souriau symplectic structures of lie group machine ... · poincaré unit disk is an homogeneous...

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Souriau Symplectic Structures of Lie Group Machine Learning on Statistical Drone Doppler & Kinematic Signatures

Frédéric BARBARESCO Key Technology Domain Processing, Control & Cognition

KTD PCC « SENSING » Segment Leader KTD PCC Representative for Thales Land & Air Systems

December 3rd 2019

ENS Ulm 1942

2 Applications of Geometric and Structure Preserving Methods

Cambridge University – Newton Institute, 03/12/19 OPEN OPEN

Drone Detection, Tracking and Recognizing

▌ Drone Recognition

The illegal use of drones requires development of systems capable of detecting,

tracking and recognizing them in a non-collaborative manner, and this with

sufficient anticipation to be able to engage interception means adapted to the

threat.

The small size of autonomous aircraft makes it difficult to detect them at long range

with enough early warning with conventional techniques, and seems more suitable

for observation by radar systems.

However, the radio frequency detection of this type of object poses other

difficulties to solve because of their slow speed which can make them confused

with other mobile echoes such as those of land vehicles, birds and movements of

vegetation agitated by atmospheric turbulences.

It is therefore necessary to design robust classification methods of its echoes to

ensure their discrimination with respect to criteria characterizing their movements

(micro-movements of their moving parts and body kinematic movements).



Drone Recognition on Radar Doppler Signature of their moving parts

▌ Drone Radar Micro-Doppler Signature

The first idea is to listen to the Doppler signature of the radar echo coming from the

drone, which signs the radial velocity variations of the reflectance parts of the

moving elements, like the blades.

Depending on the speed of the drone, the number of moving elements and their

speed of rotation, and the roll & pitch attitude characterizing the angle of

observation, the Doppler signature of the drone will be modified.

Other factors may also vary this signature as the payload that will vary the blades

rotation speeds, or as the wind according to which the drone will change the

engine speeds of each blade and the attitude of the drone.

The size of the radar radio frequency sensor analysis box, which depends on the

beam width (related to the size of the antenna) and the distance resolution (linked

to the bandwidth), can also be adapted in the cases of drones close to each other

as in coordinated flights or in swarms, which will mix the Doppler signatures of

several objects, sometimes with echoes from the ground at very low altitude.



Drone Recognition on Kinematics Signature of their body trajectory

▌ Drone Radar Kinematics Signature

To improve the classification performance of drones when the blade Doppler

signature is more difficult to characterize (blade fairing, carbon blades, ...), we

consider, in addition to Doppler signatures, the drone kinematic, characterizing its

- speed / acceleration / jerk

- curvature/torsion of its trajectory

Kinematics Data are provided through Invariant Extended Kalman Filter (IEKF)

Radar Tracker based on local Frenet-Seret model.


Existing THALES AI Technologies for Drone Recognition



Drone Recognition by Radar Micro-Doppler Signature

▌ Drone Radar Micro-Doppler Deep Learning

Use of Radar simulator to learn on Hybrid data

(simulated and real data)

Micro-Doppler time/frequency spectrum

▌ Complex-valued CNN

Fourier transform is a convolution by the Fourier

atoms. We can learn a Fourier-like complex filter

bank.

▌ HPD neural networks

Covariance matrix has HPD (Hermitian Positive

Defnite) structure

Statistical analysis of manifold-valued data :

Information Geometry


Cambridge University – Newton Institute, 03/12/19 OPEN

Fully CNN, SPDNet/HPDNet Architecture and Complex ConvNet (THALES/Sorbonne University PhD of Daniel Brooks)

▌ Adaptation for SPD/HPD matrix

▌ Adaptation for Complex convolution & Fully Convolution Network (time axis)



Drone Recognition by Radar Micro-Doppler Signature: Results

▌ Validation on NATO Database

10 classes (7 drones and birds)

– → 10 classes

D:/Users/T0004940/ex-Bureau/KTD PCC/Objectifs DT LAS/KTD PCC Board/InnovDays 2019/video_wc_model.mp4



References

Daniel A. Brooks, Olivier Schwander, Frédéric Barbaresco, Jean-Yves Schneider,

Matthieu Cord, A Hermitian Positive Definite neural network for micro-Doppler

complex covariance processing, International Radar Conference, Toulon,

Septembre 2019

Daniel A. Brooks, Olivier Schwander, Frédéric Barbaresco, Jean-Yves Schneider,,

Matthieu Cord, Complex-valued neural networks for fully-temporal micro-Doppler

classification, International Radar Symposium (IRS), Ulm, Juin 2019

D. A. Brooks, O. Schwander, F. Barbaresco, J. Schneider, and M. Cord. Exploring

Complex Time-series Representations for Riemannian Machine Learning of Radar

Data. In ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech

and Signal Processing (ICASSP), pages 3672–3676, May 2019

D. Brooks, F/ Barbaresco, Y. Ziani, J.Y. Schneider, C. Adnet, IA & réseaux de

neurones profonds pour la reconnaissance Radar de drones sur critères Micro-

Doppler et Cinématique, CYBERWEEK, CE&SAR « IA et Défense » conference,

Rennes, November 2019



Drone Recognition by Radar Kinematics Signature

▌ Drone Radar Kinematics XGBoost Learning

Drone trajectories and Kinematics are

simulated by auto-pilot : ENAC Paparazzi UAV

Birds trajectories and kinematics are

characterized by birds on GPS: MOVEBANK

Statistics features extraction (ordered statistics,

L-moments, quantiles, …) from time series of

drone : speed / acceleation / jerk, 2D horizontal speed module, 3D speed module,

2D horizontal curvature, 3D curvature, torsion

logarithm

Python time series statistics: lmoments, tsfresh

D:/Users/T0004940/ex-Bureau/KTD PCC/Objectifs DT LAS/KTD PCC Board/Drones-ThereSiS/DronesParisDefense.avi



Gradient Boosting: XGBOOST

▌ Gradient Boosting

Construct classifiers iteratively, each new one

focusing on the errors made by the previous ones.

The final prediction is the classifiers majority vote.

Great performance even in the high dimension

setting ; can be parallelized.

better results than Random Forests

▌ XGBoost

XGBoost (Chen and Guestrin, 2016)

Several hyperparameters to tune : Number of

trees, Learning rate, examples proportion to build

each tree, variables proportion to build each tree,

Maximum depth of a tree, Minimum number of

examples in a region to make a new split



XGBOOST Learning on Drones/Birds Kinematics



References

▌ Random Forest and XGBoost

Gérard Biau and Benoît Cadre. “Optimization by gradient boosting”. In:

arXiv preprint arXiv:1707.05023 (2017)

Gérard Biau and Erwan Scornet. “A random forest guided tour”. In: Test 25.2 (2016), pp. 197–227

Gérard Biau Erwan Scornet, Johannes Welbl, Neural Random Forests , arxiv

arXiv:1604.07143

Gérard Biau, Erwan Scornet, A Random Forest Guided Tour, arXiv:1511.05741

Erwan Scornet, On the asymptotics of random forests, arXiv:1409.2090

▌ Recognition on Kinematic Data

R. Ginoulhac, F.Barbaresco & al, Target Classifcation Based On Kinematic Data From AIS/ADS-B, Using Statistical Features Extraction and Boosting, IRS, Ulm, Juin 2019

R. Ginoulhac, F. Barbaresco & al, Coastal Radar Target Recognition Based On Kinematic Data (AIS) with Machine Learning, International Radar Conference, Toulon, Sept. 2019

Gérard Biau, SCAI

Erwan Scornet X/CMAP


Motivations for Lie Group Machine Learning

ENS Ulm 1942



Lie Group Machine Learning for Drone Recognition

▌ Drone Recognition on Micro-Doppler by SU(1,1) Lie Group Machine Learning

Verblunsky/Trench Theorem: all Toeplitz Hermitian Positive Definite Covariance

matrices of stationary Radar Time series could be coded and parameterized in a

product space with a real positive axis (for signal power) and a Poincaré polydisk (for Doppler Spectrum shape).

Poincaré Unit Disk is an homogeneous space where SU(1,1) Lie Group acts transitively. Each data in Poincaré unit disk of this polydisk could be then coded by

SU(1,1) matrix Lie group element.

Micro-Doppler Analysis can be achieved by SU(1,1) Lie Group Machine Learning.

▌ Drone Recognition on Kinematics by SE(3) Lie Group Machine Learning

Trajectories could be coded by SE(3) Lie group time series provided through Invariant Extended Kalman Filter (IEKF) Radar Tracker based on local Frenet-Seret model.

Drone kinematics will be then coded by time series of SE(3) matrix Lie Groups

characterizing local rotation/translation of Frenet frame along the drone trajectory.



Drone Recognition by Lie Group Machine Learning: SU(1,1) & SE(3)

Surveillance in all battlefield spaces

Counter UAS Radar

Ground Observer 12 (G012)

Gamekeeper 16U (AVEILLANT)

* 1

0 1 1

: ( )

, ,...,

n

n n

THDP n R D

R P

2 2

* *(1,1) / 1, ,SU

/ 1D z x iy z

1 1 2 2, ,...,

0 1 0 1 0 1

n nt t t KINEMATICS

MICRO-DOPPLER

1 1 2 2

* ** * * *

1 1 2 2

, ,...,n n

n n

2(3) / ,det 1T TSO I

3(3) / (3),0 1

tSE SO t



Kalman Filter on SE(3) Lie Group (Thales/Mines ParisTech Pilté PhD): IEFSKF: Invariant Extented Frenet-Serret Kalman Filter (1/3)

▌ The state based on Frenet-Serret Model

the state is 𝑋𝑡 = 𝑅𝑡 , 𝑥𝑡 , 𝛾𝑡 , 𝜏𝑡 , 𝑢𝑡

▌ The kinematic Model

The kinematic model is based on the Frenet-Serret frame evolution, and on the fact that the target is not allowed to slide during turns:

𝑑𝑥𝑡𝑑𝑡

= 𝑅𝑡(𝑣𝑡 + 𝑤𝑡𝑥),

𝑑𝑅𝑡𝑑𝑡

= 𝑅𝑡 𝜔𝑡 + 𝑤𝑡𝜔

×,𝑑𝛾𝑡𝑑𝑡

= 0 + 𝑤𝑡𝛾,𝑑𝜏𝑡𝑑𝑡

= 0 + 𝑤𝑡𝜏 ,𝑑𝑢𝑡𝑑𝑡

= 0 + 𝑤𝑡𝑢

With 𝑣𝑡 = 𝑢𝑡, 0,0𝑇 , 𝜔𝑡 = 𝜏𝑡, 0, 𝛾𝑡

𝑇, 𝑎 × ∈ ℝ3×3 is the skew symmetric matrix associated to 𝑎 ∈ ℝ3.

We can put part of the state (the rotation and translation, dim 6) into a matricial form:

𝜒𝑡 =𝑅𝑡 𝑥𝑡01,3 1 , 𝜇𝑡 =

𝑅𝑡 𝜔𝑡 × 𝑣𝑡01,3 0

, 𝒅

𝒅𝒕𝝌𝒕 = 𝝌𝒕(𝝁𝒕+𝒘𝒕

𝝌)

keep the other part (dim 3) in vectorial form: 𝑧𝑡 = 𝛾𝑡 , 𝜏𝑡, 𝑢𝑡 , 𝒅

𝒅𝒕𝒛𝒕 = 𝟎 +𝒘𝒕

𝒛

The cartesian measurement equation is 𝒀𝒏 = 𝒙𝒕𝒏 + 𝑽𝒏 = 𝝌𝒕𝒏𝒅 + 𝑽𝒏




▌ IEFSKL Filter

Evolution equation: 𝒅𝝌𝒕

𝒅𝒕= 𝝌𝒕(𝝁𝒕+𝒘𝒕

𝝌),

𝒅𝒛𝒕

𝒅𝒕= 𝟎 +𝒘𝒕

𝒛

The 𝜒-part of the state belongs to a matrix Lie group (𝑆𝐸(3)) and the 𝑧-part of the

state belongs to a vectorial space.

The error 𝜂𝑡 : 𝜼𝒕𝝌= 𝝌𝒕

−𝟏𝝌 𝒕 and 𝜼𝒕𝒛 = 𝒛 𝒕 − 𝒛𝒕 , 𝜼𝒕 =

𝜼𝒕𝑹

𝜼𝒕𝒙

𝜼𝒕𝜸

𝜼𝒕𝝉

𝜼𝒕𝒖

=

𝑹𝒕𝑻𝑹 𝒕

𝑹𝒕𝑻(𝒙 𝒕 − 𝒙𝒕)

𝜸 𝒕 − 𝜸𝒕𝝉 𝒕 − 𝝉𝒕𝒖 𝒕 − 𝒖𝒕

The update step of the IEKF whenever a measurement 𝑌𝑛 is available and 𝑡 = 𝑡𝑛:

𝝌 𝒕𝒏+ = 𝝌 𝒕𝒏𝒆𝒙𝒑 𝑳𝒏

𝝌(𝝌 𝒕𝒏

−𝟏𝒀𝒏)

𝒛 𝒕𝒏+ = 𝒛 𝒕𝒏 + 𝑳𝒏

𝒛 𝝌 𝒕𝒏−𝟏𝒀𝒏

with 𝑳𝒏 the Kalman gain

[1] M.Pilté & S.Bonnabel & F.Barbaresco, An Innovative Nonlinear Filter for Radar Kinematic Estimation of

Maneuvering Targets in 2D, 2017 [2] A.Barrau & S.Bonnabel The Invariant Extended Kalman Filter as a stable observer, 2016




Linearized error 𝝃𝒕, such that 𝜼𝒕𝝌≈ 𝑰 + 𝝃𝒕

𝝌

×, and 𝜼𝒕

𝒛 ≈ 𝝃𝒕𝒛, 𝝃𝒕 =

𝝃𝒕𝑹

𝝃𝒕𝒙

𝝃𝒕𝜸

𝝃𝒕𝝉

𝝃𝒕𝒖

∈ ℝ𝟗

Linearized error evolution equation : 𝒅𝝃𝒕

𝒅𝒕= 𝑨𝒕𝝃𝒕 +𝒘𝒕 with 𝑨𝒕 independent of 𝑹 𝒕, 𝒙 𝒕

The Kalman gain 𝑳𝒏 is computed by integrating the Riccati equation :

𝒅

𝒅𝒕𝑷𝒕 = 𝑨𝒕𝑷𝒕 + 𝑷𝒕𝑨𝒕

𝑻 + 𝑸𝒕

𝑺𝒏 = 𝑯𝑷𝒕𝒏𝑯𝑻 + 𝑹 𝒕𝒏

𝑻 𝑵𝒏𝑹 𝒕𝒏

𝑳𝒏 = 𝑷𝒕𝒏𝑯𝑻𝑺−𝟏

𝑷𝒕𝒏+ = 𝑰𝟗 − 𝑳𝒏𝑯 𝑷𝒕𝒏

𝑸𝒕 is the covariance of the process noise, and 𝑵𝒏 is the covariance of the

measurement noise, 𝑯 is the measurement matrix.



Lie Group Model (IEKF) versus Vector Space Model (EKF)

Vector Space Model

(EKF)

Lie Group Model

(IEKF)

Kinematic Model 𝒅

𝒅𝒕𝑿𝒕 = 𝒇 𝑿𝒕, 𝒘𝒕

𝒅

𝒅𝒕𝝌𝒕 = 𝝌𝒕(𝝂𝒕+𝒘𝒕)

State Prediction 𝒅

𝒅𝒕𝑿 𝒕 = 𝒇 𝑿 𝒕

𝒅

𝒅𝒕𝝌 𝒕 = 𝝌 𝒕𝝂 𝒕

Error definition 𝜼𝒕 = 𝑿 𝒕 − 𝑿𝒕 𝜼𝒕 = 𝝌𝒕−𝟏𝝌 𝒕

Error evolution 𝒅𝜼𝒕𝒅𝒕

= 𝒇 𝑿 𝒕 − 𝒇 𝑿𝒕 𝒅𝜼𝒕𝒅𝒕

= 𝝂 𝒕𝜼𝒕 − 𝜼𝒕𝝂𝒕 − 𝜼𝒕𝒘𝒕

(autonome)

Linearized error evolution 𝒅𝝃𝒕

𝒅𝒕= 𝑭𝒕 𝑿 𝒕 − 𝑿𝒕 + 𝑸𝒕 = 𝑭𝒕𝝃𝒕 + 𝑸𝒕

(𝐹𝑡 depend on predicted space)

𝒅𝝃𝒕

𝒅𝒕= 𝑨𝒕𝝃𝒕 + 𝑸𝒕

(𝐴𝑡 independant of 𝑥 𝑡, 𝑅 𝑡)

Covariance definition 𝑷𝒕 = 𝑽𝒂𝒓(𝝃𝒕) 𝑷𝒕 = 𝑽𝒂𝒓 𝝃𝒕

Covariance prediction 𝒅𝑷𝒕

𝒅𝒕= 𝑭𝒕𝑷𝒕 + 𝑷𝒕𝑭𝒕 + 𝑸𝒕

𝒅𝑷𝒕

𝒅𝒕= 𝑨𝒕𝑷𝒕 + 𝑷𝒕𝑨𝒕 + 𝑸𝒕


Souriau Lie Group Thermodynamics

ENS Ulm 1942



Structuring Principles for Learning : Calculus of Variations

Fermat's principle

of least time

Maupertuis's

principle of

least length

Pierre

de Fermat

Pierre

Louis

Maupertuis

Joseph

Louis

Lagrange

(Euler)

Lagrange

Equation

Simeon

Denis

Poisson

Poisson

Bracket,

Poisson Geometry

Structure

Henri

Poincaré

Elie

Cartan

(Euler)

Poincaré

Equation

Poincaré

Cartan

Integral

Invariant

Jean- Marie

Souriau

Souriau

Moment

Map,

Souriau

Symplectic

2 Form,

Lie Groups

Thermodynamics

Jean-Michel

Bismut

Random

Mechanics



http://www.jmsouriau.com/structure_des_systemes_dynamiques.htm

http://www.springer.com/us/book/9780817636951

▌ Introduction of symplectic

geometry in mechanics

▌ Invention of the “moment map”

▌ Geometrization of Noether's

theorem

▌ Barycentric decomposition

theorem

▌ The total mass of an isolated

dynamic system is the class of

cohomology of the default of

equivariance for the moment map

▌ Lie Groups Thermodynamics

(Chapter IV)

Le Livre de J.M. Souriau « Structure des systèmes dynamiques », 1969









Gallileo Group & Alebra & V. Bargman Central extensions

▌ Symplectic cocycles of the Galilean group: V. Bargmann (Ann. Math. 59,

1954, pp 1–46) has proven that the symplectic cohomology space of the

Galilean group is one-dimensional.

▌ Gallileo Lie Group & Algebra

▌ Bargmann Central extension:

1100

10

1

'

'

t

x

e

wuR

t

x

)3(

, and ,

'

..'

3

SOR

ReRwux

ett

wtuxRx

1000

12

010

0

2

fu

Ru

e

wuR

t

xxso

RR

:)3(

, and ,

000

003



Souriau Model of Lie Groups Thermodynamics

▌ Souriau Geometric (Planck) Temperature is an element of Lie Algebra of Dynamical

Group (Galileo/Poincaré groups) acting on the system

▌ Generalized Entropy is Legendre Transform of minus logarithm of Laplace Transform

▌ Fisher(-Souriau) Metric is a Geometric Calorific Capacity (hessian of Massieu Potential)

▌ Higher Order Souriau Lie Groups Thermodynamics is given by Günther’s Poly-

Symplectic Model (vector-valued model in non-equivariant case)

Souriau formalism is fully covariant, with

no special coordinates (covariance of

Gibbs density wrt Dynamical Groups)


Lie Group Co-adjoint Orbits & Homogeneous Symplectic Manifold



Lie Groups Tools Development: From Group to Co-adjoint Orbits

Group/Lie Group Foundation Henri Poincaré – Fuchsian Groups

Felix Klein – Erlangen Program (Homogeneous Manifold)

Sophus Lie – Lie Group

Evariste Galois/Louis Joseph Lagange – Substitution Group

Lie Group Classification Carl-Ludwig Siegel – Symplectic Group

Hermann Weyl – Conformal Geometry, Symplectic Group

Elie Cartan – Lie algebra classification, Symmetric Spaces

Willem Killing – Cartan-Killing form, Killing Vectors

Lie Group Representation Bertram Kostant – KKS 2-form, Geometric Quantization

Alexandre Kirillov – Representation Theory, KKS 2-form

Jean-Marie Souriau – Moment Map, KKS 2-form, Souriau Cocycle

Valentine Bargmann – Unitary representation, Central extension

Harmonic Analysis on Lie Group & Orbits Method Pierre Torasso & Michèle Vergne – Poisson-Plancherel Formula

Michel Duflo – Extension of Orbits Method, Plancherel & Character

Alexandre Kirillov – Coadjoint Orbits, Kirillov Character

Jacques Dixmier – Unitary representation of nilpotent Group

Lie Group & Statistical Physics Jean-Michel Bismut – Random Mechanics

Jean-Marie Souriau – Lie Group Thermodynamics, Souriau Metric

Jean-Louis Koszul – Affine Lie Group & Algebra representation



Lie Group

▌ GROUP (Mathematics)

A set equipped with a binary operation with 4 axioms:

Closure

Associativity

Identity

invertibility

▌ LIE GROUP

A group that is a differentiable manifold, with the property that the

group operations of multiplication and inversion are smooth maps:

A Lie algebra is a vector space with a binary operation called the Lie bracket that satisfies axioms:

, then a b G a b G

, , then a b c G a b c a b c

such that e G e a a e a

, such that •a G b G b a a b e

1, then : then ( , ) is smoothx y G G G G x y x y

eT Gg .,. : g g g

, , , ; , 0 ; , ,

Jacobi Identity: , , , , , , 0

, for Matrix Lie Group

ax by z a x z b y z x x x y y x

x y z z x y y z x

x y xy yx



▌ Lie Group Adjoint Representation

the adjoint representation of a Lie group is a way of representing its elements

as linear transformations of the Lie algebra, considered as a vector space

▌ Lie Group Co-Adjoint Representation

the coadjoint representation of a Lie group , is the dual of the adjoint

representation ( denotes the dual space to ):

1

***

* and ( )g XgK Ad Ad K X ad

1

* *, , , then , ,g gg G Y F Ad F Y F Ad Y g g

1

:

( )

g g e

g

Ad d

X Ad X gXg

g g

Coadjoint operator and Coadjoint Orbits (Kirillov Representation)

1

:

g

G Aut G

g h ghg

gAd

*

gAd*g g

:

, ( ) ,

e e e

e X

ad T Ad T G End T G

X Y T G ad Y X Y



Coadjoint operator and Coadjoint Orbits (Kirillov Representation)

▌ Co-adjoint Orbits as Homogeneous Symplectic Manifold by KKS 2-form

A coadjoint orbit:

carry a natural homogeneous symplectic structure by a closed G-invariant 2-form:

.

The coadjoint action on is a Hamiltonian G-action with moment map

▌ Souriau Foundamental Theorem « Every symplectic manifold is a coadjoint

orbit » is based on classification of symplectic homogeneous Lie group

actions by Souriau, Kostant and Kirillov

* 1 * *, subset of ,F gAd F g Fg g G F g g

* *, , , , , ,X Y FK F K F B X Y F X Y X Y g

F*g

* 1 *, ,F gAd F g Fg g G F gg G

*

, , ,

,

F Fad X ad Y F X Y

X Y F

g, g

Lie Group Coadjoint Orbit (action of Lie Group on dual Lie algebra)

Homogeneous Symplectic Manifold (a smooth manifold with a closed

differential 2-form , such that d=0, where the Lie Group acts transitively)


Fisher-Koszul-Souriau Metric and Geometric Structures of Inference and Learning

ENS Ulm 1942



Fisher Metric and Fréchet-Darmois (Cramer-Rao) Bound

▌ Cramer-Rao –Fréchet-Darmois Bound has been introduced by Fréchet in

1939 and by Rao in 1945 as inverse of the Fisher Information Matrix:

▌ Rao has proposed to introduced an invariant metric in parameter space of

density of probabilities (axiomatised by N. Chentsov):

1

ˆˆˆ

IER

I

*

2

,

)(log

ji

ji

zpEI

dIdddIddgds

dzzp

zpzpds

zpzpDivergenceKullbackds

ji

jiji

ji

jiijTaylor

d

d

).(.)(

log

,_

,

*

,

,

*2

2

2

22

)(

dsds

Ww

w



▌Fisher Matrix for Gaussian Densities:

Fisher matrix induced the following differential metric :

Poincaré Model of upper half-plane and unit disk

2

1

2

10

ˆ ˆ( ) avec ( ) et 2

0

T mI E I

.2

im

z 1

iz

iz

22

2

2

1.8

dds

22 2

22

2 2 2

2. . 2.

2

T dm d dmds d I d d

Distance Between Gaussian Density with Fisher Metric



1 monovariate gaussian = 1 point in Poincaré unit disk

.2

im

z 1

iz

iz

11,m

22 ,m

1m2m

1

2

Moyenne= barycentre géodésique

2

(1) (2)2

1 1 2 2 (1) (2)

(1) (2)(1) (2)

(1) (2)*

1 ( , ), , , 2. log

1 ( , )

with ( , )1

d m m

2

2

2

22 .2

ddmds

22

2

2

1.8

dds

Fisher Metric in

Poincaré Half-Plane

Poincaré-Fisher metric In Unit Disk



Machine Learning & Gradient descent

▌ Gradient descent for Learning

Information geometry has been derived from invariant geometrical structure

involved in statistical inference. The Fisher metric defines a Riemannian metric as

the Hessian of two dual potential functions, linked to dually coupled affine

connections in a manifold of probability distributions. With the Souriau model, this

structure is extended preserving the Legendre transform between two dual

potential function parametrized in Lie algebra of the group acting transitively on

the homogeneous manifold.

Classically, to optimize the parameter of a probabilistic model, based on a

sequence of observations , is an online gradient descent with learning rate ,

and the loss function :

ty

1

T

t t

t t t

l y

t ˆlog /t t tl p y y



Information Geometry & Machine Learning

▌ Information Geometry & Natural Gradient

This simple gradient descent has a first drawback of using the same non-adaptive

learning rate for all parameter components, and a second drawback of non

invariance with respect to parameter re-encoding inducing different learning

rates. S.I. Amari has introduced the natural gradient to preserve this invariance to be insensitive to the characteristic scale of each parameter direction. The

gradient descent could be corrected by where is the Fisher information

matrix with respect to parameter , given by:

1( )I I

2

( / ) ( / )

log / log / log /with

ij

ij y p y y p y

i j i jij ij

I g

p y p y p yg E E

1

1 1( )

T

t t

t t t t

l yI



Information Geometry & Machine Learning : Legendre structure

▌ Legendre Transform, Dual Potentials & Fisher Metric

S.I. Amari has proved that the Riemannian metric in an

exponential family is the Fisher information matrix defined by:

and the dual potential, the Shannon entropy, is given

by the Legendre transform:

2,

with ( ) logy

ij

i j ij

g e dy

( ) ( )( ) , ( ) with and i i

i i

SS



Fisher Metric and Souriau 2-form: Lie Groups Thermodyamics

▌ Statistical Mechanics, Dual Potentials & Fisher Metric

In geometric statistical mechanics, J.M. Souriau has developed a “Lie groups

thermodynamics” of dynamical systems where the (maximum entropy) Gibbs

density is covariant with respect to the action of the Lie group. In the Souriau model, previous structures of information geometry are preserved:

In the Souriau Lie groups thermodynamics model, is a “geometric” (Planck) temperature, element of Lie algebra of the group, and is a “geometric”

heat, element of dual Lie algebra of the group.

2, ( )

2( ) with ( ) log

U

M

I e d

*( ) ( )( ) , ( ) with and

S QS Q Q Q

Q

g g

g Q

*g

*:U M g

Jean-Marie Souriau



Fisher-Souriau Metric and its invariance

▌ Statistical Mechanics & Invariant Souriau-Fisher Metric

In Souriau’s Lie groups thermodynamics, the invariance by re-parameterization in information geometry has been replaced by invariance with respect to the

action of the group. When an element of the group acts on the element

of the Lie algebra, given by adjoint operator . Under the action of the group

, , the entropy and the Fisher metric are invariant:

g g

gAd( )gAd S Q I

( )( )

( )

g

g

g

S Q Ad S QAd

I Ad I

g

*( ) ( )( ) , ( ) with and

S QS Q Q Q

Q

g g

2, ( )

2( ) with ( ) log

U

M

I e d



Fisher-Souriau Metric Definition by Souriau Cocycle é Moment Map

▌ Statistical Mechanics & Fisher Metric

Souriau has proposed a Riemannian metric that we have identified as a

generalization of the Fisher metric:

The tensor used to define this extended Fisher metric is defined by the

moment map , from (homogeneous symplectic manifold) to the dual

Lie algebra , given by:

This tensor is also defined in tangent space of the cocycle (this

cocycle appears due to the non-equivariance of the coadjoint operator ,

action of the group on the dual lie algebra):

1 2 1 2 with , , , , ,I g g Z Z Z Z

1 11 2 1 2 2 2 1 2with , , , ( ) where ( ) ,Z ZZ Z Z Z Q ad Z ad Z Z Z

g *g

( )J x M*g

,( , ) , with ( ) : such that ( ) ( ), , X Y XX YX Y J J J J x M J x J x X X *

g g

*

gAd

*( ) ( )g gQ Ad Ad Q g

, : with ( ) ( )

X,Y ( ),

eX Y X T X e

X Y

g g



Fisher-Souriau Metric as a non-null Cohomology extension of KKS 2 form (Kirillov-Kostant-Souriau 2 form)

▌ Souriau definition of Fisher Metric is related to the extension of KKS 2-form

(Kostant-Kirillov-Souriau) in case of non-null Cohomogy:

1 2 1 2 1 2with , , , , Z Z Z Z Q Z Z

1 2 1 2 with , , , , ,I g g Z Z Z Z

,( , ) , with ( ) : such that ( ) ( ), , X Y XX YX Y J J J J x M J x J x X X *

g g

, : with ( ) ( )

X,Y ( ),

eX Y X T X e

X Y

g g

Souriau-Fisher Metric

Equivariant KKS 2 form Non-null cohomology: aditional term from Souriau Cocycle

*( ) ( )g gQ Ad Ad Q g Souriau Fundamental Equation of Lie Group Thermodynamics

, , , 0Z Q Z

~

Ker



Fundamental Souriau Theorem

G

e

*g

)(g

g

)(gAd

)(MZ

*

gAd

)(Q

Q

*Q gQAdAdQQ gg )()( **

)(gAdQ

Q : Heat, element of dual Lie Algebra

: (Planck) température element of Lie algebra

Lie Group Lie Algebra

Dual Lie Algebra



Souriau-Fisher Metric & Souriau Lie Groups Thermodynamics: Bedrock for Lie Group Machine Learning

Gibbs canonicalensemble

*g *

g

R R

,1 g QQs ,

Q

)(gAd

e

g

G

gQAd g )(*

TEMPERATURE

In Lie Algebra

HEAT

In Dual Lie Algebra

Logarithm of Partition Function

(Massieu Characteristic Function)

Entropy

Entropy invariant under the

action of the group

)(),(, 11 QQQQQs

*g

)(Q

g )(1 Q

Legendre Clairaut

2

)(,2

2

2log

)()(

M

U

g

de

AdII

0,,~

,,, 2121 ZZZZg

Fisher Metric


Representation Theory & Orbits Methods: Fourier Transform for Non-Commutative Harmonic analysis

Alexandre Kirillov

Jacques Dixmier



Fourier/Laplace Transform and Representation Theory

Fourier analysis, named after Joseph Fourier, who showed that representing a

function as a sum of trigonometric functions greatly simplifies the study of heat

transfer and addresses classically commutative harmonic analysis. Classical

commutative harmonic analysis is restricted to functions defined on a topological

locally compact and Abelian group G (Fourier series when G = Rn/Zn, Fourier

transform when G = Rn, discrete Fourier transform when G is a finite Abelian group).

The modern development of Fourier analysis during XXth century has explored the

generalization of Fourier and Fourier-Plancherel formula for non-commutative

harmonic analysis, applied to locally compact non-Abelian groups.

This has been solved by geometric approaches based on “orbits methods” (Fourier-Plancherel formula for G is given by coadjoint representation of G in dual

vector space of its Lie algebra) with many contributors (Dixmier, Kirillov, Bernat,

Arnold, Berezin, Kostant, Souriau, Duflo, Guichardet, Torasso, Vergne, Paradan,

etc.)



Dixmier/Kirillov/Duflo/Vergne Representation Theory

▌ Classical Commutative Harmonic Analysis

▌ Fourier Transform

▌ Fourier-Plancherel formula

1 2 1 2

/ : Fourier series, : Fourier Transform

:G group character (linked to ) :

/ 1

ˆ / . ( ) ( ) ( )

n n n n

ikx

G G

G Ue

U z z

G g g g

ˆˆ :

ˆ ( ) ( )G

G

g g dg

1

ˆ

:

ˆ ( ) ( )G

G

g g g d

ˆ

ˆ( )G

e d



Dixmier/Kirillov/Duflo/Vergne Representation Theory

▌ Character-Distribution

(Schwarz) Distribution on : with

▌ Character Formula: Fourier transform on Lie algebra via Exponential map

▌ Kirillov Character :

▌ Fourier Transform:

U ( )Ug

G

g dg U ( ) Ug tr G

exp( )U ( )U XX dX g

1 ,

exp( )exp( ) trU ( )i f X

U XX j X e d f

,

exp( )( ) ( )trUi f X

Xe d f j X

1/2/2 /2

det / 2

X Xad ad

X

e ej X

ad


Gibbs Density on Poincaré Unit Disk from Souriau Lie Groups Thermodynamics and SU(1,1) Coadjoint Orbits



Poincaré Unit Disk and SU(1,1) Lie Group

The group of complex unimodular pseudo-unitary matrices :

the Lie algebra is given by:

with the following bases :

with the commutation relation:

(1,1)SU

2 2

* *(1,1) / 1, ,

a bG SU a b a b

b a

1,1g su

*/ ,

irr

ir

g

1 2 3, ,u u u g

1 2 3

0 0 1 01 1 1 , ,

0 1 0 02 2 2

i iu u u

i i

3 2 1 3 1 2 2 1 3, , , , ,u u u u u u u u u



Poincaré Unit Disk and SU(1,1) Lie Group

Dual base on dual Lie algebra is named

The dual vector space can be identified with the subspace of

of the form:

Coadjoint action of on dual Lie algebra is written

* * * *

1 2 3, ,u u u g

* *(1,1)g su

(2, )sl

*0 1 0 1 0

/ , ,1 0 0 0 1

z x iy ix y z x y z

x iy z i

g

g G * g .g



Coadjoint Orbit of SU(1,1) and Souriau Moment Map

The torus induces rotations of the unit disk

leaves 0 invariant. The stabilizer for the origin 0 of unit disk is maximal compact

subgroup of .

B. Cahen has observed that and is diffeomorphic to the unit

disk

The moment map is given by:

0,

0

i

i

eK

e

KK (1,1)SU

/ 1D z z

*

3

2* ** * *

1 2 32 2 2

:

1 ( )

1 1 1

J D ru

zz z z zz J z r u u u

z i z z

*

3 /ru G K

Benjamin Cahen, Contraction de

SU(1,1) vers le groupe de Heisenberg, Travaux mathématiques, Fascicule XV, pp.19-43, (2004)




Group act on by homography:

This action corresponds with coadjoint action of on .

The Kirillov-Kostant-Souriau 2-form of is given by:

and is associated in the frame by with:

2* ** * *

1 2 32 2 2

:

1 ( )

2 1 1 1

nJ D

zn z z z zz J z u u u

z i z z

G D * * * *. .

a b az bg z z

b a a z b

G

n

n , , , , , and n nX Y X Y X Y g

n

*

22

1n

indz dz

z




is linked to the natural action of on (by fractional linear transforms) but

also the coadjoint action of on .

could be interpreted as the stereographic projection from the two-sphere

onto :

2* ** * * *

1 2 3 32 2 2

1( ) ,

1 1 1

zz z z zJ z r u u u ru z D

z i z z

J G D

*

3 /ru G KG1J

2S

Charles-Michel Marle, Projection stéréographique et moments, hal-

02157930, version 1, Juin 2019

The coadjoint action of

is the upper

sheet of the

two-sheet hyperboloid

(1,1)G SU3 0x

* * * 2 2 2 2

1 1 2 2 3 3 1 2 3:x u x u x u x x x r



Covariant Gibbs Density by Souriau Thermodynamics

We can use Kirillov representation theory and his character formula to compute

Souriau covariant Gibbs density in the unit Poincaré disk.

For any Lie group , a coadjoint orbit has a canonical symplectic form

given by KKS 2-form .

If is finite dimensional, the corresponding volume element defines a -invariant

measure supported on , which can be interpreted as a tempered distribution.

The Fourier transform :

is Ad -invariant. When is an integral coadjoint orbit, Kirillov has proved

that this Fourier transform is related to Kirillov character by:

G*g

G

*

, *1( ) with and

!

i x dx e d xd

g

g g

G*g

1/2

sinh / 2( ) ( ) where ( ) det

/ 2

xad x

x j x e j xad x




is called Kirillov character of a unitary representation associated to the orbit.

We will consider the universal covering of , the Lie algebra is:

the Ad-invariant form allows to identify the following operator

and , could be considered analogously as rest mass, as energy, and

as the momentum vector.

The coadjoint orbits are the rest mass shells. Let Poincaré

unit disk, for any , there is a corresponding action of the universal

covering of on (with the holomorphic cotangent bundle of

unit disk), with the invariant symplectic form:

(1,1)PSU*

* *(1,1) / ,iE p

E pp iE

g su

22 2m E p Ad*Ad m E

1 2p p ip

/ 1D w w 0m

/2m

*2*

22

log 2

1

dw dwcurv i dw i

w

(1,1)PSU




The moment map is equivariant isomorphism( coadjoint orbit for )

In case , the Kirillov character formula is given by:

where

which reduces to :

m

2 0, 0m E

2/2

2: , , 2 ,1

1

m

m

mJ w D curv p E iw w

w

1m *

1

1

.,

.1.

exp ( ). m

m

x iE pi

x p iE

m

xj x e

x

1/2/ 2 / 2 sinh( )

( ) det sinh // 2 / 2

x x xj x ad ad

x x x

2

2

1( 1)

1 *

22 2

1( )

1 1

wm xmx

w

x

D

ej x e dw dw

e w




Souriau-Gibbs covariant density is given by:

This density is invariant under the action of SU(1,1)/K

2

2 2

*2* 2

2 22 2

* *

12

1 1,

1122

1 11 1

( ) ( )

w wim m

ix w w

wix wwwm xm im

w ww w

Gibbs x i x i

i x i x

m m

e ep w

j x e j x e


Conclusion & perspectives



Supervised & Non-Supervised Learning on Lie Groups

Souriau-Fisher Metric on Coadjoint Orbits

Extension of Fisher Metric for Lie Group through homogeneous Symplectic Manifolds on Lie Group Co-Adjoint Orbits

Mean-Shift on Lie Groups with Souriau-Fisher Distance

Extension of Mean-Shift for Homogeneous Symplectic Manifold and Souriau-Fisher Metric Space

Geodesic Natural Gradient on Lie Algebra

Extension of Neural Network Natural Gradient from Information Geometry on Lie Algebra for Lie Groups Machine Learning

Souriau Maximum Entropy Density on Co-Adjoint Orbits

Covariant Maximum Entropy Probability Density for Lie Groups defined with Souriau Moment Map, Co-Adjoint Orbits Method & Kirillov Representation Theory

Exponential Map for Geodesic Natural Gradient on Lie Algebra based on Souriau Algorithm for Matrix Characteristic Polynomial

Fréchet Geodesic Barycenter

by Hermann Karcher Flow

Extension of Mean/Median on Lie Group by Fréchet Definition of Geodesic Barycenter on Souriau-Fisher Metric Space, solved by Karcher Flow

LIE GROUP SUPERVISED LEARNING LIE GROUP NON-SUPERVISED LEARNING

Lie Group

Machine Learning

Souriau Exponential Map on Lie Algebra

Symplectic Integrator preserving Moment Map

Extension of Neural Network Natural Gradient to Geometric Integrators as Symplectic integrators that preserve moment map



Extension of Mean-Shift for Lie Group (e.g. with SO(3))



Koszul Book on Souriau Work



https://www.mdpi.com/journal/entropy/special_issues/Lie_group

Special Issue

"Lie Group Machine

Learning and Lie Group

Structure Preserving

Integrators"



Joint Structures and Common

Foundation of Statistical Physics,

Information Geometry and

Inference for Learning 26th July to 31st July 2020

https://franknielsen.github.io/SPIG-LesHouches2020/

Geometric Structures of Statistical Physics & Information

- Statistical Mechanics and Geometric Mechanics

- Thermodynamics, Symplectic and Contact Geometries

- Lie groups Thermodynamics

- Relativistic and continuous media Thermodynamics

- Symplectic Integrators

Physical structures of inference and learning

- Stochastic gradient of Langevin's dynamics

- Information geometry, Fisher metric and natural gradient

- Monte-Carlo Hamiltonian methods

- Varational inference and Hamiltonian controls

- Boltzmann machine



SOURIAU 2019

▌ SOURIAU 2019

Internet website : http://souriau2019.fr

In 1969, 50 years ago, Jean-Marie Souriau published

the book "Structure des système dynamiques", in which using the ideas of J.L. Lagrange, he formalized the "Geometric Mechanics" in its modern form based on Symplectic Geometry

Chapter IV was dedicated to "Thermodynamics of Lie groups" (ref André Blanc-Lapierre)

Testimony of Jean-Pierre Bourguignon at Souriau'19 (IHES, director of the European ERC)

https://www.youtube.com/watch?v=93hFolIBo0Q&t=3s

https://www.youtube.com/watch?v=beM2pUK1H7o

http://souriau2019.fr/

http://souriau2019.fr/




Main references




▌ Main references

Bargmann, V. : Irreducible unitary representations of the Lorentz group. Ann.

Math. 48, pp.588-640, (1947).

Souriau, J.-M. : Mécanique statistique, groupes de Lie et cosmologie, Colloques

int. du CNRS numéro 237. In Proceedings of the Géométrie Symplectique et

Physique Mathéma-tique, Aix-en-Provence, France, 24–28, pp. 59–113, (1974)

Souriau, J.-M. : Structure des systèmes dynamiques, Dunod, (1969).

Kirillov, A.A. : Elements of the theory of representations, Springer-Verlag, Berlin,

(1976).

Marle, C.-M. : From Tools in Symplectic and Poisson Geometry to J.-M. Souriau’s

Theo-ries of Statistical Mechanics and Thermodynamics. Entropy, 18, 370, (2016).

Barbaresco, F. : Higher Order Geometric Theory of Information and Heat Based

on Poly-Symplectic Geometry of Souriau Lie Groups Thermodynamics and Their

Contextures: The Bedrock for Lie Group Machine Learning. Entropy, 20, 840,

(2018).




▌ Main references

Cishahayo C., de Bièvre S. : On the contraction of the discrete series of SU(1;1),

Annales de l’institut Fourier, tome 43, no 2, p. 551-567, (1993).

Cahen, B.: Contraction de SU(1, 1) vers le groupe de Heisenberg. Mathematical

works, Part XV,Séminaire de Mathématique Université du Luxembourg, 19–43,(2004).

Cahen, M., Gutt, S. and Rawnsley, J. : Quantization on Kähler manifolds I,

Geometric in-terpretation of Berezin quantization, J. Geom. Phys. 7,45-62, (1990).

Dai,J. : Conjugacy Classes, Coadjoint Orbits and Characters of Diff+S1, PhD

dissertation, The University of Arizona, Tucson, AZ, 85721, USA, (2000).

Dai J., Pickrell D. : The orbit method and the Virasoro extension of Diff+(S1): I. Orbital

integrals, Journal of Geometry and Physics, n°44, pp.623-653, (2003).

Knapp, A. : Representation Theory of Semisimple Groups: An Overview based on

Exam-ples, Princeton University press, (1986).

Frenkel, I. : Orbital theory for affine Lie algebras, Invent. Math. 77, pp. 301–354,

(1984).




▌ Main references

Libine, M. : Introduction to Representations of Real Semisimple Lie Groups,

arXiv:1212.2578v2, (2014).

Guichardet, A. : La methode des orbites: historiques, principes, résultats. Leçons

de ma-thématiques d’aujourd’hui, Vol.4, Cassini, pp. 33-59, (2010).

Vergne, M. : Representations of Lie groups and the orbit method, Actes Coll. Bryn

Mawr, p.59-101, Springer, (1983).

Duflo, M. ; Heckman, G. ; Vergne, M.: Projection d'orbites, formule de Kirillov et

formule de Blattner, Analyse harmonique sur les groupes de Lie et les espaces

symétriques, Mé-moires de la Société Mathématique de France, Série 2, no. 15, p.

65-128, (1984).

Witten, E: Coadjoint orbits of the Virasoro group, Com.. Math. Phys. 114, p. 1–53,

(1988).

Pukanszky, L. : The Plancherel formula for the universal covering group of SL(2,R),

Math. Ann. 156, pp.96-143, (1964).




▌ Main references

Clerc, J.L.; Orsted B.: The Maslov Index Revisited, Transformation Groups, vol. 6,

n°4, pp.303-320, (2001).

Foth, P.; Lamb M. : The Poisson Geometry of SU(1,1), Journal of Mathematical

Physics, Vol. 51, (2010).

Perelomov, A.M. : Coherent States for Arbitrary Lie Group, Commun. math. Phys.

26, pp. 222-236, (1972).

Ishi, H.: Kolodziejek, B: Characterization of the Riesz Exponential Familly on

Homogeneous Cones. arXiv:1605.03896, (2018).

Tojo, K.; Yoshino, T. : A Method to Construct Exponential Families by

Representation Theory. arXiv:1811.01394, (2018).

Frédéric Barbaresco, Lie Group Machine Learning and Gibbs Density on Poincaré

Unit Disk from Souriau Lie Groups Thermodynamics and SU(1,1) Coadjoint Orbits.

In: Nielsen, F., Barbaresco, F. (eds.) GSI 2019. LNCS, vol. 11712, SPRINGER, 2019




▌ Main references

Frédéric Barbaresco, Application exponentielle de matrice par l’extension de

l’algorithme de Jean-Marie Souriau, utilisable pour le tir géodésique et

l’apprentissage machine pour les groupes de Lie, Colloque GRETSI 2019, Lille, 2019

Charles-Michel Marle. From Tools in Symplectic and Poisson Geometry to J.-M.

Souriau’s Theories of Statistical Mechanics and Thermodynamics. MDPI Entropy,

18, 370, 2016

Charles-Michel Marle, Projection stéréographique et moments, hal-02157930,

version 1, Juin 2019

souriau symplectic structures of lie group machine ... · poincaré unit disk is an homogeneous...

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