall, 1938]counts the number of discordant pairsbetween permutations, i.e.,

nd(σ, σ′) =∑i<j

1σ(i)<σ(j)1σ′(i)>σ′(j)

+ 1σ(i)>σ(j)1σ′(i)<σ′(j) .

The number of concordant pairsbetween permutations is

nc(σ, σ′) =

(n

2

)− nd(σ, σ′) .

E.g.,

index e 1 2 3 4

rank σ 2 3 4 1rank σ′ 3 1 4 2

nd(σ, σ′) = 1 + 1 + 0 = 2

nc(σ, σ′) =4(4− 1)

2−2 = 4

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Kendall and Mallows Kernels for Permutations

The Kendall tau coefficient is definedas

Kτ (σ, σ′) =nc(σ, σ′)− nd(σ, σ′)(n

2

) .

The Mallows measure is defined forany λ ≥ 0 by

KλM(σ, σ′) = e−λnd (σ,σ′) .

E.g.,

index e 1 2 3 4

rank σ 2 3 4 1rank σ′ 3 1 4 2

Kτ (σ, σ′) =4− 2

6=

1

3

KλM(σ, σ′) = e−2λ, λ ≥ 0

Theorem (Main theorem)

These two similarity measures for permutations are positive definitekernels.

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Kendall and Mallows Kernels for Permutations

The Kendall tau coefficient is definedas

Kτ (σ, σ′) =nc(σ, σ′)− nd(σ, σ′)(n

2

) .

The Mallows measure is defined forany λ ≥ 0 by

KλM(σ, σ′) = e−λnd (σ,σ′) .

E.g.,

index e 1 2 3 4

rank σ 2 3 4 1rank σ′ 3 1 4 2

Kτ (σ, σ′) =4− 2

6=

1

3

KλM(σ, σ′) = e−2λ, λ ≥ 0

Theorem (Main theorem)

These two similarity measures for permutations are positive definitekernels.

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Kendall and Mallows Kernels for Permutations

The Kendall kernel is defined as

Kτ (σ, σ′) =nc(σ, σ′)− nd(σ, σ′)(n

2

) .

The Mallows kernel is defined for any λ ≥ 0 by

KλM(σ, σ′) = e−λnd (σ,σ′) .

Theorem (Main theorem)

These two kernels for permutations are positive definite.

Proof.

Consider the explicit kernel mapping

Φ : Sn → R(n2), σ 7→(

sgn(σ(i)− σ(j)))

1≤i<j≤n.

The Kendall and Mallows kernel correspond respectively to a linearand Gaussian kernel on a

(n2

)-dimensional embedding of Sn.

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Kendall and Mallows Kernels for Permutations

The Kendall kernel is defined as

Kτ (σ, σ′) =nc(σ, σ′)− nd(σ, σ′)(n

2

) .

The Mallows kernel is defined for any λ ≥ 0 by

KλM(σ, σ′) = e−λnd (σ,σ′) .

Theorem (Main theorem)

These two kernels for permutations are positive definite.

Theorem ([Knight, 1966])

These two kernels for permutations can be evaluated in O(n log n)time.

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Convolution Kendall Kernel for Partial Rankings

Two interesting types of partial rankings are interleavingpartial ranking

xi1 � xi2 � · · · � xik , k ≤ n.

and top-k partial ranking

xi1 � xi2 � · · · � xik � Xrest, k ≤ n.

Partial rankings can be uniquely represented by a set ofpermutations compatible with all the observed partial orders.

Theorem

For these two particular types of partial rankings, the convolutionkernel [Haussler, 1999] induced by Kendall kernel

K ?τ (R,R ′) =

1

|R||R ′|∑σ∈R

∑σ′∈R′

Kτ (σ, σ′)

can be evaluated in O(k log k) time.

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Convolution Kendall Kernel for Partial Rankings

Two interesting types of partial rankings are interleavingpartial ranking

xi1 � xi2 � · · · � xik , k ≤ n.

and top-k partial ranking

xi1 � xi2 � · · · � xik � Xrest, k ≤ n.

Partial rankings can be uniquely represented by a set ofpermutations compatible with all the observed partial orders.

Theorem

For these two particular types of partial rankings, the convolutionkernel [Haussler, 1999] induced by Kendall kernel

K ?τ (R,R ′) =

1

|R||R ′|∑σ∈R

∑σ′∈R′

Kτ (σ, σ′)

can be evaluated in O(k log k) time.14 / 25

Stabilized Kendall Kernel for Quantitative Vectors

−3 −2 −1 0 1 2 3

−1.0

−0.5

0.0

0.5

1.0

xi − xj

ΦijΨij

Kendall mapping for quantitative vectors isdiscrete-valued and very sensitive to “almostties”, i.e.,

Φ : Rn → R(n2), x 7→(1xi>xj − 1xi<xj

)1≤i<j≤n

.

We propose a noise-corrupted kernel mapping instead(similarly to [Muandet et al., 2012])

Ψ(x) = EΦ(x + ε︸ ︷︷ ︸x

) =(P (xi > xj)− P (xi < xj)

)1≤i<j≤n

.

Kendall kernel stabilized alternative is given by

G(x, x′

)= Ψ(x)>Ψ(x′) = EKτ (x, x′) .

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Mallows Kernel vs. Diffusion Kernel over Sn

Figure : Cayley graph of S4.

Diffusion kernel[Kondor and Lafferty, 2002] isdefined by

Kβdif(σ, σ

′) = [eβ∆]σ,σ′ ,

where ∆ is the graph laplacian.

Mallows kernel is written as

KλM(σ, σ′) = e−λnd (σ,σ′) ,

where nd(σ, σ′) = dG(σ, σ′) theshortest path distance on graph.

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Gene Expression Data

Datasets

Dataset No. of features No. of samples (training/test)C1 C2

Breast Cancer 1 23624 44/7 (Non-relapse) 32/12 (Relapse)Breast Cancer 2 22283 142 (Non-relapse) 56 (Relapse)Breast Cancer 3 22283 71 (Poor Prognosis) 138 (Good Prognosis)

Colon Tumor 2000 40 (Tumor) 22 (Normal)Lung Cancer 1 7129 24 (Poor Prognosis) 62 (Good Prognosis)Lung Cancer 2 12533 16/134 (ADCA) 16/15 (MPM)

Medulloblastoma 7129 39 (Failure) 21 (Survivor)Ovarian Cancer 15154 162 (Cancer) 91 (Normal)

Prostate Cancer 1 12600 50/9 (Normal) 52/25 (Tumor)Prostate Cancer 2 12600 13 (Non-relapse) 8 (Relapse)

Methods

Kernel machines Support Vector Machines (SVM) and KernelFisher Discriminant (KFD) with Kendall kernel, linear kernel,Gaussian RBF kernel, polynomial kernel.

Top Scoring Pairs (TSP) classifiers [Tan et al., 2005].

Hybrid scheme of SVM + TSP feature selection algorithm.17 / 25

Results

●

●

●

SV

Mkd

tALL

SV

Mlin

earT

OP

SV

Mlin

earA

LL

SV

Mkd

tTO

P

SV

Mpo

lyA

LL

KF

Dkd

tALL

kTS

P

SV

Mpo

lyTO

P

KF

Dlin

earA

LL

KF

Dpo

lyA

LL TS

P

SV

Mrb

fALL

KF

Drb

fALL

AP

MV

0.4

0.6

0.8

1.0

acc

Kendall kernel SVM

Competitiveaccuracy!

Insensitive to Cparameter!

No featureselection!

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Results

1e−02 1e+00 1e+02

0.3

0.4

0.5

0.6

0.7

0.8

0.9

BC1

C parameter

acc

●

SVMlinearALLSVMkdtALLSVMpolyALLSVMrbfALLKFDlinearALLKFDkdtALLKFDpolyALLKFDrbfALL

Kendall kernel SVM

Competitiveaccuracy!

Insensitive to Cparameter!

No featureselection!

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Results

1 5 10 50 500 5000

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

PC1

Number k of top gene pairs

acc

●

SVMlinearTOPSVMkdtTOPSVMpolyTOPkTSPSVMlinearALLSVMkdtALLSVMpolyALLTSPAPMV

Kendall kernel SVM

Competitiveaccuracy!

Insensitive to Cparameter!

No featureselection!

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Results

●

● ●

●

●●

●

●

● ●

●

●

●

●

●

●

●

●●

●

1e+01 1e+02 1e+03 1e+04 1e+05

0.60

0.62

0.64

0.66

0.68

0.70

MB

Noise window size a

cvac

c

● SVMkdtALLalt−−exactSVMkdtALLalt−−MCapprox (D=1)SVMkdtALLalt−−MCapprox (D=3)SVMkdtALLalt−−MCapprox (D=5)SVMkdtALLalt−−MCapprox (D=7)SVMkdtALLalt−−MCapprox (D=9)SVMkdtALL

Kendall kernel SupportMeasure Machines[Muandet et al., 2012]

Improvedaccuracy!

−3 −2 −1 0 1 2 3

−1.0

−0.5

0.0

0.5

1.0

xi − xj

ΦijΨij

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Conclusion

Poster Session 6E Tonight!

IF you are dealing with ranking-related problems,

IF your problem can be formulated in a way that some kernelmachine can cope with,

DO throw Kendall and Mallows kernel into that kernel machine!

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Acknowledgments

This work was supported by theEuropean Union 7th FrameworkProgram through the Marie CurieITN MLPM grant No 316861, andby the European Research Councilgrant ERC-SMAC-280032.

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References I

Haussler, D. (1999).

Convolution kernels on discrete structures.

Technical Report UCSC-CRL-99-10, UC Santa Cruz.

Kendall, M. G. (1938).

A new measure of rank correlation.

Biometrika, 30(1/2):81–93.

Knight, W. R. (1966).

A computer method for calculating Kendall’s tau with ungrouped data.

J. Am. Stat. Assoc., 61(314):436–439.

Kondor, I. R. and Lafferty, J. (2002).

Diffusion kernels on graphs and other discrete input spaces.

In Proceedings of the Nineteenth International Conference on MachineLearning, volume 2, pages 315–322, San Francisco, CA, USA. MorganKaufmann Publishers Inc.

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References II

Muandet, K., Fukumizu, K., Dinuzzo, F., and Scholkopf, B. (2012).

Learning from distributions via support measure machines.

In Pereira, F., Burges, C. J. C., Bottou, L., and Weinberger, K. Q.,editors, Adv. Neural. Inform. Process Syst., volume 25, pages 10–18.Curran Associates, Inc.

Tan, A. C., Naiman, D. Q., Xu, L., Winslow, R. L., and Geman, D.(2005).

Simple decision rules for classifying human cancers from gene expressionprofiles.

Bioinformatics, 21(20):3896–3904.

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