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Applications Radarde la Géométrie de l'information
associée aux matrices de covariances : traitements spatio-temporels
Frédéric BARBARESCOThales Air Operations
Domaine Surface Radar, Direction TechniqueDépartement Développements Avancés
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Radar Applications ofInformation Geometry
based on covariance matrices : Space-Time Processing
Frédéric BARBARESCOThales Air Operations
Domaine Surface Radar, Direction TechniqueDépartement Développements Avancés
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Radar Signal Processing :Doppler Processing
4 /4 / Doppler Sensors : Radar, Sonar & Lidar
Radar (RAdio Detection And Ranging) Technology
107 Years old (C. Hülsmeyer, 1904)
Sonar (SOund Navigation And Ranging)
(piezoelectric effect) 95 years old (Paul Langevin & Constantin Chilowski, 1916)
Lidar (LIght Detection And Ranging)Technology
(laser) 51 years old (T. Maiman, 1960)
5 /5 / All these sensors use Doppler-Fizeau Effects
Woldemar Voigt (1850 - 1919)
Armand Hippolyte Louis Fizeau (1819 – 1896)
Christian Andreas Doppler (1803- 1853)
freq Radial Velocity(Doppler Spectrum Mean)
Var(freq) Turbulence (Doppler Spectrum Width)
6 /6 / Radar Processing based on Covariance Matrix
Doppler Processing(Time Covariance Matrix)
Antenna Processing(Space Covariance Matrix)
STAP Processing(Space-Time Covariance Matrix)
Polarimetric Radar Processing(Polarimetry Covariance Matrix)
7 /7 /
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Military Air Defense Application
Detection of slow/stealth targets in inhomogeneous Clutter
New requirements in Air Defense to detect low altitude or surface targets at low elevation. Target Doppler is very close to fluctuating Clutter Doppler (Ground & Sea Clutters)
Detection of asymmetric & stealth targets in Ground Clutter Microlight Airplane General Aviation UAV & Micro UAV Micro Helicopter
Detection of small targets in Sea Clutter Wooden & inflatable canoe Jetski Unmanned Boat Naval Micro Helicopter Periscope
Detection of low RCS targetsDetection of teneous Doppler SignalIncrease Range & Reactivity
8 /8 /
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Civil Air Traffic Control Application
Monitoring of turbulences: New Requirement in ATC (SESAR)
In Turbulences, signal is no longer characterized by Doppler velocity Mean but by Doppler Spectrum Width & “shape”
Atmospheric Air turbulences Eddy Dissipation Rate
Turbulent Kinetic Energy
Airplane Wake-Vortex turbulences (A380, B747-8) Circulation
Decay Rate
Windshear in Final Approach Headwind
Crosswind
Improve Safety by mitigating weather hazardsIncrease Capacity by reducing safe separations
9 /9 /
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Challenges of Doppler Radar Processing: Detection on Ground
Detection of asymmetric & stealth targets in Inhomogenesous Ground Clutter
Classical Doppler Filter Banks (or FFT) are not efficient with very short bursts (<16 pulses) : Low Resolution of Doppler Filters with short Bursts (Low
sidelobes / high loss, wide filter)
If Target Doppler is between two Doppler filters, energy is spread on adjacent filters. Gain between 2 filters is lower thangain at filter center ("Straddling loss")
Ground Clutter Energy is not limited to zero-Doppler filter but pollution is spread over all filters due to poor Filter-Banks Resolution & Doppler side lobes in case of very short Bursts.
Filter 0 Filter 1 Filter 7Filter 2 Filtre 6Filter 3 Filter 4 Filter 5
Pollution of all Doppler Filters in case of Burst with low number of pulses
0.5 0.6 0.7 0.8 0.9 1e normalisee (V/Va)
0 0.1 0.2 0.3 0.4 0.50
0
0
0
0
0
0
0
Frequence normalisee
S/N gain [dB]
0-10-20-30-40-50-60 V/ Vamb
10 /10 /
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Challenges of Doppler Radar Processing:Detection in Sea
Detection of slow targets in Sea Clutter
Sea Clutter is highly inhomogeneous Doppler fluctuation
Time/space Fluctuation
Sea Clutter is dependant of Sea current
Surface wind
fetch
Bathymetry
Sea Clutter is corrupted by Spikes due to breaking waves
“Moutonement”
Offshore SeaDoppler Spectrum
Close to the Shore Sea Doppler Spectrum
(Breaking waves)
11 /11 /
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Challenges of Doppler Radar Processing: Atm. turbulences
Monitoring of atmospheric turbulences
Air turbulence is characterized by Spread of fluctuating speeds
This composition of different Doppler speeds in Radar cells generates a Widen Doppler Spectrum
Speed variance of Doppler Spectrum Width are related to 2 measures of turbulence : EDR: Eddy Dissipation Rate
TBE: Turbulent Kinetic Energy
12 /12 /
Monitoring of Wake-Vortex Turbulences
Wake vortex generate to contra-rotative roll-up spirals
Mean speed depends on cross-wind Wake-Vortex has spiral geometry with
increasing speed in the core and decreasing speed outside the core
Wake-Vortex Speed and structure depend on Wake-Vortex age/decay phase
Wake-Vortex Strength is characterized by Circulation in m2/s
Monitoring of Windshear
Inversion of speed in range or in altitude Microburst in the same radar cell
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Challenges of Doppler Radar Processing: Wake Turbulences
Wake-Vortex Doppler Spectrum
Wind-ShearDoppler Spectrum
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Probability Metrics: Information Geometry
& Optimal Transport
14 /14 /
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Probability Metrics: Distance between 2 random variables
Question in Probability and Statistic: Could we define distance between 2 random variables ?
Or equivalently, could we define distance between 2 probability densities of these 2 random variables ?
)1(
)1(1
)1(2
)1(1
)1(
n
n
zz
zz
Z
)2(
)2(1
)2(2
)2(1
)2(
n
n
zz
zz
Z ???, )2()1( ZZd
)1()1( /Zp )2()2( /Zp ???, )2()1( d
15 /15 / 2 Definitions of Probability Metric
What is the good geometry : geometry with best properties for our applications in Radar Processing
Maurice René Fréchet
1957 (CRAS)Fréchet-Levy Distance
Optimal Transport Theory
1939 (IHP Lecture)Fréchet Bound (Cramer-Rao)
Information Geometry
1948 (Annales de l’IHP)Les éléments aléatoires de nature quelconque
dans un espace distanciéExtension of Probability/Statistic in abstract space
1928
16 /16 / Information Geometry: Fréchet-Darmois-Cramer-Rao Bound
Information Geometry :
Fisher Information Matrix and Fréchet-Darmois-Cramer-Rao Bound
19451943
(1939 IHP Lecture)
1ˆˆ
IE
FDCR Bound
*
2
, /ln)(
jiji
XpEI
Fisher Information Matrix R.A. Fisher
17 /17 / Information Geometry : Rao-Chentsov metric
Information Geometry :
Rao-Chentsov Metric defined with Kullback divergence
Kulback-Leibler Divergence :
Rao-Chentsov Metric (invariance by parameter changes)
jijiji ddgdIddXpXpKds
,
*,
2 )(/,/
jiji Ig , , )(
1945Calyampudi Radhakrishna Rao Nikolai Nikolaevich Chentsov
1964 (axiomatisation)
dxxqxpxpeEESupqpK qp
//ln/ln),(
18 /18 / Dual Information GeometryDual Coordinates systems & Potential functions
Potential Functions are Dual and related by Legendre transformation :
enΗ)(ηΗηΗΦ
nTrΘΨ
mmmη,ΗΗ
Σ)m,(Σθ,ΘΘ
T
T
T
2log2detlog21log2~~)log(2detlog22~~
,~2~
scoordinate Dual
1111
1112
11
ΘΗΦ
θηΦ
Η
ΘΨ
ηθΨ
~
~
and~
~
TT ΘΗTrH,Θ
ΨH,ΘΦ
~~with
~~~~
pEΗΦ log~~ Entropy
ji
*ij
jiij ΗΗ
ΦgΘΘΨg
~ and
~ 22
Hessians are convexe and define Riemannian metrics :
19 /19 / Combinatorial/Variational Foundation of Kullback Divergence
Combinatorial Fundation of Kullback Divergence Kullback Divergence can be naturally introduced by combinatorial elements and
stirling formula :
)(log),(
eEESupqpK qp
in
M
i i
ni
MMM nqNqqnnnP
i
1121 !
!,...,/,...,,
iq
M
ii Nn
1 Nnp i
i
n when ..2..! nenn nn
),(log.log11
qpKqppP
NLim
M
i i
iiMN
Let multinomial Law of N elements spread on M levels
with priors , and
Sirling formula gives :
Variational Foundation of Kullback Divergence Donsker and Varadhan have proposed a variational definition of Kullback divergence :
20 /20 / Kullback Divergence & VARADHAN’s Variational Approach
)(log),(
eEESupqpK qp
Donsker and Varadhan have proposed a variational definition of Kullback divergence :
),()1ln(),()()()(ln
)()(ln)()(ln)(
)()(ln)( :Consider
qpKqpKqpq
qppeEE
qp
qp
0)()(ln)()(ln)(),(
)()()(with
)()(ln)(
)(ln)(ln)(
)(
)(
qppeEEqpK
eqeqq
qqp
eEeEeEE
qp
qpqp
This proves that the supremum over all is no smaller than the divergence
Using the divergence inequality,
Link with « Large Deviation Theory » & Fenchel-Legendre Transform which gives that logarithm of generating function are dual to Kullback Divergence :
)(
(.)
)(
(.)
)(
log)(),(
)(log)()(),(
),()()()(log
xVqp
V
xV
V
p
xV
eEVESupqpK
dxxqedxxpxVSupqpK
qpKdxxpxVSupdxxqe
21 /21 / Optimal Transport Theory: Wasserstein distance
Wasserstein distance
Framework of optimal transport theory
Particular cases of Wasserstein distance
Case n = 1 : Monge-Kantorovich-Rubinstein distance
Case n=2 : Fréchet distance
)(,)(,,,
),(),(,
/1
/1
),(
YlawXlawYXdEInfW
yxdyxdInfW
nnn
n
nn
YXEInfQPdWYX
,
1 ,),(
2
,2 ,),( YXEInfQPdW
YX
Cédric Villani, Field Medal 2010, Book on « Optimal Transport
Theory : Old & New »
22 /22 / Optimal Transport Theory : Fréchet-Wasserstein distance
Fréchet distance
In 1957, Maurice René Fréchet has introduced a distance, based on Paul Levy’s paper (particular case of Wasserstein distance)
dxdyyxHyxYXEInfGFdYX
),()(, 22
,
2
23 /23 / Optimal Transport Theory : Fréchet-Wasserstein distance
Fréchet Distance
Fréchet’s paper from 1957 in CRAS :
Paul Levy’s letter to Maurice Fréchet (2nd of April 1958) … J’ai ainsi pu apprécier ce que vous aviez fait, en prenant comme point de départ de votre
mémoire ce que vous appelez ma première définition de la distance de deux lois de probabilité(en fait ce n’était pas la première). Vous l’avez d’ailleurs généralisée, en ce sens que je ne l’avais associée qu’à une de vos définitions de deux variables aléatoires. Et j’ai beaucoup admirécomment avec votre quatrième définition, vous arrivez à faire quelque chose de maniable d’une idée qui pour moi était surtout théorique, vu la difficulté de déterminer le minimum de la distance de deux variables aléatoires ayant les répartitions marginales données.
24 /24 / 4th definition of Fréchet distance
4th Fréchet Distance
Extreme Fréchet Copulas
4th Fréchet’s Distance
)(),(),(with
),(,
1
122
yGxFMinyxH
dxdyyxHyxGFd
)(),(),(
0,1)()(),(with
),(),(),(
1
0
01
yGxFMinyxHyGxFMaxyxH
yxHyxHyxH
25 /25 / Application for Multivariate Circular Gaussian Law of 0 mean
Model : Multivariate Circular Gaussian Law of zero mean
Rao-Chentsov(-Siegel) Metric & distance
Fréchet-Levy(-Wasserstein) distance
RRE
mZmZRe
RRmZp RRTr
n ˆ
ˆ with
det1,/
1ˆ
0m
22/12/1212
FdRRRdRRTrds
Case with
0det...detwith
ln..log,
2/12/11
222/12/12
XYXYX
n
kkFXYXYX
RRIRRR
RRRRRd
2/12/12/12 .2, XYXYXYX RRRTrRTrRTrRRd
26 /26 / Properties of each geometry
Information Geometry
Geodesic
Space with sectional Negative Curvature
Invariance by parameters changes
Wasserstein Geometry
Geodesic
Space with sectional Positive Curvature
YXYX
Xt
XYXXXRRRt
X
RRRRRRRRRReRt XYX
)2/1( and )1( , )0()( 2/12/12/12/12/1log2/1 2/12/1
12/12/12/12/12/1
,
,,)(
with
.)1(.)1(
YXXXYXXYX
YXkYYXkt
RRRRRRRD
DtItRDtItR
Existence and Unicity of
Barycenter only proved in case of
negative curvature by Elie
Cartan during 20’s
27 /27 / Information Geometry : Multivariate Gaussian Case, m=0
Information Geometry for Multicariate Gaussian Laws(cas m=0)
Rao-Chentsov distance
Particular case of Carl-Ludwig Siegel case (in the framework of symplectic geometry)
Siegel Metric
Invariant by all automorphisms of SHn
Particular Case :
0det...det avec
ln..log,
2/12/11
222/12/12
XYXYX
n
kkXYXYX
RRIRRR
RRRRRd
0Im/),( Y(Z)CnSymiYXZSH n
iYXZZdYdZYTrdsSiegel avec ... 112
nTT IBCDADCZBAZZM avec )( 1
2120dRRTrds
RYX
C.L. Siegel
28 /28 / Example : Monovariate Gauss-Laplace LawGauss-Laplace Law
Fisher Information Matrix for Gaussian Law :
mIEI
T and )(ˆˆ with
2001
.)( 12
2
22
2
2
2
22
2.2.2..
ddmddmdIdds T
Rao-Chentsov Metric from Information Geometry
H. Poincaré
.2
imz 1
iziz
22
22
1.8
dds
This metric is the Poincaré metric (model of hyperbolic geometry)
)*2()1(
)2()1()2()1(
2
)2()1(
)2()1(
22112
1),(with
),(1),(1log.2,,,
mmd
29 /29 / Poincaré Space in Art (Escher, Irène Rousseau )
30 /30 / Poincaré & Siegel Upper Half Plane & Disk
*1*1*2
22
22
11
1
dwwwdwwwds
w
dwds
izizw
1 iIZiIZW
Poincaré Upper Half Plane Poincaré Unit disk
Siegel Upper Half Plane Siegel Unit disk
),( and ),(with
112
CnHPDYCnHermXiYXZ
ZddZYYTrds
0 and avec
*112
2
2
2
222
yiyxzdzdzyyds
ydz
ydydxds
dWWWIdWWWITrds 112
31 /31 / Siegel Space
0Im/),( Y(Z)CnSymiYXZSH n
Siegel Upper Half Space
X
0Y
ZdYdZYTrds 112
n
iiiZZd
1
221
2 1/1log,
kkk YiXZ .
1k
2k
212 dRRTraceds
kkkk RNWRiZ ,0 if .
1k
2k
n
kkRRd
1
221
2 log,
12121
1212121,
ZZZZZZZZZZR
0...det 2/112
2/11 IRRR
0.,det 21 IZZR
L.K. Hua
C.L. Siegel
F. Berezin
32 /32 /Information Geometry based on Poincaré’s hyperbolic geometry
259 Letters between Mittag-Leffler & Poincaré
• « Acta Mathematica » was founded by Gösta Mittag-Leffler in 1882• Henri Poincaré has published paper on Fuchsian Group in first volume of Acta Mathematica: Henri Poincaré (1882) "Théorie des Groupes Fuchsiens", Acta Mathematica v.1, p.1.
33 /33 /Information Geometry based on Poincaré’s hyperbolic geometry
• Henri Poincaré (1882) "Théorie des Groupes Fuchsiens", Acta Mathematica v.1, p.1., 1882 published by Mittag-Leffler
34 /34 /
« At one point Siegel thought that too many unnecessary things were being published, so he decided not to publish anything at all »George PolyaThe Polya Picture Album, Encounters of a Mathematician, Birkäuser
Carl Ludwig SiegelWith George Polya
Carl Ludwig siegel
35 /35 / Information Geometry : Multivariate Gaussian Case, m=0
Information Geometry for Multivariate Gaussian Laws(cas m=0)
Geodesic :
Properties of this space
Symetric Space as studied by Elie Cartan : Existence of bijective geodesic isometry
Bruhat-Tits Space : semi-parallelogram inequality
Cartan-Hadamard Space (Complete, simply connected withnegative sectional curvature Manifold)
10 with ),(.))(,( ,tRRdttRd YXX
YXYX
Xt
XYXXXRRRt
X
RRRRRRRRRReRt XYX
)2/1( and )1( , )0()( 2/12/12/12/12/1log2/1 2/12/1
2/12/12/12/12/11-),( avec )(X ABAAABABABAXG BA
Xx)d(x,x)d(x,xd(x,z)),xd(x
xzxx
224
que tel ,2
22
122
21
21
36 /36 /
x1x
2x
z
U
VU VU V
Information Geometry : Multivariate Gaussian Case, m=0
Symetric Space as studied by Elie Cartan : Existence bijective geodesic isometry
Bruhat-Tits Space : semi-parallelogram inequality
2222 22 VUVUVU
),( AGB BA
BGA BA ),( )(X-1),( BABAXG BA
X
2/1/212/12/12/1 ABAAABA
1,0)( 2/12/12/12/1
tABAAAt t
A)0(
B)1(
22
21
2221 224 )d(x,x)d(x,xd(x,z)),xd(x
E. J. Cartan
M. Berger
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Information Geometry & Fréchet-Karcher
Gradient Flow
38 /38 /
For detection of Slow & Stealth/Small Target in inhomogeneous clutter, we need simultaneously :
High Doppler Resolution with short Bursts Robust CFAR in inhomogeneous clutter & closely separated targets
Proposed Solution : OS-HR-Doppler-CFAR
Avoid drawbacks of Doppler Filters / FFT in case of short bursts Take advantages of Robust Ordered Statistic of OS-CFAR (Ordered
Statistic CFAR, Median-CFAR)
Challenges to define OS-HR-Doppler-CFAR :
Can we order « Doppler Spectrums » : NO there is no total order of covariance matrices R1>R2>…>Rn
There is only Partial « Lowner » order : R1>R2 if and only if R1-R2 Positive Definite
Can we define « Median » of « Doppler Spectrums » : YES !!! In a « Metric Space », the median is defined as the point that minimizes the « geodesic »
distance to each point (compared to the mean that minimizes the square distance to each point)
We can define a deterministic or stochastic gradient flow that converges fastly to « median spectrum » (Modified Karcher Flow : THALES Patent)
Transport Optimal : barycentre de Fréchet-WassersteinOS-HD-Doppler-CFAR
39 /39 /
In Rn, the center of mass is defined for finite set of points
Arithmetic mean:
This point minimizes the function of distances:
The median (Fermat-Weber Point) minimizes :
M
ii
xcenter xxdMinx
1
2 ),(arg
1x
2x
3x
4x
5x
M
ii
xcenter xxMinx
1arg
M
iicenter x
Mx
1
1
Miix ,...,1
M
iicenter x
Mx
1
1
M
ii
xcenter xxdMinx
1
2 ),(arg
Center of Mass : Arithmetic Mean and Median in Rn
M
ii
xmedian xxdMinx
1),(arg
1x
2x
3x
4x
5x
M
ii
i
xmedian
xx
xxMinx1
arg
mxEMinmmmedian 2mxEMinm
mmean
M
ii
xmedian xxdMinx
1),(arg
40 /40 /
Median Mean
x1
x2
x3
x1
x2
x3
Median Mean
x1
x2
x3
x1
x2
x3
x’2 x’2
outliers outliers
MEDIAN MEAN
Sensitivity to outliers : Median versus Mean
41 /41 / Extension of barycenter in metric space
Right Triangle
h2=a2+b2
Naive Mean of N Right Triangles :
Let N Right triangles :
« Arithmetic » Mean is not a Right triangle
Solution : Fréchet Mean (Center of Mass)
Consider the surface h2=a2+b2 in Space of coordinates (a,b,h)
Fréchet Mean/Barycenter/Center-of-Mass
2221 with ,, iii
Niiii bahhba
222
111
,,1,1,1
BAH
HBAhN
bN
aN
N
ii
N
ii
N
ii
22221
2
,,
/,, surfaceon defined (.,.)
,,,,,arg,,
bahhbad
HBAhbadMinHBA
geodesic
N
iiiigeodesicHBA
a
b
h
One Right Triangle could be represented by 1 point on
surface h2=a2+b2
42 /42 / Cartan Center of Mass and Karcher Flow
Cartan Center of Mass
Elie Cartan has proved that the following functional :
is strictly convexe and has only one minimum (center of mass of A for distribution da) for a manifold of negative curvature
Karcher Flow
Hermann Karcher has proved the convergence of the following flow to the Center of Mass :
E. J. Cartan
H. Karcher
A
daamdmf ),(: 2
)()0( avec )(.exp)(1 nnnnmnnn mfmfttmn
)(exp 1 A
m daaf
43 /43 / probability on a manifold : Emery’s exponential barycenter
Maurice René Fréchet, inventor of Cramer-Rao bound in 1939, has also introduced the entire concept of Metric Spaces Geometry and functional theory on this space (any normed vector space is a metric space by defining but not the contrary). On this base, Fréchet has then extended probability in abstract spaces.
In this framework, expectation of an abstract probabilisticvariable where lies on a manifold is introduced by Emery as an exponential barycenter :
In Classical Euclidean space, we recover classical definition of Expectation E[.] :
xyyxd ),(M. R. Fréchet, “Les éléments aléatoires de nature quelconque dans un espace
distancié”, Annales de l’Institut Henri Poincaré, n°10, pp.215-310, 1948
xgEb )(xg x
0)(exp 1 dxPxgM
b
nn RX
Rp
n dxxpxgdxPxgxgEpqqRqp )()()()()()(exp, 1
M. Emery & G. Mokobodzki, “Sur le barycentre d’une probabilité sur une variété”, Séminaire de Proba. XXV, Lectures note in Math. 1485, pp.220-233, Springer, 1991
44 /44 / Mean & Median of N matrices HPD(n)
Mean of N Hermitian Positive Definite Matrices HPD(n)
Solution given by Karcher Flow with Information Geometry metric
Median (Fermat-Weber Point) of N matrices HPD(n)
PhD Yang Le supervised by Marc Arnaudon (Univ. Poitiers/Thales)
N
kk
N
kk
Xmoyenne XBXBXdMinArgX
1
22/12/1
1
2 ..log,
2/1
log2/1
11
2/12/1
n
XBX
nn XeXX
N
knkn
N
kk
N
kk
Xmediane XBXBXdMinArgX
1
2/12/1
1..log,
knnn
XBXXBX
nn BXkSXeXX nSk nkn
nkn
/ with 2/1loglog
2/11
2/12/1
2/12/1
45 /45 / Median on a Manifold: Karcher-Cartan-Fréchet
Gradient flow on Surface/Manifold
Gradient Flow :Pushed by Sum of Normalized Tangent vectors of
Geodesics
Fréchet-Karcher Barycenter : Sum of Normalized Tangent vectors
of Geodesics is equal to zero
A m
m
MinA
daaahdaamdmh
)(exp)(exp),(
21:
1
1
Sum of Normalized Tangent Vectors
)(exp
exp.exp
11
1
1
M
k km
kmmn x
xtm
n
n
n
Compute point on the surface In the direction of sum of Normalized
Tangent Vector
46 /46 / Mean : Karcher Barycenter
2/1log2/12/1 2/12/12/1 2/12/1
)( XeXXXBXXt XBXttkk
k
2/12/12/12/10 log)( XXBXX
dttd
ktk
N
k
N
kkt
k XXBXXdt
td1
2/1
1
2/12/12/10 0log)(
1B 2B
3B
4B
5B
6B
47 /47 / Comparison of Mean & Median Doppler Spectrum
Raw Doppler Spectrum
Mean Doppler Spectrum
Median Doppler Spectrum
Range axis
Doppler axis
No preservation of discontinuities
Perturbation by outlier
48 /48 /
Diffusion Fourier Equation on 1D graph (scalar case) Approximation par un Laplacien discret :
with arithmetic mean of adjacent points :
Dicrete Fourier Heat Equation for Scalar Values in 1D :
By Analogy, we can define Fourier Heat Equation in 1D grapf of HDP(n) matrices :
nnnnnn uu
xxuu
xuu
xtu
xu
tu
ˆ21
211
2
2
2/ˆ 11 nnn uuuJ. Fourier
Other approach to smooth spectrum : Fourier Heat Equation
2,,1,.2 with ˆ.).1(x
tuuu tntntn
tntntntntntntntn
tntntntntntnXXX
tntn
XXXXXXXX
XXXXXXeXX tntntn
,12/1,12/1
,12/12/1
,1,12/1,1
2/1,1,
2/1,
2/1,,
2/1,
2/1,
2/1,
ˆlog2/1,1,
ˆwith
ˆ2/1,,
2/1,
49 /49 / Isotropic Diffusion of Doppler Spectrum
50 /50 / Anisotropic Diffusion of Doppler Spectrum
51 /51 /
Geodesic between matrix P and Geodesic between Q & R :
The geodesic projection is a contraction :
Geodesic Projection
Q
R
PP
Q
2/1
2/12/1 2/12/12/12/12/1,)( PPQRQQQPPtstts
s
QPdistQPdist MM ,)(),(
M : geodesically convex closed submanifold of
Hadamard Manifold SPdistP
MSM ,minarg)(
M(P)
M(Q)
P
Q
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Optimal Transport Theory&
Gradient Flow
53 /53 / Optimal Transport : Fréchet-Wasserstein distance
Wasserstein Distance for Multivariate Gaussian Laws
Fréchet-Wasserstein distance
Proof If we set
Solution is given by:
YXYX RTrRTrRTrYEXEYXE
YXYXTrEYXE
,22
2
2)()(
0
Y
XW R
RR
YX
W
2/12/12/12/12/1
0..
1
XXYXX
RRRRRRRTrSup
YX
2/12/12/1 ...2 XYX RRRTrTr
XRRRRRY XXYXX ... 2/12/12/12/12/1
54 /54 / Optimal Transport : Fréchet-Wasserstein distance
Wasserstein Distance for Multivariate Gaussian Laws
Fréchet-Wasserstein distance
Geodesic If we set :
Optimal Transport : transport from to
2/12/12/122 .2,,, XYXYXYXYYXX RRRTrRTrRTrmmRmRmd
12/12/12/12/12/1,
YXXXYXXYX RRRRRRRD
YXkYYXkt
XYt
DtItRDtItR
mtmtm
,,)(
)(
.)1(.)1(
.)1(
)1()1()0()0(2)()()()(2 ,,,).(,,, RmRmWstRmRmW ttss
YYk RmNI ,, # YY RmN , XX RmN ,
XYYX mmyDyx ,)(
XYYXY mmvDmvv ,2
1)(
YYYXXx myRmymxRmx 11
55 /55 / Optimal Transport : Fréchet-Wasserstein Distance
Characteristics of this space (case m=0)
Wasserstein metric :
Tangent Space and Exponential Map :
Properties of this space
Alexandrov space
space Positive Sectional Curvature (geodesiquely convexe and simply connected)
YRXTrYXg YRN Y..),(),0(
)(,,0 ,0).(exp tRRN YYRNXt
XtItRXtItR kYktRY.)1(.)1()(,
VTYRN ,0
).(exp ,0 XtYRN
YRRN ,0
2222 )1(),0().1()(,.)0(,).1()(, dtttdtdttd
56 /56 / Optimal Transport : Fréchet-Wasserstein Barycenter
Wasserstein Barycenter for Multivariate Gaussian Laws(Case m=0)
Fréchet-Wasserstein Barycenter
Solution of Fréchet-Wasserstein Barycenter for N Multivariate Gaussian Laws of zero meams Constraint :
Iterative Solution Iterative solution (convergence for d=2, d>3 convergence conditionaly to the
initiation)
2
,2
1
22 ),( with ),( YXEInfWWInf
YX
N
kk
NkkRN 1,0
RRRRN
kk
1
2/12/12/1
with 2/12/1
1
2/1)(2)()1(ii
N
k
nk
nn RKKKKK
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Stationnary Signal &Toeplitz Constraint on
Covariance Matrix :Gradient Flow through Partial
Iwasawa & CAR Model
58 /58 /
Covariances matrices are structured matrices that verify the following constraints : Toeplitz Structure (for stationary signal) :
Hermitian structure :
Positive Definite Structure :
How to built a flow that preserves the Toeplitz structure ?
0121
*10
*212
*101
*1
*2
*10
0121
*10
*212
*101
*1
*2
*10
rrrrrrrrr
rrrrrrr
rrrrrrrrr
rrrrrrr
R
n
n
n
n
n
1nR
conjuguate and d transpose: with nn RR
00det such that ,0, nnnn λ.IRZRZCZ
Additional Constraint : Toeplitz Structure
kknn rzzEn * ,
If you remember first example on « Right Triangle », in this case Pythagore constraint is replaced by HPD & Toeplitz constraints
59 /59 / Preservation of Toeplitz Structure
We use Partial Iwasawa Decomposition = Complex AR model
Information Geometry metric (metric = Hessian of Entropy)
1111
121
.1
.1..nnnn
nnnnnnn AAA
AWWR
11
21 .1
nnn
1011)(
nn
nn
Aμ
AA *)( .VJV
Tnn P 110
)(
)*()(
2 ~n
jn
iijg
100P
..ln.1ln).(.logdetlog,~0
1
1
210 PenkneR)P(R
n
kknn
with
1
122
22
0
0)()(2
1)(.
n
ii
inij
nn
din
PdPndgdds
TNN
NNkknn
N
knkn
Nkn aaAbbEbzaz )()(
12
0,*
1
)( and with
K. Iwasawa
E. Kähler
S. Bergman
1t with coefficien reflection : kk
60 /60 / Complex Autoregressive Model : Regularized Burg Algorithm
Regularized Burg Algorithm (THALES Patent)
)(.)1()( )1(.)()(
11 , .
1
).()2.( with ..2)1()(1
...2)1().(2 to1For : (n) tep .
1
)(.1ech.) nb. : (N 1 , )()()(f
:tion Initialisa .
1*
1
11
)(
)*1()1()(
)(0
221
)(
1
1
0
2)1()(21
21
1
1
1
)1()1()(*11
)0(0
1
20
00
kfkbkbkbkfkf
a
,...,n-k=aaa
a
nkakbkf
nN
aakbkfnN
MnSa
kzN
P
,...,N k=kzkbk
nnnn
nnnn
nn
n
nknn
nk
nk
n
nkN
nk
n
k
nk
nknn
N
nk
n
k
nkn
nk
nknn
n
N
k
61 /61 / Median AR model Through Median Reflection Coefficients k
n,3
n,1n,2
nmedian,
Classical Karcher Flow in Unit Disk
1,3 n
1,1 n
Duale Karcher Flow
εμl/ avec μμ
γw l,n
m
lkk k,n
k,nnn
1
nnmedian w1,
*nk,n
nk,nk,n .wμ
wμμ
11
1,2 n
*nmedian,n
nmedian,nmedian,n wμ
wμμ
11
62 /62 / Median AR model Through Median Reflection Coefficients k
63 /63 / Alternative to Karcher Flow : Arnaudon Stochastic Flow
Random selection at each iteration of 1 point in the
disk + evolution along the geodesic
nnrand
nnrandnn γw
),(
),(
)(exp
exp.exp
)(1
)(1
1nrandm
nrandmnmn x
xtm
n
n
n
M. Arnaudon, C. Dombry, A.Phan, L.Yang, ”Stochastic algorithms for computing means of probability measures”, http://hal.archives-ouvertes.fr/hal-00540623
64 /64 /
nnrand
nnrandnn γw
),(
),(
)(exp
exp.exp
)(1
)(1
1nrandm
nrandmnmn x
xtm
n
n
n
Arnaudon Stochastic Flow
65 /65 / OS-HR-Doppler-CFAR
FFTI&Q
log|.|
Scalar OS CFAR
>S
log|.|
Scalar OS CFAR
>S
OU
I&QRegularized
AR
Model
-
-
Robust
Rao
Distance
Median CFAR on
>S
CfenêtreTFAiimiiP
,,1,0 ,...,,
2
1*
,,
,,
2
,0
,0,,1,0,,1,0 ].[1
arglog,...,,,,...,,
m
n mediannjn
mediannjn
j
medianmedianmmedianmedianimii μ
thn)(mP
PmPPdist
Classical
Chain
OS-HRD
CFAR
66 /66 / Figures of Merit (sea clutter)
COR curve
Red : OS-HR-DOPPLER-CFAR based on CAR median
Black : Classical Doppler Filter & OS-CFAR
Gain of
New processing
chain
First tests on real recorded
sea clutter
67 /67 / Figure of Merit for Closely Separated Targets (sea clutter)
COR Curves
Gain of
New processing
Chain
Red : OS-HR-DOPPLER-CFAR based on CAR median
Black : Classical Doppler Filter & OS-CFAR
First tests on real recorded
sea clutter
68 /68 /In Progress: tests on real Air Defense Radar Ground Clutter records
Doppler Filters
High ResolutionDoppler Filters
Waveform 1 Waveform 2
69 /69 /
Robustness is given by : Metric & Distance between Doppler Spectrums take into account variances of
estimation (Rao’s metric of Information Geometry is given by « Fisher Information Matrix » used in Cramer-Rao Bound)
HR Doppler estimation is « Regularized » (Regularized Burg Algorithm) because with short bursts, we cannot estimate AR model order or dimension of « signal space » (e.g. MUSIC)
Detector is not based on whitening filters (sub-optimal because filter weights have high variances with short bursts) but on robust distance
Median is a basic tool of Robust statistic : low sensitivity to outliers (Fréchethas studies main advantages of “Median” compared to “Mean”)
Jointly Robust Doppler Estimation Robust Distance
Robust Statistics Robust Detector
ROBUSTNESS of OS-HRD-CFAR
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Information Geometry for turbulences monitoring
71 /71 /
THOMASSET Cyrille : 01/04/2011
Monitoring of Atmospheric Turbulences
Distance between AR model of order 1 and Regularized AR model of maximum order
Laminar Flow
Turbulent Flow2
1
2 11
ln21)(
n
k k
kknS
72 /72 /
STAR2000 (I&Q)
Upgrade of STAR2000 ATC Radar Weather Channel
Rain Rate
Atmospheric Turbulence
Wind Radial Speed
73 /73 / WakeWake--Vortex MonitoringVortex Monitoring
74 /74 /
Wake Vortex Roll-up(Arrival)
DepartureArrival
East Configuration
WakeWake--Vortex Monitoring : Vortex Monitoring : ArrivalArrival
75 /75 /
Wake Vortex Roll-up(Departure)
DepartureArrival
West Configuration
WakeWake--Vortex Monitoring : Vortex Monitoring : DepartureDeparture
76 /76 / Opportunity Trials : ATC PSR Radar records of A380
T T+ 4s T+ 8s
0.5 km
A380 AirplaneRange ~ 4 km
HR Doppler Entropy
77 /77 / BOR-A Installation : 06/05/2011
78 /78 / BOR-A First tests
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SPACE-TIME COVARIANCE MATRIX MEDIAN
COMPUTATION: STAP PROCESSING
80 /80 / Extension to STAP : Toeplitz-Block-Toeplitz Matrices
Covariance Matrix
Space or Time Covariance Matrix
Space-Time Covariance Matrix
Entropy Hessian metric
Karcher/Fréchet barycenter/Median
Partial Iwasawa Decomposition
Anistropic Fourier Diffusion on graph
Median in Poincaré’s disc
Hermitian Positive Definite Matrix Group : HPD(n) nICnSpCnPSp 2/),(),(
Symplectic Quotient Group :
Toeplitz structure
Multi-variate Entropy Hessian metric
Partial Iwasawa Decomposition
Median in Siegel’s disk
Block-Toeplitz structure
Complex Autoregressive
Polar decomposition in Poincaré’s disc
Mostow decomposition in Siegel’s disc
Maslov-Leray Index
INFORMATION GEOMETRY
LIE GROUP & RIEMANNIAN SYMMETRIC SPACES GEOMETRIES
K-Mean via G-PCA in SDn
K-Median in SD
Geodesic PCA (G-PCA)
enRRΦ logdetlog~
ji
jiRΦg
~2
Robust Space-Time Processing
81 /81 / Extension to STAP : Toeplitz-Block-Toeplitz Matrices
Previous results can be extented to Block-ToeplitzMatrices :
0
,
01
1
01
10
1, ~~
RRRR
RRRR
RRRRR
Rn
nnp
n
n
np
*1~
n
n
R
RVR
000
000
p
p
p
JJ
J
Vwith
82 /82 / Extension to STAP : Toeplitz-Block-Toeplitz Matrices
From Burg-like parameterization, we can deduced thisinversion of Toeplitz-Block-Toeplitz matrix :
nnnnpnn
nnnnp AARA
AR
....
1,
11,
npnnp
npnnnpnnnp RAR
RAARAR
,,
,,1
1, ....
p
pn
p
pnnp
nn
p
n
nn
n
-n
nn
nnn
IJAJ
JAJ
AA
A
AA
RαAA
*11
*11
1
11
01
011
1
.0
and
,.1with
83 /83 / Extension to STAP : Toeplitz-Block-Toeplitz Matrices
Kähler potential defined by Hessian of multi-channel/Multi-variate entropy :
npji
npnpnp
RHessg
csteµRTrcsteRRΦ
,
,,, logdetlog~
0
1
1, det..log.detlog).(~ RenAAIknR
n
k
kk
kknnp
1
1
1120
10
2 )(.n
k
kk
kk
kkn
kk
kk
kkn dAAAIdAAAITrkndRRTrnds
84 /84 / Multi-Channel Burg Algorithm
Multi-Channel Burg Algorithm :
)()(
)()(
)1()(
with
)()()()()1()(2
2
)()()( with )()1()(
)1()()(
and
0010
01100
with )()()(
)()()(
,,...,,0,,...,,,...,,
1
111
11
1*11
*
00*111
111
0
0
*
0
*11
*12
*11
11
12
1121
11
kkEP
kkEP
kkEP
kkkkkkA
JJPPJJPPAJJPPTrMin
kZkkkJJAkk
kAkk
IAJlnkJZkJAk
lkZkAk
IJJAJJAJJAAAAAAAA
bn
bn
bn
fn
fn
fn
bn
fn
fbn
nN
k
bn
bn
nN
k
fn
fn
nN
k
bn
fn
nn
fn
bn
fbTn
fbn
nn
bn
fn
A
bff
nnn
bn
fn
bn
nn
fn
fn
nn
l
nl
bn
n
l
nl
fn
nnn
nn
nn
nn
nnnn
nn
nn
Siegelnnn
nn
nn DiskAIAA
111
11
11 .
85 /85 / Mostow Decomposition & Berger Fibration
Georges Daniel Mostow
(Yale University & US Academy of Sciences)
Marcel Berger
(IHES & French Academy of Sciences)
Mostows decomposition may be found in Georges Giraud’s paper of
1921
86 /86 / Mostow Decomposition (& Berger Fibration)
Mostow Theorem :
Every matrix of can be decomposed :
whereis unitaryis real antisymmetricis real symmetrix
Can be deduce from
Lemma : Let and two positive definite hermitianmatrices, there exist a unique positive definite hermitian matrix such that :
Corollary : if is Hermitian Positive Definite, there exist a unique real symmetric matrix such that :
M CnGL ,SiAeUeM
UAS
A B
X BXAX
MS
SS eMeM 1*
87 /87 /
MMPPPPPPS
PPPPPeePeP SSS
with log.21
:corrolary Lemma
2/12/12/1*2/12/1
2/12/12/1*2/12/12212*
Mostow Decomposition
Mostow Theorem :
All matrix of can be decomposed in :
is unitary, is real antisymmetric réelle and is real symmetric
Proof :
M CnGL ,SiAeUeM
U A S
SS
SSiASSSiAS
SiASSiA
ePePeeeeeeeeP
eeeMMPeUeM
212*
2222*
2
MMPPeei
A
Peee
SS
SSiA
with log21
:y injectivit lleexponentie 2
iASeMeU
88 /88 / Siegel Disc Automorphism
Automorphism of Siegel Disc given by :
All automorphisms given by :
Distance given by:
Inverse automorphism given by :
nSD
2/100
100
2/100)(
0ZZIZZIZZZZIZZ
tZn UZUZCnUUSDAut )()(/,),(
0
2/100
2/100
01
01
100
2/100
2/100
with
)()(0
ZZIZZIG
ZGIZGZ
ZZIZZZZIZZIG
Z
)(Φ1)(Φ1
log21,,,
WW
WZdSDWZZ
Zn
89 /89 / Iterated Computation of Median in Siegel disk
2/12/1
1111
1
2/112/11
1
,1
,
,,,2/1
,2/12/1
,*,
2/1,
2/1,,
2,
2,,,,
10010
11
)()(
then1For
with
with log.2/1
: w
until on Iterate
0
: 1For
Plane-Half Siegelin
,,,,,,
nnmedian,nnn
nnmedian,nGmedian,nmedian,nGmedian,n
nnk,nnnk,nnnk,n
k,nGk,n
Fnk
m
lkk
nknn
nknknknknknknknknk
S
nk
SS
nkiA
nkk,nSiA
nknk
Fn
mm,,median,
iii
m
GGIWGGIG
GGIGGWWWW
GGIWGIGWGGIW
W W,...,m k
εHl/ HγG
WWPPPPPPS
ith eWeeWeUHeeUW
Gn
,...,WW,...,WW et Wtion :Initialisa
iIZiIZW,...,mi
,...,ZZ
n
n
n
nknknknknknk
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Extensionfor Polarimetric Data
91 /91 / STOKES VECTORS
)Im(2)(2
*123
*122
22
211
22
210
2
1
zzszzRes
zzszzs
zz
E
92 /92 / POLARIMETRY : POINCARE UNIT SPHERE
2;
2
2
arctan1
2
ss
2
arctan22
21
3
sss
2sin2sin2cos2cos2cos
Im2Re2
0
0
0
0
*12
*12
22
21
22
21
3
2
1
0
ssss
zzzz
zzzz
ssss
S
23
22
21
20 ssss
4;
4
93 /93 / Extension : Arnaudon Flow for Polar Data
Arnaudon Stochastic Flow on Sphere
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Conclusion
95 /95 /
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Conclusion
What are the benefits of Information Geometry
High Doppler Resolution Property Improve detection of slow targets Increase monitoring accuracy of turbulences
Robustness Property very short Burst (few number of pulses) with “Median” to be robust to outliers and clutter edges With metric that take into account parameters correlations/variances
Future Extension
For Robust STAP Robust Estimation of Secundary Data Covariance Matrix
For Polarimetric Data Processing Stochastic Flow on Unit Poincaré Sphere
Other actors
EADS, MBDA, ONERA in Europe MIT LL in US NUDT & BIT (China), Australian labs
96 /96 / QUESTIONS ?
Mosaïque d’Irène Rousseau(Le disque de Poincaré)
More Information :. Séminaire Léon Brillouin sur les « Sciences géométriques de l’information »http://www.informationgeometry.org/Seminar/seminarBrillouin.html. Séminaire Franco‐Indien CEFIPRA/THALES/Ecole‐Polytechnique « Matrix Information Geometries »http://www.informationgeometry.org/MIG/http://www.informationgeometry.org/MIG/MIG‐proceedings.pdfhttp://www.lix.polytechnique.fr/~schwander/resources/mig/slides/
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GENERAL THEORY OFCOMPLEX SYMMETRIC
SPACES
98 /98 / Classical Symmetric Bounded Domains
Symmetric bounded domains in Cn are particular cases of symmetric spaces of noncompact type.
Elie Cartan has proved that there is :
4 types of classical symmetric bounded domains 2 exceptional types (group of motion E6 and E7)
Classical Symmetric Bounded Domains (extension of Poincaré Disk)
021
1ZZ
:such that row 1 and columnsn with matricesComplex : IV Type
porder of matrices symmetric-skewComplex : III Type
porder of matrices symmetricComplex : II Type
columns q and rows p with matricesComplex : I Type):(
MatrixComplex r rectangula:
2
t
ZZZZ
Ω
Ω
Ω
ΩconjugatetransposedIZZ
Z
t
IVn
IIIp
IIp
Ip,q
99 /99 / Kernel function of Symmetric Bounded Domains
Luo-Geng Hua has computed the kernel functions for all classical domains :
Particular case (p=q=n=1) : Poincaré Unit Disk
domain theof volumeeuclidean theis where
, : IV Typefor 2ZZ11,
1 , : III Type
1 , : II Type
, : I Type
for det1,
-**t*
*
nΩZWWWWZK
pΩ
pΩ
qpΩ
ZWIWZK
IVn
IIIp
IIp
Ip,q
2*
*
*11111
11,
1/
zwwzK
zzCzΩΩΩΩ IVIIIIII,
100 /100 / Analytic automorphisms of Bounded Domains
Groups of analytic automorphisms of these domains are locally isomorphic to the group of matrices which preservefollowing forms:
n
tIVn
p
p
p
ptIIIp
p
p
p
ptIIp
p
pIp,q
II
HHAHAHAHAΩ
II
LI
IHLALAHAHAΩ
II
KI
IHKAKAHAHAΩ
AI
IHHAHAΩ
00
,, , : IV Type
00
,0
0,, , : III Type
00
,0
0,, , : II Type
1det,0
0, , : I Type
2*
*
*
*
122211211
2221
1211
AZAAZAgZ
AAAA
A
101 /101 / Analytic automorphisms of Bounded Domains
All classical domains are circular following from Cartan’s general theory, and the point 0 is distinguished for the potential :
Berezin quantization is based on the construction of the Hilbert Space of functions analytic in :
ZZI
KZZKZZ detlog0,0
,ln,*
*
ZgZZgjZZKZgjzgjgZgZK
ZZdK
ZZKhc
ZZdK
ZZKZgZfhcgf
h
h
),( with ),(),(),(),(
),()0,0(),()(
),()0,0(),()()()(,
***
*/1*
1
*/1*
102 /102 / Berezin Quantification of Poincaré Unit Disk
The most elementary example of Berezian quantification is, in the case of complex dimension 1, given by the Poincaré unit Disk with volume element :
Map from path on D to automorphy factor :
*22 )1.(2/1 dzdzzi
*
2
*
2
**2
22**
1
)()(
)(lnRe2)(1ln)( : potentialKähler
1 where with )1,1(
/)1,1(1/
zzzF
zzgzF
zFazbgzFzzF
baabba
gSUg
SSUzCzD
)()(
1ln1ln))0(()0(
1*
*1
2
1
21*1*22
gFgFabba
g
babgFabgba
103 /103 /
Extension for Siegel Unit Disk :
)()(
lnRe2)())((lndetlog)( : potentialKähler
)(
0 where
00
with and with
/
**
**
1**
*
*
**
ZFZgFZBAtraceZFZgF
ZZItraceZZIzFAZBBAZZg
BAABIBBAA
II
JJJggBBBA
g
IZZZSD
t
t
t
n
Berezin Quantification of Siegel Unit Disk
BBItraceBBIgFABg lndetln))0(()0( 1*
104 /104 / Cartan Decomposition on Poincaré Unit Disk
Lemma of Cartan for radial coordinates in Poincaré Disk :
)(ln21ln)(
.)(
1.)2coth(..)2(8
)()()(
)()()()(
and 0
0with
:ion DecompositCartan
1 where)( and with )1,1(
1/
2
2
2
22
22222
21*)(
22****
chzzF
shdshdds
ethabzshebchea
tchtshtshtch
de
eu
udug
baazbbazzg
abba
gSUg
zzD
LB
ii
i
i
i
105 /105 / Iwasawa Decomposition on Poincaré Unit Disk
Iwasawa coordinates in Poincaré Disk :
eu
euisheb
euichea
Ntchtshtshtch
DK
ii
CCgCghNDKhg
baazbbazzg
abba
gSUg
zzD
i
i
with
22/
22/
101
and )()()()(
, 2/cos2/sin2/sin2/cos
11
21 , )( with : Dec. Iwasawa
1 where)( and with )1,1(
1/
22/
22/
1
22****
106 /106 / Hua-Cartan Decomposition on Siegel Unit Disk
Lemma of Hua for radial coordinates in Siegel Disk (Hua-Cartan) :
)()()(diag ,Let
)(with
0)()(0
exp)()()()(
)(
)(
norder of matricescomplex unitary and exist there0
00
0 , )(
)()()()()()()()(
0 with
21
210
20
21*
212
2
2
00
00
0
*0
*
00
00**
210
210
1121
n
t
n
t
t
t
n
n
nnn
thththdiagPZZeigenABPPUUABZ
diagZ
ZZ
ABBA
VBUB
VAUA
VUV
VABBA
UU
gnSpABBA
g
shshshdiagBchchchdiagA
107 /107 / Iwasawa Decomposition on Siegel Unit Disk
Iwasawa coordinates in Siegel Disk :
SBAiBUB
SBAiAUA
ABBA
KANhSiISi
SiSiINh
ABBA
Ah
SISI
NN
eediageediag
BABA
A
CU
UC
UUUUiUUiUU
gUU
Ughg
iIIiII
CCgCghABBA
g
AZBBAZg(Z)IZZZSD
n
n
n
2121
with
.2/.2/
.2/.2/)( ,)(
norder ofmatrix real , 0
/
00
00
00
21norder unitary ,
00
)(/
21 with )( ,
and /
0001
0001
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11
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00
00
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1**
1
1
108 /108 / Iwasawa/Cartan Coordinates on Siegel Unit Disk
Iwasawa/Cartan coordinates relation in Siegel Disk :
1
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0001
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~2
with
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: Iwasawa
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.2/.2/
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SBAiAUA
VBUB
VAUA
ABBA
g
BASSBA
ABBA
MMABBA
SiISiSiSiI
M
tt
t
t
SSS
109 /109 / Special Berezin Coordinates
For every symmetric Riemannian space, there exist a dual space being compact. The isometry groups of all the compact symmetric spaces are described by block matrices (the action of the group in terms of special coordinates is described by the same formula as the action of the group of motions of the dual domain).
Berezin coordinates for Siegel domain :
Remark :
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with , :ly equivalentor
, 1**
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110 /110 / Invariant metric in Special Berezin Coordinates
Let M be a classical complex compact symmetric space. The invariant volume and invariant metric in terms of special Berezin coordinates have the form :
Link with : For arbitrary Kählerian homogeneous space, the logarithm of the density for the invariant measure is the potential of the metric
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WWWWFgdWdWgds
WWdWWFWWd nL
det),( where
,ln with
,,,
*
*
*2
,
*,
2
***
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Information surSéminaire Inter-disciplinaire
Léon Nicolas Brillouin sur « les Sciences Géométriques
de l’Information »
112 /112 / Séminaire Léon Brillouin
Corpus thématique : « Science géométrique de l’information »
Laboratoire d’accueil : IRCAM (Arshia Cont), Paris Animation : Arshia Cont (IRCAM), Frank Nielsen (Sony Research), F.
Barbaresco (Thales) Les laboratoires
IRCAM (Arshia Cont, Gerard Assayag, Arnaud Dessein) Polytechnique (Schwanger) Mines ParisTech (Pierre Rouchon, Silvere Bonnabel, Jesus Angulo) Telecom ParisTech (Hugues Randriam) SUPELEC (Mérouane Debbah, Romain Couillet) UTT Troyes (Hichem Snoussi) Univ. Poitiers (Marc Arnaudon, Le Yang) Univ. Montpellier (Michel Boyom, Paul Bryand) Observation de Nice (Cédric Richard) INRIA (Jean-Paul Zolesio, Rama Cont, Xavier Pennec) Thales (Frédéric Barbaresco, Jean-Francois Marcotorchino, François Gosselin) …
113 /113 / Séminaire Léon Brillouin « Science géométrique de l’information »
« La Science et la Théorie de l’Information », L. Brillouin, 1956
« Théorie scientifique de l’information d’une part,
mais aussi application de la théorie de l’information
à des problèmes de science pure. »
Théorie & géométrie de l’information
Probabilité & statistique
Géométrie Riemannienne (symétrique, Kählerien, métrique)
Géométrie & Groupes de Lie
Géométrie symplectique
Géométrie complexeGéométrie
discrète
Géométrie algébrique & arithmétique
Géométrie non commutative
Physique Statistique Thermo-
dynamique & Géométrie (Souriau,
Ruppeiner)
Science géométrique
de l’information
Physique quantique
114 /114 /
Site web: http://www.informationgeometry.org/Seminar/seminarBrillouin.html
9 Juin 2011 : Session « stochastic geometry & information geometry »
New Perspectives in Stochastic Geometry par Wilfrid S Kendall (univ. De Warwick, UK, [email protected] )
Pecularities of the q-exponential function in phi-exponential functions par Asuka Takatsu (IHES et Univ. De Nagoya, [email protected] )
5 Juillet 2011 : Session Digiteo & EPFL
Minimisation de l’Entropie Géométrique par Alfred Hero (Chaire DIGITEO lab, [email protected] )
Christophe Vignat (Ecole Polytechnique de Lausanne)
Octobre 2011 : Session « Entropies et Divergences » (29 Avril)
Estimateurs statistiques par minimisation des -divergences duales par Michel Broniatowski (Univ. UPMC/Paris-6, [email protected] )
Novembre 2011 : Session Franco-Italienne
Algebraic and Geometric Methods in Statistics par Paolo Gibilisco (univ. De Rome, [email protected] )
Learning the Fréchet Mean over the Manifold of Covariance Matrices par Simone Fiori (UniversitàPolitecnica delle Marche, [email protected] )
Decembre 2011 : Session historique (date à définir)
Œuvre de Léon Brillouin par Rémy Mosseri (UPMC Paris-6, [email protected] ) Léon Brillouin et les débuts de la théorie de l’information par Philippe Jacquet (INRIA & X/LIX,
[email protected] ) Le concept d’Entropie en Physique par Roger Balian (CEA, [email protected] )
Séminaire Brillouin : sciences géométriques de l’information
115 /115 / Projet PEPS
PEPS « Géométrie de l’Information »
Titre long du Projet : La géométrie de l’information : un cadre général nouveau pour l’analyse et le traitement de
signaux multiformes
Titre court du Projet : InfoGeo
Mots clés : Géométrie de l’information, analyse et traitement du signal, audio, image, radar,
télécommunications, mathématiques appliquées.
Résumé du Projet : Ce projet centré autour de la géométrie de l’information cherche à regrouper des acteurs de
différentes disciplines : audio,image, radar, télécommunications, mathématiques appliquées. L’objectif est d’amener les outils de la géométrie de l’information vers un nouveau champ d’applications : l’analyse et le traitement des signaux multiformes, pour lequel nous souhaitons formuler un cadre théorique générique. Afin de réunir les multiples compétences nécessaires, l’IRCAM, le LIX et Thales se sont associés et ont récemment initié un groupe de travail pluridisciplinaire autour de la géométrie de l’information avec des manifestations scientifiques prévues pour 2011. Le projet présenté permettra de soutenir le lancement de ce groupe de travail et de le consolider en étendant le réseau d’acteurs nationaux impliqués.
Coordinateur : Arshia cont (IRCAM) Equipes participantes : F. Nielsen (Sony Research), F. Barbaresco (Thales),
A. Dessein (IRCAM)
116 /116 / Séminaire Franco-Indien « Matrix Information Geometries »
« Matrix & Information Geometries », MIG’11
Lieu : Ecole Polytechnique et Thales Research & Technology Date : 23 au 25 Février 2011 Site web : http://www.informationgeometry.org/MIG/ Résumés : http://www.informationgeometry.org/MIG/MIG-proceedings.pdf Slides: http://www.lix.polytechnique.fr/~schwander/resources/mig/slides/ Photos : http://www.lix.polytechnique.fr/~schwander/resources/mig/pictures/ Financement :
CEFRIPA CEFRIPA (http://www.cefipra.org) : Programme bilatéral de coopération scientifique entre
l'Inde et la France financé respectivement par : le Département Science et Technologie (DST) du Ministère indien de la science et de la
technologie la Direction générale de la coopération internationale et du développement du Ministère
français des affaires étrangères.
Laboratoires indiens : Dehli Indian Institute of Statistics (Prof. Rajendra Bhatia) 2 IITs : Guwahati (Prof. Amit Kumar Mishra), Kharagpur Sociétés indiennes : The BuG Design, Honeywell Technology de Bangalore
Laboratoires français : Membres du séminaire Brillouin
117 /117 / Séminaire Franco-Indien « Matrix Information Geometries »
118 /118 / Séminaire Franco-Indien « Matrix Information Geometries »
119 /119 / Séminaire Franco-Indien « Matrix Information Geometries »
120 /120 / Séminaire Franco-Indien « Matrix Information Geometries »
R. Bhatia, « Positive Definite Matrices », Princeton university Press, 2007
TABLE OF CONTENTS:
Preface vii
Chapter 1: Positive Matrices 1
1.1 Characterizations 1
1.2 Some Basic Theorems 5
1.3 Block Matrices 12
1.4 Norm of the Schur Product 16
1.5 Monotonicity and Convexity 18
1.6 Supplementary Results and Exercises 23
1.7 Notes and References 29
Chapter 2: Positive Linear Maps 35
2.1 Representations 35
2.2 Positive Maps 36
2.3 Some Basic Properties of Positive Maps 38
2.4 Some Applications 43
2.5 Three Questions 46
2.6 Positive Maps on Operator Systems 49
2.7 Supplementary Results and Exercises 52
2.8 Notes and References 62
Chapter 3: Completely Positive Means 65
3.1 Some Basic Theorems 66
3.2 Exercises 72
3.3 Schwarz Inequalities 73
3.4 Positive Completions and Schur Products 76
3.5 The Numerical Radius 81
3.6 Supplementary Results and Exercises 85
3.7 Notes and References 94
Chapter 4: Matrix Means 101
4.1 The Harmonic Mean and the Geometric Mean 103
4.2 Some Monotonicity and Convexity Theorems 111
4.3 Some Inequalities for Quantum Entropy 114
4.4 Furuta's Inequality 125
4.5 Supplementary Results and Exercises 129
4.6 Notes and References 136
Chapter 5: Positive Definite Functions 141
5.1 Basic Properties 141
5.2 Examples 144
5.3 Loewner Matrices 153
5.4 Norm Inequalities for Means 160
5.5 Theorems of Herglotz and Bochner 165
5.6 Supplementary Results and Exercises 175
5.7 Notes and References 191
Chapter 6: Geometry of Positive Matrices 201
6.1 The Riemannian Metric 201
6.2 The Metric Space Pn 210
6.3 Center of Mass and Geometric Mean 215
6.4 Related Inequalities 222
6.5 Supplementary Results and Exercises 225
6.6 Notes and References 232
121 /121 / GDR Maths & Entreprises
122 /122 / Mini-Symposia SMAI 2011
Mini-Symposia pour Congrès SMAI 2011, 5e Biennale Française des Mathématiques Appliquées et Industrielles Lieu : Guidel, Bretagne Date : 23 au 27 Mai 2011 Site web : http://smai.emath.fr/smai2011/index.php Thème : Géométrie de l’Information
Thème : La géométrie de l'information est un thème émergeant qui consiste à traiter des phénomènes aléatoires en regardant la géométrie différentielle des espaces de probabilités correspondants. Les applications sont nombreuses en machine learning, théorie de l'information, traitement du signal et aussi géométrie différentielle. Il s'agit ici de donner une introduction aux techniques et aux motivations de la géométrie de l'information. Ce thème intéresse tout autant des mathématiciens purs qu'appliqués, des informaticiens et des industriels.
Programme : Arshia Cont (IRCAM) Frank Nielsen (Ecole Polytechnique, Sony) Xavier Pennec (INRIA, Sophia) Frédéric Barbaresco (THALES)
123 /123 / GRETSI’11 : session spéciale SS2
Session spéciale SS2 « Science géométrique de l'information », au XXIIIème colloque GRETSI 2011
Lieu : Bordeaux Date : 5 au 8 Septembre 2011 Site web : http://www.gretsi2011.org/sessions-speciales.html Programme :
Frédéric Barbaresco (THALES), « Science géométrique de l’Information : Géométrie des matrices de covariance, espace métrique de Fréchet et domaines bornés homogènes de Siegel »
Olivier SCHWANDER (Ecole Polytechnique), Frank NIELSEN (Sony), « Simplification de modèles de mélange issus d’estimateur par noyau »
Silvère Bonnabel (Ecole des Mines de Paris), « Convergence des méthodes de gradient stochastique sur les variétés riemanniennes »
Arnaud Dessein, Arshia Cont (IRCAM), « Segmentation statistique de flux audio en temps-réel dans le cadre de la géométrie de l’information »
V. Devlaminck (Université de Lille), « Modèles sous-jacents à certaines techniques d’interpolation géodésique dans l'espace des matrices de cohérence en optique de polarisation »
Pierre Formont (Supelec), Frédéric Pascal (Sondra), Jean-Philippe Ovarlez (ONERA), Gabriel Vasile(INPG Grenoble), Laurent Ferro-Famil (Université de Rennes), « Apport de la géométrie de l'information pour la classication d'images SAR polarimétriques
Xavier Pennec (INRIA), Marco Lorenzi (IRCCS, Italie), « Which parallel transport for the statistical analysis of longitudinal deformations? »
Hichem Snoussi (UTT Troyes), « Filtrage particulaire sur les vari´et´es riemanniennes » Marc Arnaudon (Université de Poitiers), Yang Le, « Algorithmes stochastiques pour calculer les p-
moyennes de mesures de probabilité et géométrie des matrices de covariance Toeplitz »
124 /124 / Sessions spéciales IRS’11
2 Sessions spéciales « New Generation of Advanced Radar Processing based on Information Geometry », au 12ème IRS 2011, International Radar Symposium
Lieu : Leipzig, Allemagne Date : 7-9 Septembre 2011 Site web : http://www.irs-2011.de Programme :
F. Barbaresco (THALES) : Geometric Radar Processing based on Fréchet Distance : Information Geometry versus Optimal Transport Theory
M. Frasca (MBDA Italy) : Optimal Cramer-Rao estimators for dimensions greater than two F. Opitz (EADS Gmbh) : Differential Geometry and Applications to Signal Processing and Tracking J.P. Ovarlez (ONERA), P. Formont (SONDRA), F. Pascal (SONDRA), G. Vasile (GIPSA) and L. Ferro-
Famil (Université de Rennes) : Contribution of Information Geometry for Polarimetric SAR Classification in Heterogeneous Areas
M. Arnaudon (Université de Poitiers), Le YANG & F. Barbaresco : Stochastic algorithms for computingp-means of probability measures, geometry of radar Toeplitz covariance matrices and applications to HR Doppler processing
Y. Cheng (NUDT, Chine), X. Wang, M. Morelande & Bill Moran (Université de Melbourne) : Bearings-Only Sensor Trajectory Optimization Using Accumulative Information
Y. Cheng (NUDT, Chine), X. Wang, M. Morelande & Bill Moran (Université de Melbourne) : Information Resolution of Joint Detection-Tracking Systems
F. Barbaresco (THALES) : Robust Statistical Radar Processing in Fréchet Metric Space: OS-HDR-CFAR and OS-STAP Processing in Siegel Homogeneous Bounded Domains
125 /125 / IGAIA’12 à Paris
Projet d’organisation du 4th IGAIA’12 à Paris
Organisateur : séminaire Brillouin (A. Dessein & A. Cont / IRCAM) Lieu : IRCAM, Paris Pour info sur 3rd IGAIA’10 :
http://www.mis.mpg.de/calendar/conferences/2010/infgeo.html
http://www.mis.mpg.de/calendar/conferences/2010/infgeo/slides.html
126 /126 / Informations générales : Livre
Géométrie Hessienne & travaux de Koszul
Soutenance de thèse de P. Byande (Prof. Boyom), 7. Déc. 2011 Livre de Hirohiko Shima, « Geometry of Hessian Structures », world
Scientific Publishing 2007
127 /127 / G. Pistone : Algebraic and Geometric Methods in Statistics