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Submitted on 23 May 2006

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Bi-invariant Means in Lie Groups. Application toLeft-invariant Polyaffine Transformations

Vincent Arsigny, Xavier Pennec, Nicholas Ayache

To cite this version:Vincent Arsigny, Xavier Pennec, Nicholas Ayache. Bi-invariant Means in Lie Groups. Application toLeft-invariant Polyaffine Transformations. [Research Report] RR-5885, INRIA. 2006. inria-00071383

https://hal.inria.fr/inria-00071383

https://hal.archives-ouvertes.fr

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Thème BIO

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Bi-invariant Means in Lie Groups. Application toLeft-invariant Polyaffine Transformations.

Vincent Arsigny — Xavier Pennec — Nicholas Ayache

N° 5885

April, 2006.

Unité de recherche INRIA Sophia Antipolis2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France)

Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65

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(TeG, +, . , [ , ])©u»gxIqtv~rgNkg3Iq;Û@¶¿ÂÇÜµ¼Ú ÔIÂ3µ¾k¼ x«\²gNskxI®gz~wqx G `qkvyg§'gNvywgHskv°«¬p

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q.fagHsv°xIwgNwkgN²Wv°xIqtzyp»skcgxsskvyx©«¬xIskcwga³Ìvygz°|Iguw´x« G §vyz°z!u`gvy¯scwv~q kskv~zygg\iÊsc»qaWwfÒu`gx«gfÃk¨>uz°g.zy|gNuwvyYwx`gkskvygNq á¿v°©rvÏsv°xIskxÃv°sqIqkqkxrv~>skvy²WvÏspuwv°ÉÊz°vywgNkv°spwâ\§cvyc©gskcwg«¬xIz°zyx>§v°|Ëv¸â ∀a, b ∈ g, [a, b] = −[b, a]

á Hskv°ÉÊxfafÒrs>sv°²Wv°sp â8r§cvycv°fawzyvygNq[a, a] = 0vyv¸â ∀a, b, c ∈ g, [a, [b, c]] + [c, [a, b]] + [b, [c, a]] = 0

áIIxuv4vyrgNHskv°sprânWvyfz°g-gÓwfawzygNqx«³v°g'uwI¨gsqÌv~qg\|vy²gN uHp

GL(n) v°sq/fÒwz°skvywzyvyN>skvy²g-qtuw|xw»q

z°vy¨gSL(n)

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M

Nx«4skcwv~q-zy|gu.v~q\scwgÄÆÁÁÜrÀÊ¼ÀÊÆ¾

x«-skcwg3qtgas§-x%fÃ>skv~gNqNÌvP gI[M, N ] = M.N − N.M

Ãi¥1kskv~zyNQscwg$³ÌvygÃzy|gNuw;x«GL(n)

vyqM(n)

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v~q\scwgYqkwuW²g38sxÐqkgx«M(n)

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fÃ>skv~gNqN wxIxfawzygsgxwHsx%³Ìvyg zy|gNuwIqwqkgg Ê G Nau`g gkg3qtgNIsgN!>uWpa.|xwx«QfÃ>skv~g3q\skvyw| x g

I²Hv~.§cs\vyq'zyzygNvÏsq¼ÕÆ>¶¸¹»À¾kÂ¥Å`¾kÂÇÂ¹»ÀÊ¼ÀE¶¿Æ¹

Ad(G)ÞÑg§v°zyz!qtgNg skcsscwgÒwx`gkskvygNqx«!skcvyqYkgNwgNqkgHsskvyx©©skcwg

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Ad(e) = Idvyv¸â ∀m ∈ G, Ad(m−1) = Ad(m)−1

v°vyv¸â ∀m, n ∈ G, Ad(m.n) = Ad(m).Ad(n)vy²wâAd : G → GL(g)

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Ad(m).v = Y m.v.m−1 Z = Dm−1Lm.DeR

m−1 .v = DmRm−1 .DeLm.v

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(wi)â8«¬xgÓwfz°gIË

E(Xi) = arg minY ∈E

∑

i

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bdcgNqkgα = 2

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α = 1>xg\xursvyqskcwg'|ggzyv°È3>skvyx.x«

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α = 1g3qkqkgHsvyz°zyp¯u`gNqkg´v°1scwv~qÒIqtg´skcwg

wvyqk»gNqkvyx±v°sÃvyqwxsaqtskv~8skzyp±xW²gÓQ\xHskpÑskxskcwg©Nqkgα > 1

±Ð²gZ§cwgα > 1

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g. exp(t.v) = exp(t.w).gÖÆ>¾

¼Ú¸Út ∈ R

Æ>¾kÂÅQ¾kÂÄ¶~ÇÂÚ Ø"w = Ad(g).v

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¾ÂÚ°¼ÀE¶¿Æ¹wÇW¶ Åg. exp(v) = exp(w).g

¶¸ÁÅQÚ ¶¿ÂÇw = Ad(g).v

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vµÂÒ¶¸¹

g¼¹`Õ

g¶¸¹ G wÂ¹ ÐÂ w¼º½Â

g. exp(v).g−1 = exp(Ad(g).v).

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m¶¸¹ V log(m)¼¹`Õ

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log(g.m.g−1) = Ad(g). log(m).

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G. exp(V ).G−1 = exp(G.V.G−1)

G. log(V ).G−1 = log(G.V.G−1)

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g. exp(t.v).g−1|t=0 = Ad(g).vá¿qkggd\x`xIqkvÏsv°xI :-x|Igdx« ¬â8 Yqtvyw|Yskcwgd«¿I8s!skcs

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Ψg : m 7→ g.m.g−1 vyqqkfaxWxskcΨg(e) = e

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log(m)»

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g. exp(t.v) = g. exp(t.v).g−1.g»¯wz°p©³ÌgNfafa

I

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G

g ∈ G

Vx

x(t) = x. exp(t.v)

y

VG

x ∈ Vx(0) = x.v ∈ TxG

x(t) = x. exp(t.w)

x(t) = x. exp(t.v)

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x»

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x. exp(v) = y %xskgskcsscwgÒ|IgxrrgNqkv~v~qgNIsv°gzyp%xHsvywgN%vy V ' "Q ¤ >sWp`xvyHs

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g. exp(t.v)" r¶¿Ä w¼>ÇÀÂÅ`¾kÆÅwÂ¾8ÀEØÆÖ

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g|t=0 = z wÂ¹

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v = DgLg−1.zwqkvyg

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B

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rg²Wv~>sgNqY«¬kxIfv + w

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v

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log(exp(v). exp(w)) = v + w + 1/2([v, w])+1/12([v, [v, w]] + [w, [w, v]])+1/24([[v, [v, w]], w]) + O((‖v‖ + ‖w‖)5).

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H(v, w) =log(exp(v). exp(w))

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w ! 1 ! ";#

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á 3Iâ

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m0 := x1

!#"%$& % '()* +, %( '-&.0/132mt+1 := mt. exp

(

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i=1 wi log(mt−1.xi)

)

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G

TmG

mxj

xi ∈ G

logm(xi) ∈ TmG

logm(xj)

m P

iwi log

m(xi) = 0

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m0á¿vE g

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xi

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Φ : m 7→ m. exp(

∑N

i=1 wi log(m−1.xi)) zy½prq;gHskz!xzyg

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Ψ : gN+1 → gÕIÂ¿ÙÐ¹`ÂÕªµØ

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∑

i wi log(xi)

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O((∑N

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‖O((

N∑

i=1

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C.(N.αN +1).R2 ≤ (1−α).R

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xi

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bdcwgNÌQz°gsYqYrg®»wg

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log(Φ(exp(v)))Ònrv°g

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E

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0

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i

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f1 = (π/4,−√

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2/2)f2 = (0,

√2, 0)

f3 = (−π/4,−

√2/2,−

√2/2)

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[−π, π[âd$z~qtsds§'xsxaskcwg.sqtz~>sv°xIÌ

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z°g«¸stÉÊv°W²>v~Hs wNcwgsdfag3ÌË (0, 0, 0)

uwv°ÉPvyW²½kv~HsfagNÌË (0,√

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√2

2−√

2

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kvy|cHskÉPvyW²½kv~Hs wNcwgsfg3ÌË (0,√

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Ti

µÂNÚ ¶¸¹`Â¼>¾ÀP¾k¼>¹Ç¸ÖÆ¾8Áa¼ÀE¶¿Æ¹wÇ¶¸¹

GL(n)¼>¹QÕÚ°ÂÀ

(wi)µÂ

N ¹`Æ¾8·

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¶ ÖÒÕÂÀ(Si) > 0,

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det(E(Ti)) = eP

iwi. ln( (Si)),¶ ÖÒÕÂÀ

(Si) < 0,ÖÆ>¾¼Ú¸Ú

i,ÀwÂ¹

det(E(Ti)) = −eP

iwi. ln(− (Si)),

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E(Ti)@jscwg§vyqkg½xIwg\x«rscwg-wkxrr»8sq

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(M) = exp(bg

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M

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1 = Πiewi (ln(det(E(Ti)

−1.Ti))).

bdcgÌÌ»qtvyw|©rgs(A.B) =

rgs(A).

wgs(B)

ln(a.b) = ln(a) + ln(b)

rgs(E(Ti)

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1 = | rgs (E(Ti))|−1. exp

(

∑

i

wi. ln (| wgs (Si)|))

,

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S1, ..., SN

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Unité de recherche INRIA Sophia Antipolis2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)

Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes4, rue Jacques Monod - 91893 ORSAY Cedex (France)

Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France)

Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)

ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)

ISSN 0249-6399