Comparing distributions: 1 geometry improves kernel two-sample - - PowerPoint PPT Presentation

Références

Comparing distributions: ℓ1 geometry improves kernel two-sample testing

• M. Scetbon1,2
• G. Varoquaux1

1Inria, Université Paris-Saclay 2CREST, ENSAE

12 décembre 2019

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Two collections of samples X, Y from unknown distributions P and Q.

McDonald's KFC

Problem : Are the two set of observations X and Y drawn from the same distribution ?

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Two collections of samples X, Y from unknown distributions P and Q.

McDonald's KFC

Problem : Are the two set of observations X and Y drawn from the same distribution ?

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Two-Sample Test Test the null hypothesis H0 : P = Q against H1 : P = Q Samples : X = {xi}n

i=1 ∼ P and Y = {yi}n i=1 ∼ Q

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Two-Sample Test Test the null hypothesis H0 : P = Q against H1 : P = Q Samples : X = {xi}n

i=1 ∼ P and Y = {yi}n i=1 ∼ Q

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Gaussian Kernel : kσ(x, y) = exp

• − x−y2

2

2σ2

• Empirical Mean Embeddings of P and Q :
• µP(T) =

n

• i=1

k(xi, T)

• µQ(T) =

n

• j=1

k(yj, T) '( '+

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Aboslute difference of the Mean Embeddings :

• S(T) = |

µP(T) − µQ(T)|

'( | '( − '+| '+

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Aboslute difference of the Mean Embeddings :

• S(T) = |

µP(T) − µQ(T)| Test locations : (Tj)J

j=1 ∼ Γ

!" !# !$ !% !&

'( | '( − '+| '+

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Test Statistic 1 with p ≥ 1 :

• dℓp,µ,J(X, Y)

p := n

p 2

J

• j=1

| µP(Tj) − µQ(Tj)| p These Statistics are derived from metrics which metrize the weak convergence : dLp,µ(P, Q) :=

• t∈Rd
• µP(t) − µQ(t)
• p

dΓ(t)

• 1/p

Theorem : Weak Convergence αn

D

− → α ⇐ ⇒ dLp,µ(αn, α) → 0

• 1. The case when p = 2 has been studied by [1, 2]

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Test Statistic 1 with p ≥ 1 :

• dℓp,µ,J(X, Y)

p := n

p 2

J

• j=1

| µP(Tj) − µQ(Tj)| p These Statistics are derived from metrics which metrize the weak convergence : dLp,µ(P, Q) :=

• t∈Rd
• µP(t) − µQ(t)
• p

dΓ(t)

• 1/p

Theorem : Weak Convergence αn

D

− → α ⇐ ⇒ dLp,µ(αn, α) → 0

• 1. The case when p = 2 has been studied by [1, 2]

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Test Statistic 1 with p ≥ 1 :

• dℓp,µ,J(X, Y)

p := n

p 2

J

• j=1

| µP(Tj) − µQ(Tj)| p These Statistics are derived from metrics which metrize the weak convergence : dLp,µ(P, Q) :=

• t∈Rd
• µP(t) − µQ(t)
• p

dΓ(t)

• 1/p

Theorem : Weak Convergence αn

D

− → α ⇐ ⇒ dLp,µ(αn, α) → 0

• 1. The case when p = 2 has been studied by [1, 2]

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67(:;#, :;) = 2 )*+,-(:;#,:;) → 0

:;# :; !<=# !<=# | !<=# − !<=# |

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Test of level α : Compute

• dℓp,µ,J(X, Y)

p and reject H0 if

• dℓp,µ,J(X, Y)

p > Tα,p = 1 − α quantile of the asymptotic null distribution. Proposition : ℓ1 geometry improves power Let δ > 0. Under the alternative hypothesis H1, almost surely there exist N ≥ 1 such that for all n ≥ N with a probability 1 − δ :

• dℓ2,µ,J(X, Y)

2 > Tα,2 ⇒ dℓ1,µ,J(X, Y) > Tα,1

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Test of level α : Compute

• dℓp,µ,J(X, Y)

p and reject H0 if

• dℓp,µ,J(X, Y)

p > Tα,p = 1 − α quantile of the asymptotic null distribution. Proposition : ℓ1 geometry improves power Let δ > 0. Under the alternative hypothesis H1, almost surely there exist N ≥ 1 such that for all n ≥ N with a probability 1 − δ :

• dℓ2,µ,J(X, Y)

2 > Tα,2 ⇒ dℓ1,µ,J(X, Y) > Tα,1

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Conclusion Under the alternative hypothesis, Analytic Kernel (e.g Gaussian Kernel) guarantees dense differences between µP and µQ We have also considered statistics based on Smooth Characteristic Functions and obtained similar results. Finally we have normalized the tests to obtain a simple null distribution and learn the locations where the distributions differ the most.

@ East Exhibition Hall B + C #6

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Conclusion Under the alternative hypothesis, Analytic Kernel (e.g Gaussian Kernel) guarantees dense differences between µP and µQ ℓ1 geometry captures better these dense differences. We have also considered statistics based on Smooth Characteristic Functions and obtained similar results. Finally we have normalized the tests to obtain a simple null distribution and learn the locations where the distributions differ the most.

@ East Exhibition Hall B + C #6

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Conclusion Under the alternative hypothesis, Analytic Kernel (e.g Gaussian Kernel) guarantees dense differences between µP and µQ ℓ1 geometry captures better these dense differences. We have also considered statistics based on Smooth Characteristic Functions and obtained similar results. Finally we have normalized the tests to obtain a simple null distribution and learn the locations where the distributions differ the most.

@ East Exhibition Hall B + C #6

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Conclusion Under the alternative hypothesis, Analytic Kernel (e.g Gaussian Kernel) guarantees dense differences between µP and µQ ℓ1 geometry captures better these dense differences. We have also considered statistics based on Smooth Characteristic Functions and obtained similar results. Finally we have normalized the tests to obtain a simple null distribution and learn the locations where the distributions differ the most.

@ East Exhibition Hall B + C #6

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Conclusion Under the alternative hypothesis, Analytic Kernel (e.g Gaussian Kernel) guarantees dense differences between µP and µQ ℓ1 geometry captures better these dense differences. We have also considered statistics based on Smooth Characteristic Functions and obtained similar results. Finally we have normalized the tests to obtain a simple null distribution and learn the locations where the distributions differ the most.

@ East Exhibition Hall B + C #6

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References I

[1] K. P. Chwialkowski, A. Ramdas, D. Sejdinovic, and A. Gretton. Fast two-sample testing with analytic representations of probability measures. In Advances in Neural Information Processing Systems, pages 1981–1989, 2015. [2] W. Jitkrittum, Z. Szabó, K. P. Chwialkowski, and A. Gretton. Interpretable distribution features with maximum testing power. In Advances in Neural Information Processing Systems, pages 181–189, 2016.

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