Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Geometry of Covariance Matrices and Computation of Median Le Yang∗ , Marc Arnaudon∗ and Frédéric Barbaresco† ∗ Laboratoire

de Mathématiques et Applications,

Université de Poitiers, Chasseneuil, France

† Thales

Air Systems, Département Stratégie,

Technologie et Innovation, Limours, France

MaxEnt 2010, Chamonix, France, July 4-9

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

1

Introduction and Background : Radar Target Detection Radar observation values Standard method Geometric method

2

Geometry of Covariance Matrices Reection coecients parametrization Riemannian metric and curvature Riemannian distance Geodesics

3

Riemannian Median Average of points in a Riemannian manifold Computation of Riemannian median

4

Simulation

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Radar observation values Standard method Geometric method

1

Introduction and Background : Radar Target Detection Radar observation values Standard method Geometric method

2

Geometry of Covariance Matrices Reection coecients parametrization Riemannian metric and curvature Riemannian distance Geodesics

3

Riemannian Median Average of points in a Riemannian manifold Computation of Riemannian median

4

Simulation

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Radar observation values Standard method Geometric method

Radar observation values Fix a direction

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Radar observation values Standard method Geometric method

Radar observation values Fix a direction Subdivide : radar cells

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Radar observation values Standard method Geometric method

Radar observation values Fix a direction Subdivide : radar cells Emit −→ Reect −→ Receive

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Radar observation values Standard method Geometric method

Radar observation values Fix a direction Subdivide : radar cells Emit −→ Reect −→ Receive

Fig. 1: Emission

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Radar observation values Standard method Geometric method

Radar observation values Fix a direction Subdivide : radar cells Emit −→ Reect −→ Receive

Fig. 1: Reection

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Radar observation values Standard method Geometric method

Radar observation values Fix a direction Subdivide : radar cells Emit −→ Reect −→ Receive

Fig. 1: Reception

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Radar observation values Standard method Geometric method

Radar observation values Fix a direction Subdivide : radar cells Emit −→ Reect −→ Receive Observation value of one radar cell Z

= (z1 , ..., zk = rk e i ϕk , ..., zn )T

k : amplitude of reected signal ϕk : phase of reected signal r

n

: number of signals emitted in one burst

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Radar observation values Standard method Geometric method

Standard method for target detection Observation value of one radar cell : Z = (z , ..., zn )T 1

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Radar observation values Standard method Geometric method

Standard method for target detection Observation value of one radar cell : Z = (z , ..., zn )T 1

Method using Fourier transform Discrete Fourier transform

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Radar observation values Standard method Geometric method

Standard method for target detection Observation value of one radar cell : Z = (z , ..., zn )T 1

Method using Fourier transform Discrete Fourier transform Identication of exeptional frequency behavior : Constant False Alarm Rate (CFAR)

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Radar observation values Standard method Geometric method

Standard method for target detection Observation value of one radar cell : Z = (z , ..., zn )T 1

Method using Fourier transform Discrete Fourier transform Identication of exeptional frequency behavior : Constant False Alarm Rate (CFAR)

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Radar observation values Standard method Geometric method

Standard method for target detection Observation value of one radar cell : Z = (z , ..., zn )T 1

Method using Fourier transform Discrete Fourier transform Identication of exeptional frequency behavior : Constant False Alarm Rate (CFAR)

Limitation : n small (for example, n = 8 or 16)=⇒ low resolution Geometry of Covariance Matrices and Computation of Median

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Radar observation values Standard method Geometric method

Geometric Method for target detection Statistical modeling hypothesis : Z = (z , ..., zn )T is a realization of a centered stationary Gaussian process. 1

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Radar observation values Standard method Geometric method

Geometric Method for target detection Statistical modeling hypothesis : Z = (z , ..., zn )T is a realization of a centered stationary Gaussian process. 1

Covariance Matrix  

n = E[zi zj ]



R

ij n

≤, ≤

1

r0

r1

 r1  = .  ..

r0

... ...

...

r1

r

Geometry of Covariance Matrices and Computation of Median

n−

... ... 1

n− r n− r

1 2

  

..  . 

r0

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Radar observation values Standard method Geometric method

Geometric Method for target detection Statistical modeling hypothesis : Z = (z , ..., zn )T is a realization of a centered stationary Gaussian process. 1

Covariance Matrix  

n = E[zi zj ]



R

ij n

≤, ≤

1

r0

r1

 r1  = .  ..

r0

... ...

...

r1

r

n−

... ... 1

n− r n− r

2

  

..  . 

r0

n ∈ THPDn : Toeplitz Hermitian positive denite

R

Geometry of Covariance Matrices and Computation of Median

1

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Radar observation values Standard method Geometric method

Principle of target detection : geometric method Observation value of one radar cell : Rn

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Radar observation values Standard method Geometric method

Principle of target detection : geometric method Observation value of one radar cell : Rn

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Radar observation values Standard method Geometric method

Principle of target detection : geometric method Observation value of one radar cell : Rn

To be precisely dened Distance between two covariance matrices

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Radar observation values Standard method Geometric method

Principle of target detection : geometric method Observation value of one radar cell : Rn

To be precisely dened Distance between two covariance matrices Average of covariance matrices

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Reection coecients parametrization Riemannian metric and curvature Riemannian distance Geodesics

1

Introduction and Background : Radar Target Detection Radar observation values Standard method Geometric method

2

Geometry of Covariance Matrices Reection coecients parametrization Riemannian metric and curvature Riemannian distance Geodesics

3

Riemannian Median Average of points in a Riemannian manifold Computation of Riemannian median

4

Simulation

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Reection coecients parametrization Riemannian metric and curvature Riemannian distance Geodesics

Reection coecients by means of autoregressive model Autoregressive model : zk + = ek + − 1

Geometry of Covariance Matrices and Computation of Median

1

Pk

i = aik zk + −i 1

1

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Reection coecients parametrization Riemannian metric and curvature Riemannian distance Geodesics

Reection coecients by means of autoregressive model Autoregressive model : zk + = ek + − Minimize prediction error : E[|ek + | ] 1

1

1

Geometry of Covariance Matrices and Computation of Median

2

Pk

i = aik zk + −i 1

1

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Reection coecients parametrization Riemannian metric and curvature Riemannian distance Geodesics

Reection coecients by means of autoregressive model P Autoregressive model : zk + = ek + − ki= aik zk + −i Minimize prediction error : E[|ek + | ] Optimal prediction coecients : (ak , ..., akk ) 1

1

1

1

1

2

1

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Reection coecients parametrization Riemannian metric and curvature Riemannian distance Geodesics

Reection coecients by means of autoregressive model P Autoregressive model : zk + = ek + − ki= aik zk + −i Minimize prediction error : E[|ek + | ] Optimal prediction coecients : (ak , ..., akk ) 1

1

1

1

1

2

1

Denition

µk = akk ∈ D = {z ∈ C : |z | < 1}

is called the k-th reection coecient.

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Reection coecients parametrization Riemannian metric and curvature Riemannian distance Geodesics

Reection coecients parametrization Change of coordinates ϕ:

n −→ R∗+ × Dn− , Rn 7−→ (r

THDP

1

is a dieomorphism.

Geometry of Covariance Matrices and Computation of Median

0

, µ1 , . . . , µn−1 )

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Reection coecients parametrization Riemannian metric and curvature Riemannian distance Geodesics

Reection coecients parametrization Change of coordinates ϕ:

n −→ R∗+ × Dn− , Rn 7−→ (r

THDP

1

is a dieomorphism.

0

, µ1 , . . . , µn−1 )

Computation of ϕ : det Sk µk = (−1)k , where det Rk

Geometry of Covariance Matrices and Computation of Median

S

k = Rk +

 1

2, . . . , k + 1 . 1, . . . , k 

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Reection coecients parametrization Riemannian metric and curvature Riemannian distance Geodesics

Reection coecients parametrization Computation of ϕ− : 1

r0

r

1

r1

= −P0 µ1 ,

+ αkT−1 Jk −1 Rk−−11 αk −1 , Q = P0 ik=−11 (1 − |µi |2 ),

k = −µk Pk −

where Pk −

= P0 ,

1



  αk −1 =  ... 

k−

r

0 ... 0 1 0 . . . 1 0 . =   



r1

2 ≤ k ≤ n − 1,

and

1

Geometry of Covariance Matrices and Computation of Median

J

k−

1



... 1 ... 0 0

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Reection coecients parametrization Riemannian metric and curvature Riemannian distance Geodesics

Riemannian metric and curvature of THPDn

Kähler potential : Φ(Rn ) = − ln(det Rn ) − n ln(π e )

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Reection coecients parametrization Riemannian metric and curvature Riemannian distance Geodesics

Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Riemannian metric and curvature of THPDn

Kähler potential : Φ(Rn ) = − ln(det Rn ) − n ln(π e ) Riemannian metric 2

ds

where (r , µ 0

1

2

=n

dr0

+

n −1 X

(n − k )

k= , . . . , µn− ) = ϕ(Rn ). 2 r 0

1

|d µk |2 , (1 − |µk |2 )2

1

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Reection coecients parametrization Riemannian metric and curvature Riemannian distance Geodesics

Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Riemannian metric and curvature of THPDn

Kähler potential : Φ(Rn ) = − ln(det Rn ) − n ln(π e ) Riemannian metric 2

ds

where (r , µ 0

1

2

=n

dr0

+

n −1 X

(n − k )

k= , . . . , µn− ) = ϕ(Rn ). 2 r 0

1

|d µk |2 , (1 − |µk |2 )2

1

Curvature THPDn is a Cartan-Hadamard manifold with −4 ≤ K ≤ 0.

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Reection coecients parametrization Riemannian metric and curvature Riemannian distance Geodesics

Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Riemannian distance of THPDn Riemannian distance x

= (P , µ1 , . . . , µn−1 ),

y

= (Q , ν1 , . . . , νn−1 ). Then the

Riemannian distance between x and y is given by  ( , )=

d x y

n

2

σ(P , Q ) +

n −1 X k=

(n − k )τ (µk , νk )

2

1/2 ,

1

νk −µk 1 1 + | −¯ µk ν k | where σ(P , Q ) = | ln( )| and τ (µk , νk ) = ln . k −µk | P 2 1 − | ν−¯ µ ν Q

1

1

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k k

Reection coecients parametrization Riemannian metric and curvature Riemannian distance Geodesics

Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Geodesics of THPDn Geodesics x

= (P , µ1 , . . . , µn−1 ),

v

= (v0 , v1 , . . . , vn−1 ) ∈ Tx . The geodesic

starting from x with velocity v is given by

ζ(t , x , v ) = (ζ0 (t ), ζ1 (t ), . . . , ζn−1 (t )),

where ζ (t ) = Pe P t and for 1 ≤ k ≤ n − 1, v0

0

2|vk |t

ζk (t ) =

(µk + e i θk )e 1−|µk |2 + (µk − e i θk ) 2|v |t

k (1 + µ ¯k e i θk )e 1−|µk |2

+ (1 − µ ¯k e i θk )

Geometry of Covariance Matrices and Computation of Median

,

θk = arg vk .

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Average of points in a Riemannian manifold Computation of Riemannian median

1

Introduction and Background : Radar Target Detection Radar observation values Standard method Geometric method

2

Geometry of Covariance Matrices Reection coecients parametrization Riemannian metric and curvature Riemannian distance Geodesics

3

Riemannian Median Average of points in a Riemannian manifold Computation of Riemannian median

4

Simulation

Geometry of Covariance Matrices and Computation of Median

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Average of points in a Riemannian manifold Computation of Riemannian median

Variational formulation Let a , ..., aN ∈ M . 1

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Average of points in a Riemannian manifold Computation of Riemannian median

Variational formulation Let a , ..., aN ∈ M . For α > 0, 1

ˆ = arg min

xα

N X

x ∈M k =

Geometry of Covariance Matrices and Computation of Median

( ,

d x a

k )α .

1

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Average of points in a Riemannian manifold Computation of Riemannian median

Variational formulation Let a , ..., aN ∈ M . For α > 0, 1

ˆ = arg min

xα

N X

x ∈M k =

( ,

d x a

k )α .

1

Two notions of average α = 2 : mean (center of mass)

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Average of points in a Riemannian manifold Computation of Riemannian median

Variational formulation Let a , ..., aN ∈ M . For α > 0, 1

ˆ = arg min

xα

N X

x ∈M k =

( ,

d x a

k )α .

1

Two notions of average α = 2 : mean (center of mass) α = 1 : median

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Average of points in a Riemannian manifold Computation of Riemannian median

Variational formulation Let a , ..., aN ∈ M . For α > 0, 1

ˆ = arg min

xα

N X

x ∈M k =

( ,

d x a

k )α .

1

Two notions of average α = 2 : mean (center of mass) α = 1 : median Which one is preferable for radars ?

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Average of points in a Riemannian manifold Computation of Riemannian median

Comparison of sensitivity for outliers

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Average of points in a Riemannian manifold Computation of Riemannian median

Comparison of sensitivity for outliers

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Average of points in a Riemannian manifold Computation of Riemannian median

Comparison of sensitivity for outliers

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Average of points in a Riemannian manifold Computation of Riemannian median

Comparison of sensitivity for outliers

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Average of points in a Riemannian manifold Computation of Riemannian median

Comparison of sensitivity for outliers

Proposition (Fletcher et al. 2009) In order to move the median of a set of N points in a Riemannian manifold, one should move at least N /2 points in this set. Geometry of Covariance Matrices and Computation of Median

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Average of points in a Riemannian manifold Computation of Riemannian median

Comparison of sensitivity for outliers

Proposition (Fletcher et al. 2009) In order to move the median of a set of N points in a Riemannian manifold, one should move at least N /2 points in this set. Median is preferable. Geometry of Covariance Matrices and Computation of Median

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Average of points in a Riemannian manifold Computation of Riemannian median

Non-dierentiable minimization problem Objective function : f (x ) =

PN

Geometry of Covariance Matrices and Computation of Median

k=

1

( ,

d x a

k)

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Average of points in a Riemannian manifold Computation of Riemannian median

Non-dierentiable minimization problem Objective function : f (x ) = Nk= d (x , ak ) Problem : f is not dierentiable at points a , . . . , aN P

1

1

grad f (x ) =

Geometry of Covariance Matrices and Computation of Median

N − exp−1 a X k k=

x

1

( ,

d x a

k)

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Average of points in a Riemannian manifold Computation of Riemannian median

Non-dierentiable minimization problem Objective function : f (x ) = Nk= d (x , ak ) Problem : f is not dierentiable at points a , . . . , aN P

1

1

grad f (x ) =

N − exp−1 a X k k=

x

1

( ,

d x a

k)

Solution : use the subgradient of f ( )=

H x

N X k = ,ak 6=x 1

− exp− x 1 ak d (x , ak )

In particular, H (x ) = 0 =⇒ x is the median Geometry of Covariance Matrices and Computation of Median

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Average of points in a Riemannian manifold Computation of Riemannian median

Subgradient algorithm

Algorithm

initialization do while

x

k+

1

x1

∈M

= expxk (−tk ( k ) 6= 0

H (xk ) ) |H (xk )|

H x

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Average of points in a Riemannian manifold Computation of Riemannian median

How the algorithm works ?

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Average of points in a Riemannian manifold Computation of Riemannian median

Convergence theorem Theorem

λ > 0 such that if (0, λ) and verifying

There exists a constant chosen in the interval

lim t = 0 k →∞ k then

(xk )k

and

∞ X

k=

converges to the median of

Geometry of Covariance Matrices and Computation of Median

t

the stepsizes

k = +∞

1

{a1 , . . . , aN }.

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(tk )k

are

Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Average of points in a Riemannian manifold Computation of Riemannian median

Convergence theorem Theorem

λ > 0 such that if (0, λ) and verifying

There exists a constant chosen in the interval

lim t = 0 k →∞ k then

(xk )k

and

∞ X

k=

converges to the median of

n:

THPD

λ = 1/4,

t

the stepsizes

k = +∞

1

{a1 , . . . , aN }.

k = 1/(4k )

t

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(tk )k

are

Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

1

Introduction and Background : Radar Target Detection Radar observation values Standard method Geometric method

2

Geometry of Covariance Matrices Reection coecients parametrization Riemannian metric and curvature Riemannian distance Geodesics

3

Riemannian Median Average of points in a Riemannian manifold Computation of Riemannian median

4

Simulation

Geometry of Covariance Matrices and Computation of Median

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Initial spectra

Fig. 1: Initial Spectra

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Median spectra

Fig. 2: Median Spectra

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Mean spectra

Fig. 3: Mean Spectra

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Target detection by median

Fig. 4: Target detection by median

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

References

L. Yang Riemannian Median and Its Estimation, submitted to LMS J. Comput. and Math. T. Fletcher et al. Robust Statistics on Riemannian Manifolds via the Geometric Median, Neuroimage 2009.

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Introduction and Background Geometry of Covariance Matrices Riemannian Median Simulation

Thank you for your attention

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