Gyrogroup Batch Normalization

Published as a con e ence pape a ICLR 2025
GYROGROUP BATCH NORMALIZATION
Ziheng Chen1∗
, Yue Song1, Xiao-Jun Wu2& Nicu Sebe1
1Uni e si y o T en o, 2Jiangnan Uni e si y
ABSTRACT
Se e al Riemannian mani olds in machine lea ning, such as Symme ic Posi i e
De ini e (SPD), G assmann, sphe ical, and hype bolic mani olds, ha e been p o en
o admi gy o s uc u es, hus enabling a p incipled and e ec i e ex ension o
Euclidean Deep Neu al Ne wo ks (DNNs) o mani olds. Inspi ed by his, his
s udy in oduces a gene al Riemannian Ba ch No maliza ion (RBN) amewo k
on gy og oups, e med Gy oBN. We iden i y he leas equi emen s o gua an ee
Gy oBN wi h heo e ical con ol o e sample s a is ics, e e ed o as pseudo-
educ ion and gy oisome ic gy a ions, which a e sa is ied by all he exis ing
gy og oups in machine lea ning. Besides, ou Gy oBN inco po a es se e al exis ing
no maliza ion me hods, including he one on gene al Lie g oups and di e en ypes
o RBN on he non-g oup SPD geome y. Las ly, we ins an ia e ou Gy oBN on
he G assmannian and hype bolic spaces. Expe imen s on he G assmannian and
hype bolic ne wo ks demons a e he e ec i eness o ou Gy oBN. The code is
a ailable a h ps://gi hub.com/Gi ZH-Chen/Gy oBN.gi .
1 INTRODUCTION
Deep Neu al Ne wo ks (DNNs) on Riemannian mani olds ha e gained inc easing in e es in a ious
machine lea ning applica ions, such as compu e ision (Huang e al.,2017;Huang & Van Gool,
2017;Huang e al.,2018;Skopek e al.,2019;Wang e al.,2022b;a;Chen e al.,2023c;Gao e al.,
2023;Wang e al.,2024b;Chen e al.,2025), na u al language p ocessing (Ganea e al.,2018;Shimizu
e al.,2020), d one classi ica ion (B ooks e al.,2019;Chen e al.,2024a), human neu oimaging (Pan
e al.,2022;Koble e al.,2022a;Ju e al.,2024;Wang e al.,2024a), medical imaging (Huang e al.,
2019;Chak abo y e al.,2020), node and g aph classi ica ion (Chami e al.,2019;Dai e al.,2021;
Zhao e al.,2023;Chen e al.,2023b;Nguyen e al.,2024;Chen e al.,2024c). As co e echniques in
DNNs, no maliza ion echniques (Io e & Szegedy,2015;Ba e al.,2016;Ulyano e al.,2016;Wu &
He,2018;Chen e al.,2023a) ha e also been ex ended in o di e en geome ies.
Howe e , mos exis ing Riemannian no maliza ion me hods a e designed o a selec ed ew geome ies
o ail o no malize he sample s a is ics. B ooks e al. (2019); Koble e al. (2022b;a) in oduced
Riemannian Ba ch No maliza ion (RBN) on SPD mani olds unde he speci ic A ine-In a ian
Me ic (AIM). Chak abo y (2020) gene alized his idea and p oposed a Riemannian no maliza ion
amewo k o e homogeneous spaces. Howe e , his app oach canno gene ally no malize he sample
s a is ics. Simila o mula ion and issue can also be ound in Lou e al. (2020, Alg. 2) and Bdei
e al. (2024, Sec. 4.2). Besides, Chak abo y (2020) also de eloped a Riemannian no maliza ion
o ma ix Lie g oups. Al hough his app oach can con ol i s - and second-o de s a is ics, i is
limi ed o a speci ic ype o dis ance (Chak abo y,2020, Sec. 3.2). Chen e al. (2024b) ook one s ep
u he and de eloped RBN o e he gene al Lie g oup, e e ed o as LieBN. Al hough LieBN can
e ec i ely no malize sample s a is ics, many geome ies do no admi a g oup s uc u e. In summa y,
he exis ing RBN me hods ail o no malize mani old- alued samples in a p incipled manne .
Recen ly, building Riemannian ne wo ks based on gy o s uc u es has shown no able success ac oss
a ious geome ies, including Symme ic Posi i e De ini e (SPD) (Nguyen,2022a;b), G assmannian
(Nguyen,2022b), hype bolic (Ganea e al.,2018), and sphe ical (Skopek e al.,2019) mani olds.
Gy o s uc u es, na u al ex ensions o ec o s uc u es, o e powe ul ma hema ical ools o building
Riemannian neu al ne wo ks. Mo eo e , gy og oups na u ally encompass Lie g oups and ex end o
non-g oup geome ies. Fo ins ance, AIM on he SPD mani old, as well as G assmannian, hype bolic,
and sphe ical mani olds, do no o m Lie g oups bu ins ead gy og oups.
∗Co espondence o [email p o ec ed]
1
Published as a con e ence pape a ICLR 2025
Figu e 1: Illus a ion o Gy oBN on he G assmannian and hype bolic spaces. As
G (1,3)
is
homeomo phic o he eal p ojec i e space
RP2
, we illus a e
G (1,3)
as he uni hemisphe e wi h
an ipodal poin s iden i ied. We se he bias as
I1,3= (1,0,0)⊤
and he scaling scala as 0.2 o be e
illus a ion. Fo he hype bolic space, we isualize he Gy oBN on he Poinca é ball model
P3
−1
,
which is he in e io o he uni sphe e in
R3
. We se bias and shi as ze o ec o and 0.7, espec i ely.
Table 1: Compa ison o p e ious RBN me hods wi h ou Gy oBN, whe e M and V deno e he sample
mean and a iance. Compa ed wi h he exis ing RBN me hods, ou Gy oBN can no malize sample
s a is ics in a p incipled manne . Besides, se e al p e ious RBN me hods wi h heo e ical con ol
o e sample s a is ics a e special cases o ou Gy oBN.
Me hod Con ollable S a is ics Applied Geome ies Inco po a ed by Gy oBN
SPDBN
(B ooks e al.,2019)M SPD mani olds unde AIM ✓
SPDBN
(Koble e al.,2022b)M+V SPD mani olds unde AIM ✓
SPDDSMBN
(Koble e al.,2022a)M+V SPD mani olds unde AIM ✓
Mani oldNo m
(Chak abo y,2020, Algs. 1-2) N/A Riemannian homogeneous space ✗
Mani oldNo m
(Chak abo y,2020, Algs. 3-4) M+V Ma ix Lie g oups unde he dis ance
d(X, Y ) = mlog X−1Y✓
RBN
(Lou e al.,2020, Alg. 2) N/A Geodesically comple e mani olds ✗
LieBN
(Chen e al.,2024b)M+V Gene al Lie g oups ✓
Gy oBN M+V Pseudo- educ i e gy og oups
wi h gy o isome ic gy a ions N/A
Based on he abo e analysis, his pape in oduces Gy oBN, an RBN amewo k o gene al gy-
og oups. We use he gy o addi ion, sub ac ion, and scala p oduc o ex end he cen e ing ( ec o
sub ac ion), biasing ( ec o addi ion), and scaling ( ec o scala p oduc ) in he Euclidean BN in o
mani olds in a p incipled manne . Fo b oade applicabili y and in-dep h heo e ical analysis, we
adap he exis ing gy og oup in o a mo e elaxed s uc u e, e med pseudo- educ i e gy og oup. Ou
heo e ical analysis shows ha pseudo- educ i e gy og oups wi h gy oisome ic gy a ions can enable
Gy oBN wi h heo e ical con ol o e sample s a is ics. Mo e impo an ly, hese equi emen s a e
sa is ied by all he exis ing gy og oups in machine lea ning. The e o e, compa ed wi h he exis ing
RBN me hods, ou Gy oBN can no malize sample s a is ics in a p incipled manne . Besides, se e al
exis ing RBN me hods a e inco po a ed by ou Gy oBN as special cases, including he LieBN on
Lie g oups, such as h ee SPD Lie g oups and o a ion ma ices, and di e en ypes o RBN on he
non-g oup SPD geome y. We p o ide a de ailed compa ison in Tab. 1. Empi ically, we ins an i-
a e ou Gy oBN on he G assmannian and hype bolic spaces, as illus a ed in Fig. 1. To he bes
o ou knowledge, ou G assmannian Gy oBN is he i s implemen a ion o G assmannian RBN.
Expe imen s on he G assmannian and hype bolic ne wo ks alida e he e ec i eness o ou Gy oBN.
Ou main heo e ical con ibu ions a e summa ized as ollows: (1) We p opose he pseudo- educ i e
gy og oup, a elaxed s uc u e o he gy og oup, and p esen ele an heo e ical analyses; (2) We
iden i y he equi emen s ha gua an ee ou Gy oBN wi h heo e ical con ol o e sample s a is ics,
i.e.,pseudo- educ ion and gy oisome ic gy a ions; (3) We p opose a Gy oBN amewo k o RBN
o e gene al gy og oups, which can be mani es ed in a ious geome ies in a plug-and-playe manne .
(4) We implemen ou Gy oBN on he G assmannian and hype bolic spaces. Ex ensi e expe imen s
on popula G assmannian and hype bolic ne wo ks alida e he e ec i eness o ou amewo k.
2
Published as a con e ence pape a ICLR 2025
Main heo e ical esul s: De . 3.1 elaxes he exis ing gy og oup in o he pseudo- educ i e gy-
og oup. P op. 3.2 e eals ha he non- educ i e G assmannian gy og oup is, in ac , pseudo- educ i e.
Thms. 3.3 and 3.5 highligh ha he in a iance o gy ono m unde gy a ions is c ucial o enabling
se e al ope a o s o ac as gy oisome ies. P op. 3.6 con i ms ha all gy og oups lis ed in Tab. 2
a e pseudo- educ i e and hei gy a ions a e gy oisome ies. These wo p ope ies a e essen ial o
enabling Gy oBN in Alg. 1 o no malize sample s a is ics, which a e o malized in Thm. 4.1. Sec. 5
discusses how se e al exis ing RBN me hods a e special cases o ou Gy oBN. Las ly, Sec. 6mani es
ou Gy oBN on he G assmannian and hype bolic spaces, whe e P op. 6.1 discuss he e icien
implemen a ion on he G assmannian. Due o page limi s, all he p oo s a e p esen ed in App. G.
2 PRELIMINARIES
This sec ion ecaps gy og oups (Unga ,2009) and se e al conc e e gy og oups in machine lea ning.
De ini ion 2.1 (Gy og oups (Unga ,2009)).Gi en a nonemp y se
G
wi h a bina y ope a ion
⊕:G×G→G
,
{G, ⊕}
o ms a gy og oup i i s bina y ope a ion sa is ies he ollowing axioms o
any a, b, c ∈G:
(G1) The e is a leas one elemen
e∈G
called a le iden i y (o neu al elemen ) such ha
e⊕a=a
.
(G2) The e is an elemen ⊖a∈Gcalled a le in e se o asuch ha ⊖a⊕a=e.
(G3) The e is an au omo phism gy [a, b] : G→G o each a, b ∈Gsuch ha
a⊕(b⊕c) = (a⊕b)⊕gy [a, b]c(Le Gy oassocia i e Law). (1)
The au omo phism
gy [a, b]
is called he gy oau omo phism, o he gy a ion o
G
gene a ed by
a, b
.
(G4) Le educ ion law: gy [a, b] = gy [a⊕b, b].
De ini ion 2.2 (Gy ocommu a i e Gy og oups (Unga ,2009)).A gy og oup
{G, ⊕}
is gy ocommu-
a i e i i sa is ies
a⊕b= gy [a, b](b⊕a)(Gy ocommu a i e Law). (2)
De ini ion 2.3 (Non educ i e Gy og oups (Nguyen,2022a)).A g oupoid
{G, ⊕}
is a non educ i e
gy og oup i i sa is ies axioms (G1), (G2), and (G3).
In ui i ely, gy og oups a e na u al gene aliza ions o g oups. Unlike g oups, gy og oups a e non-
associa i e bu ha e gy oassocia i i y cha ac e ized by gy a ions. Since all gy a ions in any (Lie)
g oup a e he iden i y map, e e y (Lie) g oup is au oma ically a gy og oup.
As shown by Nguyen & Yang (2023), gi en
P
,
Q
and
R
in a mani old
M
and
∈R
, he gy o
s uc u es can be de ined as:
Gy o addi ion: P⊕Q= ExpP(PTE→P(LogE(Q))) ,(3)
Gy o scala p oduc : ⊙P= ExpE( LogE(P)) ,(4)
Gy o in e se: ⊖P=−1⊙P= ExpE(−LogE(P)) ,(5)
Gy a ion: gy [P, Q]R= (⊖(P⊕Q)) ⊕(P⊕(Q⊕R)),(6)
Gy o inne p oduc : ⟨P, Q⟩g =⟨LogE(P),LogE(Q)⟩E,(7)
Gy o no m: ∥P∥g =⟨P, P⟩g ,(8)
Gy odis ance: dg y(P, Q) = ∥⊖P⊕Q∥g ,(9)
whe e
E
is he gy o iden i y elemen , and
LogE
and
⟨·,·⟩E
is he Riemannian loga i hm and me ic
a E. A bijec ion ω:G→Gis called gy oisome y, i i p ese es he gy odis ance
dg y(ω(P), ω(Q)) = dg y(P, Q).(10)
No e ha he gy o scala p oduc
⊙
is equi ed o a gy og oup o o m a gy o ec o space (Nguyen,
2022b). Al hough his pape only in ol es gy og oups, we also ecap gy o ec o spaces in App. C.
Se e al geome ies in machine lea ning admi a gy o s uc u e de ined in Eq. (3)-Eq. (9) and o m a
(non educ i e) gy og oup, such as A ine-In a ian Me ic (AIM) (Pennec e al.,2006), Log-Euclidean
Me ic (LEM) (A signy e al.,2005), and Log-Cholesky Me ic (LCM) (Lin,2019) on he SPD
mani old
Sn
++
(Nguyen,2022a), O hono mal Basis (ONB) pe spec i e
G (p, n)
(Bendoka e al.,
2024) and p ojec o pe spec i e
G (p, n)
(Bendoka e al.,2024) o he G assmannian (Nguyen,
3
Published as a con e ence pape a ICLR 2025
2022a;Nguyen & Yang,2023), Poinca é ball
Pn
K
o he hype boloid (Unga ,2009;Ganea e al.,
2018), and p ojec ed hype sphe e
Dn
K
o he hype sphe e (Skopek e al.,2019). Besides, he
gy og oups p oposed by Nguyen (2022b) on he SPD mani old unde he LEM and LCM coincide
wi h he Lie g oups p oposed by A signy e al. (2005); Lin (2019).
We deno e
MK
as
Pn
K(K < 0)
,
Dn
K(K > 0)
and
Rn(K= 0)
, espec i ely.
MK
is known as
he Cons an Cu a u e Space (CCS) (Do Ca mo & Flahe y F ancis,1992). We summa ize all he
necessa y gy o p ope ies in Tab. 2.
Table 2: Gy og oup p ope ies on se e al geome ies. Rela ed no a ions a e de ined in App. C.3.2.
Geome y Symbol P⊕Qo x⊕y E ⊖Po ⊖xLie g oup Gy og oup Re e ences
AIM Sn
++ ⊕AI P1
2QP 1
2InP−1✗✓(Nguyen,2022b)
LEM Sn
++ ⊕LE mexp(mlog(P) + mlog(Q)) InP−1✓ ✓ (A signy e al.,2005)
(Nguyen,2022b)
LCM Sn
++ ⊕LC ψ−1
LC(ψLC(P) + ψLC(Q)) InψLC(−ψLC(P)) ✓ ✓
(Lin,2019)
(Nguyen,2022b)
(Chen e al.,2024e)
G (p, n)e
⊕G mexp(Ω)Qmexp(−Ω) e
Ip,n mexp(−Ω)e
Ip,n mexp(Ω) ✗Non- educ i e (Nguyen,2022a)
G (p, n)⊕G mexp(Ω)V Ip,n mexp(−Ω)Ip,n (Nguyen & Yang,2023)
MK⊕K(1−2K⟨x,y⟩−K∥y∥2)x+(1+K∥x∥2)y
1−2K⟨x,y⟩+K2∥x∥2∥y∥20−x✗(✓ o K=0)✓
(Unga ,2009)
(Ganea e al.,2018)
(Skopek e al.,2019)
3 PSEUDO-REDUCTIVE GYROGROUPS
As shown in Tab. 2, he G assmannian does no sa is y le educ ion (G4) in De . 2.1. Howe e , we
ind ha a elaxed e sion o (G4) is necessa y o gua an ee he sample no maliza ion o e gy og oups.
The e o e, his sec ion p oposes an in e media e be ween he gy og oup and he non educ i e one,
called pseudo- educ i e gy og oups.
Unless speci ically emphasized, he gy o s uc u e in his pape is de ined as Eq. (3)-Eq. (9). Gi en a
gy og oup {G, ⊕}, he le gy o ansla ion by P∈Gis de ined as
LP:G→G, wi h LP(Q) = P⊕Q, ∀Q∈G. (11)
I any le gy o ansla ion is a gy oisome y, we can use gy o ansla ion o cen e samples. Nguyen
& Yang (2023) show ha any le gy o ansla ion on he SPD and G assmannian mani olds is a
gy oisome y. Howe e , he p oo elies on he le cancella ion law o gy og oups, which does no
hold o non educ i e gy og oups. The e o e, he p oo is ques ionable o he G assmannian. This
subsec ion p oposes an in e media e s uc u e, e e ed o as pseudo- educ i e gy og oups, which
can suppo le cancella ion law in gene al and, he e o e, gy oisome y o le gy o ansla ion. We
illus a e he de i a ion logic in Fig. 2.
In a iance o gy ono m
unde any gy a ion
Le cancella ion law
Le gy o ansla ion law
Axiom (G1-3)
Pseudo- educ ion
Gy oisome ies o any
gy a ion and le
gy o ansla ion
Gy oisome y o he
gy oin e se
Gy ocommu a i i y
Axiom (G1-3)
Le educ ion (G4)
Ou s:
P e ious:
O
Figu e 2: The concep ual compa ison o de i a ion logic o gy oisome ies o ou wo k agains
p e ious wo k (Nguyen & Yang,2023), whe e he le gy o ansla ion law is p esen ed in Lem. G.1.
The p e ious wo k p o es he esul s on he SPD and G assmannian mani olds in a case-by-case
manne . In con as , we elax he le educ ion in o pseudo- educ ion and gi e a gene al heo e ical
amewo k. Ou amewo k also co ec s he p oo o he G assmannian cases.
De ini ion 3.1 (Pseudo- educ i e Gy og oups).A g oupoid
{G, ⊕}
is a pseudo- educ i e gy og oup
i i sa is ies axioms (G1), (G2), (G3) and he ollowing pseudo- educ i e law:
gy [X, P] =
1
, o any le in e se Xo Pin G, (12)
4
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whe e
1
is he iden i y map.
Eq. (12) can be in ui i ely iewed as he in e media e be ween educ ion and non- educ ion. Fo
gy og oups, Eq. (12) can be di ec ly ob ained om he le gy oassocia i i y (G3) and educ ion
(G4) (Unga ,2009, p. 12). Howe e , he e is no heo e ical gua an ee ha Eq. (12) holds o gene al
non- educ i e gy og oups. The e o e, we name Eq. (12) as pseudo- educ ion. Ne e heless, o he
speci ic non- educ i e G assmannian, i is indeed pseudo- educ i e.
P oposi ion 3.2. [
↓
]
G (p, n)
and
G (p, n)
o m pseudo- educ i e and gy ocommu a i e gy og oups.
Ou pseudo- educ i e gy og oup na u ally gene alizes he anilla gy og oup, as i sha es mos o he
basic p ope ies o gy og oups (Unga ,2009, Thms. 1.13 - 1. 14), which a e de iled in Thm. D.1.
The mos ela ed p ope y in Thm. D.1 is he le cancella ion law, one o he key p e equisi es o
gy o ansla ion as gy oisome y. No e ha he le cancella ion on he gy og oup comes om le
gy o associa i i y and Eq. (12) (Unga ,2009, p. 12). The e o e, le cancella ion does no gene ally
hold o non- educ i e gy og oups, bu exis s in pseudo- educ i e gy og oups. Nex , we poin ou an
i s a emen abou gy oisome y, which will be use ul in he ollowing.
Theo em 3.3. [
↓
]Gi en a pseudo- educ i e gy og oup
{G, ⊕}
,
gy [P, Q]
p ese es gy ono m o
any P, Q ∈G, i gy [P, Q]is a gy oisome y o any P, Q ∈G.
We ind ha he isome y o any gy a ion is a key p e equisi e o o he gy o ope a o s as isome ies.
De ini ion 3.4 (Gy o Le -in a iance).The gy odis ance o gy og oup is gy o le -in a ian i any
le gy o ansla ion is a gy oisome y.
Theo em 3.5 (Gy oisome ies).[
↓
]Gi en a pseudo- educ i e gy og oup
{G, ⊕}
wi h any
gy [·,·]
as
a gy oisome y, hen we ha e he ollowing:
1. The gy odis ance (Eq. (9)) is gy o le -in a ian ;
2. Symme y o he gy odis ance: ∀P, Q ∈G, dg y(P, Q)=dg y(Q, P);
3. I {G, ⊕}is gy ocommu a i e, hen he gy oin e se is a gy oisome y;
P oposi ion 3.6. [
↓
]Fo e e y (pseudo- educ i e) gy og oup in Tab. 2, he gy odis ance is iden-
ical o he geodesic dis ance ( he e o e symme ic). The gy oin e se, any gy a ion and any le
gy o ansla ion a e gy oisome ies.
C edi o he p oo .
Fo he G assmannian and SPD mani olds, he isome ies o gy a ion, gy oin-
e se, and le gy o ansla ion ha e been p o en by Nguyen & Yang (2023, Thms. 2. 12 - 2. 14 and
2.16 - 2. 18). Ne e heless, in he p oo o Thms. 2. 12 - 2. 14, he au ho s iew he non- educ i e
G assmannian as a gy og oup, as hey use he le cancella ion p ope y o he gy og oup. Fo una ely,
ou P op. 3.2 and Thm. D.1 shows ha he G assmannian s ill enjoys le cancella ion. The e o e, all
he esul s on isome ies in hei Thms. 2.12-2.14 a e co ec . Ne e heless, hese esul s on he SPD
and G assmannian mani olds can be di ec ly ob ained by ou Thms. 3.3 and 3.5, as hese gy og oups
a e pseudo- educ i e and he gy a ions p ese e he gy ono m. Besides, we show he exp essions
o he gy odis ance o all gy og oups in Tab. 2, and he associa ed gy oisome ies on
MK
, bo h o
which is none- i ial. The de ailed p oo is p esen ed in App. G.4.
4 GYROBN ON GENERAL PSEUDO-REDUCTIVE GYROGROUPS
P op. 3.6 shows ha se e al geome ies enjoy isome ic gy o ansla ion, o e ing a heo e ical
ounda ion o no malizing samples o e gy og oups in a p incipled manne . Inspi ed by his, his
sec ion de elops Riemannian Ba ch No maliza ion (RBN) o gene al pseudo- educ i e gy og oups,
e e ed o as Gy oBN. In he ollowing, {M,⊕}is assumed as a pseudo- educ i e gy og oup wi h
he gy o s uc u e de ined as Eq. (3)-Eq. (9)1.
4.1 EUCLIDEAN BATCH NORMALIZATION REVISITED
As he co e ope a ions o di e en Euclidean no maliza ion a ian s (Io e & Szegedy,2015;Ba e al.,
2016;Ulyano e al.,2016;Wu & He,2018) a e simila , his pape ocuses on BN. Gi en a ba ch o
ac i a ions {xi...N }, he co e ope a ions o BN can be exp essed as:
∀i≤N, xi←γxi−µ
√ 2+ϵ+β(13)
1In ou Gy oBN, ⊙is no equi ed o comply wi h he axioms o he gy o ec o space (De . C.4).
5

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whe e
µ
,
2
,
γ
, and
β
a e he sample mean, sample a iance, scaling pa ame e , and biasing pa ame e ,
espec i ely.
4.2 GYROBN
To gene alize he Euclidean BN in o gy og oups, we i s de ine sample mean, sample a iance,
cen e ing, biasing, and scaling o e gy og oups. Then, we in oduce ou Gy oBN amewo k wi h a
heo e ical analysis o he abili y o no malize sample s a is ics.
We de ine he gy omean as he F éche mean unde he gy odis ance:
M= FM({Pi}) = a gmin
Q∈M
1
NXN
i=1 d2
g y (Pi, Q)(14)
The gy o a iance is he F éche a iance, i.e., he minimiza ion o he igh es hand side o Eq. (14).
When he dis ance in Eq. (14) is he geodesic dis ance, he F éche mean and a iance a e known as
he Riemannian mean and a iance. Al hough he gy omean and gy o a iance a e no necessa ily he
same as he Riemannian ones, P op. 3.6 indica es he equi alence o he gy og oups in Tab. 2.
Easy compu a ion shows ha he cen e ing and biasing in he Euclidean BN (Eq. (13)) can be iewed
as gy o addi ion (Eq. (3)) in
Rn
, and he scaling can be iewed as gy o scala p oduc (Eq. (4)),
scaling in he angen space a he iden i y elemen . Inspi ed by his, we de ine he no maliza ion o e
gy og oups by gy o addi ion and gy o scala p oduc . Gi en a ba ch o ac i a ions
{Pi...N ∈ M}
, we
de ined he co e ope a ions o Gy oBN as
∀i≤N, ˜
Pi=
Biasing
z}|{
B⊕



Scaling
z }| {
s
√ 2+ϵ⊙

Cen e ing
z }| {
⊖M⊕Pi



,(15)
whe e
M∈ M
and
2
a e gy omean and gy o a iance,
B∈ M
is he biasing pa ame e ,
s∈R
is
he scaling pa ame e , and ϵis a small alue o nume ical s abili y.
Theo em 4.1 (Homogenei y).[
↓
]Supposing
{M,⊕}
is a pseudo- educ i e gy og oup wi h any
gy a ion
gy [·,·]
as a gy oisome y, o
N
samples
{Pi...N ∈ M}
, we ha e he ollowing p ope ies:
Homogenei y o gy omean: FM({B⊕Pi}) = B⊕FM({Pi}),∀B∈ M,(16)
Homogenei y o dispe sion om E:1
NXN
i=1 d2
g y( ⊙Pi, E) = 2
NXN
i=1 d2
g y(Pi, E),(17)
The mos impo an p ope y o he Euclidean BN (Eq. (13)) lies in i s abili y o no malize da a
dis ibu ion by he con ol o e sample mean and a iance. Simila ly, ou o mula ion in Eq. (15) can
also no malize gy omean and gy o a iance. Speci ically, gi en a pseudo- educ i e gy og oup wi h
isome ic gy a ions, Eq. (16) indica es ha he cen e ing and biasing can ans e he gy omean, while
Eq. (17) can scale he sample a iance, since a e cen e ing, he esul ing gy omean is he iden i y
elemen E.
To inalize ou Gy oBN, we de ine he unning mean upda es o e gy og oups as he bina y ba ycen e
based on gy odis ance:
Ba η(P1, P2) = a gminQ∈M ηd2
g y (P1, Q) + (1 −η)d2
g y (P2, Q),wi h η∈[0,1].(18)
No ably, when he gy odis ance is iden ical o he geodesic dis ance, he bina y ba ycen e can be
calcula ed by geodesic.
Wi h all he abo e ing edien s, he gene al amewo k o ou Gy oBN is p esen ed in Alg. 1. Thm. 4.1
indica es ha gi en a pseudo- educ i e gy og oup wi h any gy a ion as a gy oisome y, ou Gy oBN
enjoys a heo e ical gua an ee o con ol o e he gy o s a is ics. Speci ically, o he gy og oups
in Tab. 2, ou Gy oBN can con ol he gy omean and gy o a iance. Besides, as he gy omean and
gy o a iance a e iden ical o he Riemannian coun e pa s, he Gy oBNs on hese gy og oups also
no malize he Riemannian s a is ics. Especially, simple compu a ion shows ha ou Gy oBN eco e s
he s anda d Euclidean BN (Io e & Szegedy,2015) when M=Rn.
6
Published as a con e ence pape a ICLR 2025
Algo i hm 1: Gy og oup Ba ch No maliza ion (Gy oBN)
Requi e : ba ch o ac i a ions {P1...N ∈ M}, small posi i e cons an ϵ, and momen um
η∈[0,1], unning mean M , unning a iance 2
, biasing pa ame e B∈ M,
scaling pa ame e s∈R.
Re u n :no malized ba ch {˜
P1...N ∈ M}
1i aining hen
2Compu e ba ch mean Mband a iance 2
bo {P1...N };
3Upda e unning s a is ics M = Ba γ(Mb, M ), 2
=γ 2
b+ (1 −γ) 2
;
4end
5(M, 2)=(Mb, 2
b)i aining else (M , 2
)
6∀i≤N, ˜
Pi=B⊕s
√ 2+ϵ⊙(⊖M⊕Pi)
5
AGYRO PERSPECTIVE FOR THE EXISTING RIEMANNIAN NORMALIZATIONS
Se e al exis ing Riemannian no maliza ion me hods on di e en geome ies enjoy heo e ical con ol
o sample mean and a iance, including LieBN on gene al Lie g oups (Chen e al.,2024b), and
SPDBNs based on he speci ic AIM geome y (B ooks e al.,2019;Koble e al.,2022b;a). This
subsec ion u he e eals ha hey a e conc e e implemen a ions o ou Gy oBN.
5.1 LIEBN AS A SPECIAL CASE OF GYROBN
Chak abo y (2020, Algs. 3-4) i s p oposed Riemannian no maliza ion on ma ix Lie g oups unde
a speci ic dis ance. Chen e al. (2024b) ex ended hei amewo k in o gene al Lie g oups, e e ed o
as LieBN, wi h a heo e ical con ol o e Riemannian mean and a iance. This subsec ion shows ha
LieBN is indeed a special case o ou Gy oBN.
LieBN is es ablished unde a le -in a ian me ic on he Lie g oup. The cen e ing and biasing a e
de ined by le g oup ansla ion, and scaling is de ined by he scaling on he angen space a he
iden i y elemen (Chen e al.,2024b, Eq.13-15). As e e y Lie g oup is au oma ically a gy og oup,
he gy o ansla ion is he exac g oup ansla ion. The e o e, he cen e ing, biasing, and scaling a e
he same unde Gy oBN and LieBN. Howe e , he mean, a iance, and unning mean upda es on
LieBN a e de ined based on he geodesic dis ance. In con as , he coun e pa s on he Gy oBN a e
based on gy odis ance. Ne e heless, he ollowing p oposi ion demons a es he equi alence o hese
ope a o s unde he Gy oBN wi h LieBN.
P oposi ion 5.1. [
↓
]Gi en a Lie g oup wi h a le -in a ian me ic, he gy odis ance and geodesic
dis ance a e iden ical. The Gy oBN is, he e o e, iden ical o he LieBN (Chen e al.,2024b, Alg. 1).
Chen e al. (2024b) implemen ed LieBN on ou le -in a ian geome ies, including SPD mani old
wi h AIM
2
, LEM and LCM, and o a ion ma ices. Acco ding o P op. 5.1, hese implemen a ions
a e immedia ely he special cases o ou Gy oBN.
5.2 AIM-BASED SPDBNS AS SPECIAL CASES OF GYROBN
Se e al RBNs on he SPD mani old we e de eloped based on AIM (B ooks e al.,2019;Koble e al.,
2022b;a). The co e ope a ions o hese app oaches can be exp essed as he ollowing:
No maliza ion: ∀i≤N, ˜
Pi=B1
2M−1
2PiM−1
2s
√ 2+ϵB1
2(19)
whe e
M
and
2
a e he Riemannian mean and a iance, i.e., he F éche mean and a iance unde he
geodesic dis ance. The unning mean is upda ed by bina y ba ycen e unde he geodesic dis ance.
P op. 3.6 indica es ha , unde he AIM geome y, he gy odis ance is iden ical o he geodesic
dis ance. The e o e, he gy omean and gy o a iance a e iden ical o he Riemannian mean and
a iance. The unning mean upda es a e also iden ical unde he gy odis ance and geodesic dis ance.
Besides, simple compu a ions show ha Eq. (19) is exac ly he speci ic implemen a ion o Eq. (15)
unde
{Sn
++,⊕AI, gAI}
, whe e
gAI
deno es AIM. The e o e, he SPDBNs de eloped by B ooks e al.
(2019); Koble e al. (2022b;a) a e also special cases o ou G yoBN.
2
AIM is le -in a ian w. . . he Lie g oup ope a ion
P⊕LieAI Q=LQL⊤
wi h
L
as he Cholesky ac o o
P=LL⊤(Thanwe das & Pennec,2022). This g oup s uc u e di e s om he one p esen ed in Tab. 2.
7
Published as a con e ence pape a ICLR 2025
Rema k 5.2.B ooks e al. (2019) only conside cen e ing and biasing. Koble e al. (2022b) use
unning mean o cen e ing du ing he aining. Koble e al. (2022a) use di e en momen um o
upda e unning s a is ics o aining and es ing and mul i-channel mechanisms o domain adap a ion.
Ne e heless, all o hem a e based on Eq. (19). The e o e, icks such as mul i-channel and sepa a e
momen um can also be applied o ou Gy oBN. This is wha we mean by claiming ha ou Gy oBN
inco po a es hei app oaches.
6 GYROBNS ON GRASSMANNIAN AND HYPERBOLIC SPACES
As indica ed by P op. 3.6 and Thm. 4.1, ou Gy oBN can be implemen ed on he G assmannian
and hype bolic spaces, wi h he abili y o no malize sample s a is ics. Ou Alg. 1allows us o
implemen Gy oBN in a plug-and-play manne . This sec ion cla i ies addi ional echnical de ails
ega ding i s applica ion in hese wo spaces. As P op. 3.6 has demons a ed he equi alence o he
gy odis ance and geodesic dis ance on hese spaces, we use he e ms "gy omean" and "gy o a iance"
in e changeably wi h hei Riemannian coun e pa s.
6.1 GRASSMANNIAN GYROBN
We ocus on he ONB pe spec i e. To he bes o ou knowledge, his is he i s RBN on he
G assmannian. Gi en a ba ch o ac i a ions
{U1···N}
o e
{G (p, n),⊕G }
, Eq. (15) can be exp essed
as
Cen e ing o he iden i y Ip,n:U1
i= mexp −[MM⊤,e
Ip,n]Ui,(20)
Scaling he dispe sion om Ip,n:U2
i= mexp s
√ 2+ϵ[U1
i(U1
i)⊤,e
Ip,n]Ip,n,(21)
Biasing owa ds pa ame e B∈ M:U3
i= mexp [B, e
Ip,n]U2
i(22)
whe e
(·) = g
Loge
Ip,n (·)
wi h
g
Log
as he Riemannian loga i hm unde he p ojec o pe spec i e, and
M
is he Riemannian mean o
{U1···N}
, and
e
Ip,n =Ip,nI⊤
p,n
is he iden i y elemen unde he p ojec o
pe spec i e. The Riemannian mean can be calcula ed by he Ka che low (Ka che ,1977). We use
Alg. 5.3 by Bendoka e al. (2024) o a s able and e icien compu a ion o he Riemannian loga i hm
equi ed in he Ka che low. Fo
[MM⊤,e
Ip,n]
and
[U1
i(U1
i)⊤,e
Ip,n]
, inspi ed by Bendoka e al.
(2024, Alg. 5.3) and Nguyen e al. (2024, P op. 3.12), we ha e he ollowing o as compu a ion.
P oposi ion 6.1. [
↓
]Gi en
U= (U⊤
1, U⊤
2)⊤∈G (p, n)
wi h
U1∈Rp×p
and
U2∈R(n−p)×p
, hen
we ha e he ollowing
[UU⊤,e
Ip,n] = 0−e
UT
2
e
U20,(23)
whe e
e
U2=U2Qa csin( ˆ
S)
ˆ
SR⊤
and
U⊤
1
SVD
:= QSR⊤
. He e
S
is in ascending o de ,
Q
and
R
a e
column-wisely lipped acco dingly, and ˆ
S=√1−S2.
Rema k 6.2.Al hough we ocus on he Gy oBN unde he ONB pe spec i e, he Gy oBN unde
he p ojec o pe spec i e can be calcula ed ia he ONB pe spec i e by he ollowing p ocess: (1)
mapping da a in o he ONB pe spec i e by
π−1:
G (p, n)→G (p, n)
; (2) no malizing da a by he
Gy oBN unde
G (p, n)
; (3) mapping no malized da a back o
G (p, n)
by
π
. Technical de ails a e
p esen ed in App. E.
6.2 HYPERBOLIC GYROBN
We ocus on he Poinca é ball model o e he hype bolic space, i.e.,
Pn
K
. The speci ic mani es a ion
can be conduc ed in a plug-in manne . We simply need o plug he equi ed ope a o s om Tab. 2and
Tab. 8in o Alg. 1. Besides, he Poinca é F éche mean can be calcula ed by Lou e al. (2020, Alg. 1)].
Rema k 6.3.As shown by Cannon e al. (1997), he e a e i e isome ic models ha one can wo k o
he hype bolic spaces. Al hough we ocus on he Poinca é ball, he Gy oBN unde o he isome ic
me ics can also be easily cons uc ed ia he Poinca é Gy oBN. The o e all p ocess is simila o
Rmk. 6.2. Fo mo e de ail, please e e o Lem. E.1 and Thm. E.2.
8
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7 EXPERIMENTS
Ou Gy oBN laye s a e model-agnos ic and can be seamlessly in eg a ed in o any ne wo k ope a ing
o e he gy ospaces lis ed in Tab. 2. This sec ion e alua es he e ec i eness o ou Gy oBN on
G assmannian and hype bolic neu al ne wo ks.
7.1 EXPERIMENTS ON THE GRASSMANNIAN
Implemen a ion. We ocus on a newly de eloped G assmannian ne wo k, Gy oG (Nguyen &
Yang,2023), which eplaces he non-in insic ans o ma ion block (FRMap + ReO h laye s) in
he classic G Ne (Huang e al.,2018) wi h G assmannian le gy o ansla ion. This modi ica ion
has demons a ed imp o ed nume ical pe o mance and s abili y (Nguyen & Yang,2023). Gy oG
is cons uc ed o e he ONB G assmannian and consis s o h ee basic blocks: le gy o ansla ion,
pooling (Huang e al.,2018), and he p ojec ion map (P ojMap) (Huang e al.,2018). The p ojec ion
map unc ions as an ac i a ion laye ha maps da a in o symme ic ma ices o classi ica ion.
Following Nguyen & Yang (2023), we e alua e ou me hod on skele on-based ac ion ecogni ion
asks, including he HDM05 (Mülle e al.,2007), NTU60 (Shah oudy e al.,2016), and NTU120 (Liu
e al.,2019) da ase s, ocusing on mu ual ac ions o NTU60 and NTU120. Fo a ai compa ison,
we also ex end Mani oldNo m (Chak abo y,2020, Alg. 1-2) and RBN (Lou e al.,2020, Alg.
2) o he G assmannian. Al hough hese wo BN me hods we e no o iginally implemen ed o e
he G assmannian, hey can be adap ed by le e aging Riemannian ope a o s such as geodesics,
exponen ial and loga i hmic maps, and pa allel anspo . The key di e ence is ha ou Gy oBN
can no malize da a dis ibu ions ac oss di e en geome ies, whe eas he o he wo me hods canno .
Fu he de ails on da ase s, implemen a ion, and aining e iciency a e p o ided in App. F.2.
Table 3: Compa ison o Gy oBN agains o he G assmannian BNs unde Gy oG backbone.
BN None Mani oldNo m-G RBN-G Gy oBN-G
Acc. Mean±s d Max Mean±s d Max Mean±s d Max Mean±s d Max
HDM05 48.97±0.24 49.23 49.67±0.76 50.41 48.64±0.77 49.49 51.89±0.37 52.43
NTU60 70.13±0.16 70.32 68.56±0.43 69.14 67.77±0.52 68.35 72.60±0.04 72.65
NTU120 53.76±0.18 53.96 51.41±0.38 51.92 50.56±0.22 50.82 55.47±0.10 55.59
Table 4: Abla ion o G assmannian Gy oBN unde a ious ne wo k a chi ec u es.
HDM05 NTU60 NTU120
A chi ec u e 1Block 2Block 3Block 4Block 1Block 2Block 3Block 4Block 1Block 2Block 3Block 4Block
Gy oG 49.23 49.09 47.02 27.36 70.32 70.14 70.23 65.03 53.96 54.1 54.59 47.59
Gy oG BN 52.43 50.62 51.56 30.29 72.65 71.93 72.25 66.67 55.59 56.15 54.63 48.9
Main esul s. We compa e ou Gy oBN wi h p e ious BN me hods unde he 1Block Gy oG
backbone, which consis s o one block o gy o ansla ion and pooling laye s ollowed by a P ojMap
laye . We apply he BN a e he pooling laye . The 5- old esul s a e p esen ed in Tab. 3. Gy oBN
consis en ly deli e s imp o ed pe o mance, enhancing he a e age pe o mance o he anilla Gy oG
by 2.92%, 2.47%, and 1.71%. In con as , G assmannian Mani oldNo m and RBN could deg ade
he anilla Gy oG ne wo k, pa icula ly on he NTU60 and NTU120 da ase s. This is p ima ily due
o Gy oBN’s heo e ical gua an ee o no malizing sample s a is ics, a capabili y lacking in p e ious
me hods such as Mani oldNo m and RBN, as summa ized in Tab. 1. Addi ionally, we obse e ha
Gy oBN mi iga es he pe o mance gap be ween aining and es ing, indica ing i s abili y o enhance
he model’s gene aliza ion. De ailed discussions a e p o ided in App. F.5. O e all, he abo e indings
highligh he e ec i eness o ou Gy oBN in acili a ing ne wo k aining.
Abla ions on he a chi ec u e. We u he alida e ou Gy oBN ac oss di e en a chi ec u es wi hin
he Gy oG baseline, which includes up o ou blocks o gy o ansla ion and pooling. Gy oBN is
applied a e he i s pooling laye , and we deno e Gy oG wi h Gy oBN as Gy oG BN. As implied
by Tab. 3, bo h Gy oG and Gy oG BN exhibi ela i ely small a iances, allowing us o conduc
abla ions using a single ial. Tab. 4 epo s he esul s ac oss all h ee da ase s. We obse e ha
Gy oBN consis en ly imp o es he pe o mance o he anilla Gy oG baseline, highligh ing he
e ec i eness o he Gy oBN amewo k. No ably, as he ne wo k dep h inc eases, he pe o mance o
he Gy oG backbone, wi h o wi hou Gy oBN, deg ades. This is because he dimensionali y o he
9
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G P oo s 30
G.1 P oo o P op. 3.2 ................................... 30
G.2 P oo o Thm. 3.3 ................................... 31
G.3 P oo o Thm. 3.5 ................................... 32
G.4 P oo o P op. 3.6 ................................... 32
G.5 P oo o Thm. 4.1 ................................... 35
G.6 P oo o P op. 5.1 ................................... 36
G.7 P oo o P op. 6.1 ................................... 37
16

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A LIMITATION
As shown in P op. 3.6, se e al geome ies, including SPD, G assmannian, hype bolic, and hype phe -
ical mani olds, ha e gy o s uc u es ha can enable Gy oBN wi h a heo e ical con ol o e sample
s a is ics. Howe e , some geome ies do no ha e gy o s uc u es, o hei gy o s uc u es a e s ill
un ouched. The e o e, Gy oBN canno be es ablished on hese mani olds. We will explo e o he
echniques o es ablish an RBN amewo k o hese non-gy o geome ies.
B NOTATIONS
Tab. 6summa izes all he no a ions in he main pape .
Table 6: Summa y o no a ions.
No a ion Explana ion
{G, ⊕}o abb e ia ed as GA gy og oup
{M,⊕, g}o abb e ia ed as MA Riemannian mani old {M, g}wi h a gy og oup s uc u e induced by g
1
iden i y map
TPMTangen space a P∈ M
gp(·,·)o ⟨·,·⟩PRiemannian me ic a P∈ M
∥·∥PThe no m induced by ⟨·,·⟩Pon TPM
dgeo(·,·)Geodesic dis ance
LogPRiemannian loga i hm a P
ExpPRiemannian exponen ia ion a P
∗,P The di e en ial map o he smoo h map a P∈ M
PTP→QPa allel anspo a ion along he geodesic connec ing Pand Q
EGy o iden i y o {M,⊕}
⊖PG oup in e se o P∈ M
⊙Gy o scala p oduc
gy [·,·]Gy a ion
⟨·,·⟩g Gy o inne p oduc
∥·∥g Gy ono m
dg y(·,·)Gy odis ance
FM F éche mean unde a gy odis ance
Ba η(·,·)Bina y ba ycen e based on a gy odis ance
⟨·,·⟩ The s anda d F obenius inne p oduc
∥·∥ No m induced by he s anda d F obenius inne p oduc
mlog Ma ix loga i hm
mexp Ma ix exponen ia ion
LCholesky decomposi ion
Dlog The diagonal elemen -wise loga i hm
ψLC Dlog ◦L
Sn
++ The SPD mani old
SnThe Euclidean space o symme ic ma ices
Pn
Kn-dimensional Poinca é ball wi h cu a u e K < 0
Rnn-dimensional Euclidean space
Dn
K,n-dimensional p ojec ed hype sphe e wi h cu a u e K > 0
MKCons an Cu a u e Spaces (CCS)
G (p, n)The G assmannian unde he ONB pe spec i e
G (p, n)The G assmannian unde he p ojec o pe spec i e
Ip,n The G assmannian iden i y unde he ONB pe spec i e
e
Ip,n The G assmannian iden i y unde he p ojec o pe spec i e
πThe Riemannian isome y om G (p, n)on o
G (p, n)
[·,·]Ma ix commu a o
[·]An elemen in G (p, n), which is a equi alen class
Inn×niden i y ma ix
PθMa ix powe o SPD ma ix P
⊕AI,⊕LE and ⊕LC Gy o addi ions on he SPD mani old unde AIM, LEM and LCM
e
⊕G and ⊕G Gy o addi ions on he G assmannian unde he ONB and p ojec o pe spec i es
⊕KGy o addi ions on he CCS
(·) (·) = g
Loge
Ip,n (·)wi h g
Log as he Riemannian loga i hm on
G (p, n)
gAI,gLE and gLC AIM, LEM, LCM on he SPD mani old
gG and egG Riemannian me ics on he G assmannian unde he ONB and p ojec o pe spec i es
anK anK= an i k > 0, eli K > 0, anK= anh
17
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C PRELIMINARIES
C.1 RIEMANNIAN GEOMETRY
Mani olds can be in ui i ely unde s ood as locally Euclidean spaces. Di e en ials gene alize he
concep o classical de i a i es. Fo a de ailed in oduc ion o smoo h mani olds, we e e eade s
o Tu (2011); Lee (2013). A Riemannian mani old is a mani old equipped wi h a Riemannian
me ic, which can be in ui i ely in e p e ed as a poin -wise inne p oduc . This me ic allows o
he adap a ion o a ious Euclidean ope a o s o he mani old se ing. Fo an in-dep h discussion on
Riemannian mani olds, see Do Ca mo & Flahe y F ancis (1992); Lee (2018).
De ini ion C.1 (Riemannian Mani olds (Do Ca mo & Flahe y F ancis,1992)).A Riemannian me ic
on
M
is a smoo h symme ic co a ian 2- enso ield on
M
, which is posi i e de ini e a e e y poin .
A Riemannian mani old is a pai
{M, g}
, whe e
M
is a smoo h mani old and
g
is a Riemannian
me ic.
The isome ies gene alize he bijec ion in he se heo y in o he Riemannian geome y. I wo
mani olds a e isome ic, hey can be iewed as equi alen . The Riemannian ope a o s in hese
wo mani olds a e also closely ela ed (Chen e al.,2024d, App. A. 2). The ollowing de ines he
Riemannian isome y.
De ini ion C.2 (Isome ies (Lee,2018)).I
{M, g}
and
{
M,eg}
a e bo h Riemannian mani olds, a
smoo h map :M →
Mis called a (Riemannian) isome y i i is a di eomo phism ha sa is ies
gp(V, W) = eg (p)( ∗,p(V), ∗,p(W)),(24)
whe e
∗,p(·) : TpM → T (p)
M
is he di e en ial map o
a
p∈ M
, and
V, W ∈TpM
a e wo
angen ec o s.
The exponen ial & loga i hmic maps and pa allel anspo a ion a e also c ucial o Riemannian
app oaches in machine lea ning. To bypass he no a ion bu dens caused by hei de ini ions, we e iew
he geome ic ein e p e a ions o hese ope a o s (Pennec e al.,2006;Do Ca mo & Flahe y F ancis,
1992). In de ail, in a mani old
M
, geodesics co espond o s aigh lines in Euclidean space. A
angen ec o
−→
xy ∈TxM
can be locally iden i ied o a poin
y
on he mani old by geodesic s a ing
a
x
wi h an ini ial eloci y o
−→
xy
, i.e.
y= Expx(−→
xy)
. On he o he hand, he loga i hmic map is he
in e se o he exponen ial map, gene a ing he ini ial eloci y o he geodesic connec ing
x
and
y
,
i.e.,
−→
xy = Logx(y)
. These wo ope a o s gene alize he idea o addi ion and sub ac ion in Euclidean
space. Fo he pa allel anspo a ion
PTx→y(V)
, i is a gene aliza ion o pa allelly mo ing a ec o
along a cu e in Euclidean space. we summa ize he ein e p e a ion in Tab. 7.
Table 7: The geome ic ein e p e a ions o Riemannian ope a o s.
Ope a ions Euclidean spaces Riemannian mani olds
S aigh line S aigh line Geodesic
Sub ac ion −→
xy =y−x−→
xy = Logx(y)
Addi ion y=x+−→
xy y = Expx(−→
xy)
Pa allelly mo ing V→VPTx→y(V)
A Lie g oup is a mani old wi h a smoo h g oup s uc u e. I is a combina ion o algeb a and geome y.
De ini ion C.3 (Lie G oups).A mani old is a Lie g oup, i i o ms a g oup wi h a g oup ope a ion
⊙
such ha m(x, y)7→ x⊙yand i(x)7→ x−1
⊙a e bo h smoo h, whe e x−1
⊙is he g oup in e se o x.
The ollowing a e some nai e examples o he Lie g oup:
1. The se o eal numbe s R, whose g oup ope a ion is he addi ion.
2.
The se o
n×n
in e ible ma ices
GL(n)
, whose g oup ope a ion is he ma ix p oduc .
This g oup is known as he gene al linea g oup.
3.
The se o
n×n
o hogonal ma ices
O(n)
, whose g oup ope a ion is he ma ix p oduc .
This g oup is a subg oup o GL(n), known as he o hogonal g oup.
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C.2 GYROVECTOR SPACES
Gy og oups in De . 2.1 gene alize g oups o non-associa i e algeb aic sys ems by gy a ions. Simila ly,
he gy o ec o space gene alizes he ec o space, which has shown imp essi e success in hype bolic
geome y (Unga ,2005;2009;2012;2014).
De ini ion C.4 (Gy o ec o Spaces (Nguyen,2022a)).A gy ocommu a i e gy og oup
{G, ⊕}
equipped wi h a scala mul iplica ion
⊙:R×G→G
is called a gy o ec o space i i sa is ies he
ollowing axioms o s, ∈Rand a, b, c ∈G:
(V1) 1⊙a=a, 0⊙a= ⊙e=e, and (−1) ⊙a=⊖a.
(V2) (s+ )⊙a=s⊙a⊕ ⊙a.
(V3) (s )⊙a=s⊙( ⊙a).
(V4) gy [a, b]( ⊙c) = ⊙gy [a, b]c.
(V5) gy [s⊙a, ⊙a] =
1
, whe e
1
is he iden i y map.
Gy o ec o spaces gene alize ec o spaces o cu ed geome ies, such as he SPD and G assman-
nian mani olds (Nguyen,2022a). While e aining amilia p ope ies like dis ibu i i y (V2) and
associa i i y (V3), hey inco po a e he complexi ies o gy a ions.
C.3 MATRIX AND VECTOR MANIFOLDS
C.3.1 DEFINITIONS
SPD: The se
Sn
++
o
n×n
SPD ma ices o m a mani old, named he SPD mani old (Pennec e al.,
2006). We ocus on h ee popula Riemannian me ics on he SPD mani old: A ine-In a ian Me ic
(AIM) (Pennec e al.,2006), Log-Euclidean Me ic (LEM) (A signy e al.,2005), and Log-Cholesky
Me ic (LCM) (Lin,2019).
G assmannian: The G assmannian is he se o
p
-dimensional subspace o
n
-dimensional ec o
space (Tu,2011), which has wo ma ix ep esen a ions (Bendoka e al.,2024):
P ojec o pe spec i e:
G (p, n) = {P∈ Sn:P2=P, ank(P) = p},
ONB pe spec i e: G (p, n) = {[U]:[U] := {e
U∈S (p, n)|e
U=UR, R ∈O(p)}},(25)
whe e
Sn
is he Euclidean space o symme ic ma ices,
S (p, n)
is he S ie el mani old, and
O(p)
is
he o hogonal g oup. By abuse o no a ions, we use
[U]
and
U
in e changeably o he elemen o
G (p, n)
. As shown by Helmke & Moo e (2012), he ONB pe spec i e
G (p, n)
is di eomo phism
o
G (p, n)by
π(U) = UU⊤,∀U∈G (p, n),(26)
whe e he
n×p
column-wise o hono mal ma ix
U
should be mo e p ecisely unde s ood as a
ep esen a i e o an equi alence class (Bendoka e al.,2024).
CCS: The Poinca é ball
Pn
K
o he hype bolic space (Unga ,2009;Ganea e al.,2018), p ojec ed
hype sphe e
Dn
K
o he hype sphe e (Skopek e al.,2019), and s anda d Euclidean space
Rn
(Zo ich
& Paniagua,2016) a e mo e gene ally called he Cons an Cu a u e Space (CCS) (Do Ca mo &
Flahe y F ancis,1992), as hey ha e cons an sec ional cu a u e
K
. The
n
-dimensional Poinca é
ball and p ojec ed hype sphe e a e ep esen ed as
Pn
K=x∈Rn:⟨x, x⟩<−1
K
wi h
K < 0
and
Dn
K=Rnwi h K > 0. When K= 0, he CCS becomes he s anda d Euclidean space Rn.
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C.3.2 GYRO AND RIEMANNIAN STRUCTURES
Fo a ma ix mani old Mand a CCS MK, we make he ollowing no a ions:
{M, g}=












{Sn
++, gAI}(The SPD mani old unde AIM)
{Sn
++, gLE}(The SPD mani old unde LEM)
{Sn
++, gLC}(The SPD mani old unde LEM)
{G (p, n), gG }(The G assmannian unde he ONB pe spec i e)
{
G (p, n),egG }(The G assmannian unde he p ojec o pe spec i e)
(27)
MK=


Pn
K, o K < 0
Rn, o K= 0
Dn
K, o K > 0
(28)
anK= an i K > 0
anh i K < 0(29)
No a ions in Tab. 2:We summa ize all he necessa y g oup ope a ions in Tab. 2wi h he ollowing
no a ions. Gi en any
P, Q ∈ M
wi h
M
as
Sn
++
o
G (p, n)
, and
x, y ∈ MK
wi h
MK
as
Pn
K(K < 0)
,
Dn
K(K > 0)
o
Rn(K= 0)
, we make he ollowing no a ions. Fo he G assmannian,
U=π−1(P)
and
V=π−1(Q)
a e he ONB ep esen a ions. We deno e ma ix exponen ial,
ma ix loga i hm, and Cholesky decomposi ion as
mexp
,
mlog
, and
L
, espec i ely. We deno e
ψLC = Dlog ◦L
, whe e
ψLC
is he diagonal loga i hm. As shown by Chen e al. (2024e), LCM and
he associa ed Lie g oup on
Sn
++
a e pulled back by
ψLC
o he Euclidean space o lowe iangula
ma ices.
In
is he
n×n
iden i y ma ix,
Ip,n = (Ip,0)⊤∈Rn×p
, and
e
Ip,n =π(Ip,n)
. Fo he
G assmannian
G (n, p)
,
Ω=[P, e
Ip,n]
, whe e
P= Loge
Ip,n (P)
and
[·,·]
is he ma ix commu a o .
We deno e ⟨·,·⟩ and ∥·∥as he s anda d (ma ix & ec o ) inne p oduc and no m.
We u he make he ollowing no a ions o he ela ed Riemannian ope a o s in Tab. 8. Gi en
P∈ M
(
x∈ MK
), we deno e he angen ec o as
V∈TPM
(
∈TxMK
). We deno e
he geodesic dis ance, Riemannian loga i hm, and Riemannian exponen ial as
dgeo
,
Log
and
Exp
,
espec i ely.
Table 8: Riemannian geome ies o se e al ma ix and ec o mani olds. Fo CCS MK, we p esen
he ope a o s o
K= 0
, as when
K= 0
, all he Riemannian ope a o s a e educed o he amilia
ec o ope a o s.
Mani olds dgeo(P, Q)o dgeo(x, y) LogPQo LogxyExpPVo Expx Re e ences
{Sn
++, gLE} ∥mlog(P)−mlog(Q)∥(mlog∗,P )−1(mlog(Q)−mlog(P)) mexp mlog(P) + mlog∗,P (V)(A signy e al.,2005)
{Sn
++, gAI}mlog Q−1
2PQ−1
2P1
2mlog P−1
2QP−1
2P1
2P1
2mexp P−1
2V P−1
2P1
2(Pennec e al.,2006)
{Sn
++, gLC} ∥ψLC(P)−ψLC(Q)∥(L−1)∗,L (ψLC(Q)−ψLC(P))) ψ−1
LC (ψLC(P)+(ψLC)∗,P (V)) (Lin,2019)
(Chen e al.,2024e)
G (p, n)∥a ccos(Σ)∥
P⊤QSVD
:= OΣR⊤
Oa c an(Σ)R⊤
(In−PP ⊤)Q(P⊤Q)−1SVD
:= OΣR⊤PR cos(Σ)RT+Osin(Σ)RT
VSVD
:= OΣR⊤
(Edelman e al.,1998)
(Bendoka e al.,2024)
G (p, n)1
2∥mlog ((In−2Q) (In−2P))∥1
2[mlog ((In−2Q) (In−2P)) , P ] mexp([V, P ])Pmexp(−[V, P ]) (Bendoka e al.,2024)
(Ba zies e al.,2015)
MK2
√|K| an−1
Kp|K|∥−x⊕Ky∥2
√|K|λK
x
an−1
Kp|K|∥−x⊕Ky∥−x⊕Ky
∥−x⊕Ky∥x⊕K anKp|K|λK
x∥ ∥
2
√|K|∥ ∥(Do Ca mo & Flahe y F ancis,1992)
(Pe e sen,2006)
(Skopek e al.,2019)
Rema k C.5 (The G assmannian and cu locus).Due o he cu locus o he G assmannian, he
loga i hm map does no exis globally (Bendoka e al.,2024). In his pape , when we use LogP(Q)
on he G assmannian, we implici ly assume
P
and
Q
a e no in each o he ’s cu locus. Besides, mo e
p ecisely, he gy o addi ion and scala mul iplica ion on G assmannian a e also no globally de ined
(Nguyen,2022a), due o he cu locus. Following Nguyen (2022a) and Nguyen & Yang (2023), we
implici ly assume he gy o ope a ions a e well-de ined on he G assmannian.
C.4 INTUITIVE EXPLANATIONS OF GYROGROUPS
This subsec ion in ui i ely explains gy og oups by con as wi h he i ial
R
. Gy og oups a e
p oposed o gene alize he concep o addi ion in Ro Rn o non-Euclidean spaces.
Gy og oups in De . 2.1 ex end he concep o g oups o non-associa i e algeb aic sys ems, i.e.,a⊕
(b⊕c)= (a⊕b)⊕c
. The gy og oups o e he mani olds in Tab. 2a e de ined by Eqs. (3), (5) and (6).
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Mo e impo an ly, hese de ini ions a e na u al gene aliza ions o Euclidean ec o ope a ions. Take
he gy o addi ion Eq. (3) as an example. When he mani old
M
is he Euclidean space
Rn
, he gy o
addi ion de ined by Eq. (3) becomes he amilia ec o addi ion.
Gy a ion & gy oassocia i i y: The gy a ion is an au omo phism
gy [a, b] : G→G
o each
a, b ∈G. I is called an au omo phism as i can p ese e he gy o addi ion:
gy [a, b](c⊕d) = gy [a, b](c)⊕gy [a, b](d),∀c, d ∈G. (30)
The gy a ion is used o gene alize he associa i i y:
a+ (b+c)=(a+b) + c, ∀a, b, c ∈R.(31)
Take he G assmannian gy og oup
{G (p, n),⊕G }
as an example. Fo any
U, V, R ∈ ⊕G
, we ha e
Non-associa i i y: V⊕G (U⊕G R)= (V⊕G U)⊕G R, (32)
Le gy oassocia i i y: V⊕G (U⊕G R) = (V⊕G U)⊕G gy G [U, V ](R).(33)
Eq. (33) di e s wi h Eq. (31) only in a gy a ion. The conc e e exp ession o he gy a ion o e he
G assmannian is p esen ed by Nguyen (2022a, De . 3.18).
Le educ ion is de ined as
gy [a, b] = gy [a⊕b, b]
o any
a
and
b
in he gy og oup
G
. I is called
a "le educ ion" because bin a⊕bcan be canceled ou .
Gy ocommu a i i y in De . 2.2 gene alizes he commu a i e p ope y (
a+b=b+a
) by he gy a ion
ope a o gy [a, b].
Non educ i e gy og oups in De . 2.3 a e elaxed gy og oups, allowing s uc u es whe e he le
educ ion law (G4) does no hold. I s p o o ype comes om he gy og oup o he G assmannian
(Nguyen,2022a, Thm. 3.20), whe e (G4) does no hold.
Rela ion be ween le educ ion, non- educ ion, & pseudo- educ ion: The le educ ion can
induce many basic p ope ies o gy og oups ( see Thm. D.1 and Unga (2009, Thms. 1.13 and
1.14)). Howe e , non- educ ion does no gua an ee se e al basic p ope ies. This d awback will
unde mine he a ionali y o he non- educ i e gy og oups. The e o e, as an in e media e, we p opose
pseudo- educ ion. The associa ed pseudo- educ i e gy og oups main ain mos o he basic p ope ies
o gy og oups. Please e e o Thm. D.1 and i s ema k o mo e de ails.
D FIRST PSEUDO-REDUCTIVE GYROGROUPS PROPERTIES
Theo em D.1 (Fi s Pseudo- educ i e Gy og oups P ope ies).Le
{G, ⊕}
be a pseudo- educ i e
gy og oup. Fo any elemen s P, Q, R, X ∈G, we ha e:
1. I P⊕Q=P⊕R, hen Q=R(Gene al Le Cancella ion law; see (9) below).
2. gy [E, P] =
1
o any le iden i y Ein G.
3. gy [X, P] =
1
o any le in e se Xo Pin G.
4. The e is P le iden i y which is P igh iden i y.
5. The e is only one le iden i y.
6. E e y le in e se is P igh in e se.
7. The e is only one le in e se, ⊖P, o P, and ⊖(⊖P) = P.
8. The le cancella ion law: ⊖P⊕(P⊕Q) = Q.
9. The gy a o iden i y: gy [P, Q]X=⊖(P⊕Q)⊕{P⊕(Q⊕X)}.
10. gy [P, Q]E=E.
11. gy [P, Q](⊖X) = ⊖gy [P, Q]X.
12. gy [P, E] =
1
.
13. The gy osum in e sion law: ⊖(P⊕Q) = gy [P, Q](⊖Q⊖P)
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C edi o he p oo . Mos o he p oo is bo owed om Unga (2009, Thms. 1.13 and 1.14), which
p esen s he co esponding p ope ies o gy og oups. A e e-analyzing hese wo heo ems, we ind
ha mos o he p oo can be eadily ex ended in o pseudo- educ i e gy og oups.
P oo .
This heo em ollows Unga (2009, Thms. 1.13 and 1.14), which p esen s some use ul
p ope ies o gy og oups. We a gue ha all he p ope ies excep
gy [a, a] =
1
a e independen
o he le educ ion law (G4), and a e he e o e sa is ied on pseudo- educ i e gy og oups. All he
p ope ies can be p o en in he same way as he ones o Thms. 1.13 and 1. 14 by Unga (2009). We
summa ize he logic in he ollowing:
• le gy oassocia i i y ⇒1
• le gy oassocia i i y + 1⇒2
• de ini ion ⇒3
• le gy oassocia i i y + 1+3⇒4
• de ini ion ⇒5
• le gy oassocia i i y + (G2) + 1+3+4+5⇒6
•1+6⇒7
• le gy oassocia i i y +3⇒8
• le gy oassocia i i y + le cancella ion in 8⇒9
• gy o iden i y in 9⇒10
•10 ⇒11
• le cancella ion in 8+ gy o iden i y in 9⇒12
• le cancella ion in 8+ gy o iden i y in 9⇒13
Rema k D.2.Fo non- educ i e gy og oups, 2and 3a e agnos ic. The e o e, e e y p ope y based
on 2o 3is no gua an eed, such as 4, and om 6 o 10. The missing o hese basic p ope ies
will unde mine he a ionali y o non- educ i e gy og oups. In con as , mos basic p ope ies o
gy og oups a e p ese ed in ou pseudo- educ i e gy og oups.
E GRASSMANNIAN GYROBN UNDER THE PROJECTOR PERSPECTIVE
Gi en wo isome ic mani olds
{M1, g1}
and
{M2, g2}
, he induced gy o s uc u es in Eq. (3)-
Eq. (9) ha e he ollowing ela ions. We i s p esen a use ul lemma.
Lemma E.1. Gi en mani olds
{M1, g1}
and
{M2, g2}
and a Riemannian isome y
:M1→ M2
,
we ha e he ollowing:
1.
The g oupoid
{M1,⊕1}
induced by
g1
is pseudo- educ i e (le -in a ian ), i he g oupoid
{M2,⊕2}induced by g2is pseudo- educ i e (le -in a ian );
2.
Any gy a ion in
{M1, g1}
p ese es gy ono m i any gy a ion in
{M2, g2}
p ese es
gy ono m;
3. p ese es he gy odis ance.
P oo .
Fi s , we e iew some ac s abou gy og oups unde he Riemannian isome y. As shown by
Nguyen & Yang (2023, Thm. 2.5),
{M1,⊕1}
sa is ies (G1-G3) in De . 2.1 i
{M2,⊕2}
sa is ies
(G1-G3). Besides, −1is an isomo phism sa is ying
−1(P⊕2Q) = −1(P)⊕1 −1(Q),(34)
−1( ⊙2P) = ⊕1 −1(P),(35)
−1(⊖2P) = ⊖1 −1(P),(36)
gy 2[P, Q](R) = gy 1[ −1(P), −1(Q)]( −1(R)).(37)
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whe e
P, Q, R ∈ M2
a e a bi a y poin s,
∈R
is a eal scala ,
⊖i
,
gy i
a e he gy o in e ses and
gy a ions on Mi o i= 1,2. is also an isomo phism wi h simila p ope ies.
We only need o p o e one di ec ion o he i condi ion o he pseudo- educ ion o no m in a iance.
We ocus on ⇒and ollow he abo e no a ions in he ollowing.
Pseudo- educ ion:
gy 2[⊖P, P] = ◦gy 1[ −1(⊖2P), −1(P)] ◦ −1(Eq. (37))
= ◦gy 1[⊖1 −1(P), −1(P)] ◦ −1(Eq. (36))
=
1
.
(38)
Gy ono m in a iance unde gy a ions: Fo simplici y, we deno e he gy ono m, iden i y elemen ,
and Riemannian loga i hm on
Mi
as
∥∥iEi
, and
Logi
, espec i ely. Fi s , he ollowing demons a es
ha he Riemannian isome y p ese es gy ono m:
∥P∥2=Log2
E2(P)E2
(1)
=Log1
−1(E2) −1(P) −1(E2)
(2)
=LogE1 −1(P)E1
= −1(P)1
(39)
(1) As :M1→ M2is a Riemannian isome y, we ha e he ollowing equa ions:
Log2
PQ= ∗, −1(P)Log1
−1(P) −1(Q),∀P, Q ∈ M2,(40)
g2
P(V, W) = g1
−1(P)(( −1)∗,P (V),( −1)∗,P (W)),∀V, W ∈TPM2,(41)
whe e (·)∗is he di e en ial map;
(2) E2= (E1).
Then we ha e he ollowing:
∥gy 2[P, Q](R)∥2
(1)
= gy 1[ −1(P), −1(Q)]( −1(R))2
(2)
=gy 1[ −1(P), −1(Q)]( −1(R))1
(3)
= −1(R)1
(4)
=∥R∥2
(42)
The abo e de i a ion comes om he ollowing.
(1) Eq. (37);
(2) Eq. (39);
(3) Any gy a ion on M1can p ese e he gy ono m;
(4) Eq. (39).
In a iance o gy odis ance unde
:Deno ing he gy odis ance on
Mi
as
di
, o any
U, V ∈ M1
,
we ha e he ollowing:
d1(U, V ) = ∥⊖1U⊕1V∥1
(1)
=∥ (⊖1P⊕1Q)∥2
(2)
=∥⊖1 (P)⊕2 (Q)∥2
= d2( (P), (Q))2
(43)
The abo e de i a ion comes om he ollowing.
(1) p ese es gy ono m;
23
Published as a con e ence pape a ICLR 2025
(2) is an isomo phism.
Gi en a ba ch o ac i a ions {P1···N}on a gy og oup {M,⊕}, we deno e he Gy oBN as
Gy oBN({Pi};B, s, ϵ, η),(44)
whe e
B∈ M
and
s
a e biasing and scaling pa ame e s,
ϵ
is a small posi i e alue, and
η
is he
momen um.
Theo em E.2. Gi en mani olds
{M1, g1}
and
{M2, g2}
and a Riemannian isome y
:M1→
M2
, o a ba ch o ac i a ion
{P1...N }
in
M1
,
Gy oBN1(Pi;B, s, ϵ, γ)
in
M1
can be calcula ed as
Gy oBN1(Pi;B, s, ϵ, γ) = −1(Gy oBN2( (Pi); (B), s, ϵ, γ)) ,(45)
whe e Gy oBN2is he Gy oBN in M2.
P oo .
This heo em is inspi ed by Chen e al. (2024b, hm. 5.3), which cha ac e izes he LieBNs
unde isome ic mani olds. As gy og oups a e na u al gene aliza ions o Lie g oups, ou Gy oBN is
expec ed o ha e simila esul s. The ollowing p oo ollows a simila logic o he one by Chen e al.
(2024b, hm. 5.3), excep ha all ope a ions a e gy o ope a ions.
Fo
i= 1,2
, we deno e Eq. (15) and bina y gy o ba ycen e on
Mi
as
ξi(·|M, 2, B, s)
and
Ba i
η(·,·)
.
Le B={P1...N }and (B) = { (P1...N )}. We only need o show he ollowing:
M2= (M1), 1= 2,(46)
ξ1(Pi|M1, 2, B, s) = −1(ξ2( (Pi)|M2, 2, (B), s)),(47)
Ba 1
η(P, Q) = −1Ba 2
η( (P), (Q)),∀P, Q ∈ M1.(48)
whe e
Mi
and
i
a e he ba ch F éche mean and a iance o e
Mi
o
i= 1,2
. Eqs. (46) and (48)
can be di ec ly ob ained by he in a iance o gy odis ance unde
(Lem. E.1). We only need o show
Eq. (47).
We ha e he ollowing:
−1(ξ2( (Pi)|M2, 2, (B), s)) = −1( (B)⊕2( ⊙2(⊖2M2⊕2 (Pi)))
(1)
= −1◦ (B⊕1( ⊙(⊖1M1⊕1Pi)))
=B⊕1( ⊙(⊖1M1⊕1Pi))
=ξ1(Pi|M, 2, B, s),
(49)
whe e =s
√ 2+ϵ. The abo e de i a ion comes om he ollowing.
(1) is an isomo phism p ese ing gy o ope a ions.
As
π−1:
G (p, n)→G (p, n)
is a Riemannian isome y, Thm. E.2 indica es ha he Gy oBN unde
he p ojec o pe spec i e can be calcula ed by he ONB pe spec i e by he ollowing p ocess:
1. mapping da a in o he ONB pe spec i e by π−1:
G (p, n)→G (p, n);
2. no malizing da a by he Gy oBN unde G (p, n);
3. mapping no malized da a back o
G (p, n)by π.
Besides, bo h Lem. E.1 o P op. 3.6 can gua an ee heo e ical con ol o e he gy omean and
gy o a iance unde he p ojec o pe spec i e.
F EXPERIMENTAL DETAILS AND ADDITIONAL ABLATIONS
F.1 SUMMARY OF OPERATORS IN GRASSMANNIAN AND HYPERBOLIC GYROBNS
The speci ic implemen a ion o Alg. 1on a gy og oup can be ca ied ou in a plug-in manne . This
in ol es simply subs i u ing he equi ed ope a o s om Tabs. 2and 8in o Alg. 1. To s eamline his
p ocess, we summa ize he discussion in Sec. 6in Tab. 9, whe e we p esen all he equi ed ope a o s
o he G assmannian and hype bolic Gy oBNs.
24
Published as a con e ence pape a ICLR 2025
Table 9: Key ope a o s in calcula ing Gy oBN on he G assmannian and hype bolic mani olds. He e
P, Q ∈G (p, n)
a e wo ONB G assmannian poin s, while
x, y ∈Pn
K
a e wo Poinca é ec o s.
O he no a ions ollow om Tabs. 2and 8.
Ope a o G (p, n)Pn
K
Iden i y elemen Ip,n 0∈Rn
P⊕G Qo x⊕Kymexp(Ω)V(1−2K⟨x,y⟩−K∥y∥2)x+(1+K∥x∥2)y
1−2K⟨x,y⟩+K2∥x∥2∥y∥2
⊖G Po ⊖Kxmexp(−Ω)Ip,n −x
⊙G Po ⊙Kxmexp( Ω)Ip,n 1
√|K| anh  anh−1(p|K|∥x∥)x
∥x∥
Ba G
γ(Q, P)o Ba K
γ(y, x) ExpG
P(γLogG
P(Q)) x⊕K(−x⊕Ky)⊙K
F éche Mean Ka che Flow (Ka che ,1977) (Lou e al.,2020, Alg. 1)
0 50 100 150
Epoch
50
60
70
80
90
Acc.
NTU60 T aining
Gy oG - T ain
Gy oG BN - T ain
0 50 100 150
Epoch
50
55
60
65
70
75
Acc.
NTU60 Tes ing
Gy oG - Tes
Gy oG BN - Tes
0 50 100 150
Epoch
50
60
70
80
90
Acc.
NTU120 T aining
Gy oG - T ain
Gy oG BN - T ain
0 50 100 150
Epoch
40
45
50
55
Acc.
NTU120 Tes ing
Gy oG - Tes
Gy oG BN - Tes
Figu e 4: T aining and es ing pe o mance on he NTU da ase s o 1Block Gy oG . Ou BN imp o es
he gene aliza ion abili ies o Gy oG .
F.2 DETAILS ON THE GRASSMANNIAN EXPERIMENTS
F.2.1 DATASETS AND PREPROCESSING
HDM05
3
(Mülle e al.,2007). I consis s o 2,273 skele on-based mo ion cap u e sequences
execu ed by di e en ac o s. Each ame consis s o 3D coo dina es o 31 join s. We emo e he
unde - ep esen ed clips, imming he da ase down o 2086 ins ances sca e ed h oughou 117 classes.
Following Nguyen & Yang (2023), we model each sequence as a 93 ×10 G assmannian ma ix.
NTU60 (Shah oudy e al.,2016). I has 56,880 sequences o 3D skele on da a classi ied in o 60
classes, whe e each ame con ains he 3D coo dina es o 25 o 50 body join s. We use mu ual ac ions
and ollow he c oss- iew p o ocol (Shah oudy e al.,2016). Following Nguyen & Yang (2023), we
model each sequence as a 150 ×10 G assmannian ma ix.
NTU120
4
(Liu e al.,2019). This da ase con ains 114,480 sequences in 120 ac ion classes. We use
mu ual ac ions and ollow he c oss-se up p o ocol (Liu e al.,2019). Following Nguyen & Yang
(2023), we model each sequence as a 150 ×10 G assmannian ma ix.
3h ps:// esou ces.mpi-in .mpg.de/HDM05/
4h ps://gi hub.com/shah oudy/NTURGB-D
25
Published as a con e ence pape a ICLR 2025
G.3 PROOF OF THM.3.5
P oo o Thm. 3.5.Gi en any P, Q, R ∈G, we make he ollowing p oo .
Gy oisome y o he le gy o ansla ion: This p ope y gene alizes Thms. 2.12 and 2.16 by Nguyen
& Yang (2023), which deal wi h he gy o ansla ions in he SPD and G assmannian mani olds. We
ha e he ollowing:
dg y(LP(Q), LP(R)) = dg y(P⊕Q, P ⊕R),
=∥⊖(P⊕Q)⊕(P⊕R)∥g ,
=∥gy [P, Q] (⊖Q⊕R)∥g (le gy o ansla ion),
=∥⊖Q⊕R∥g (gy oisome y o he au omo phism),
= dg y(Q, R).
(63)
Symme y o he gy odis ance:
dg y(P, Q) = ∥⊖P⊕Q∥g ,
=∥⊖(⊖P⊕Q)∥g (Eq. (5) and Eq. (7)),
=∥gy [⊖P, Q](⊖Q⊕P)∥g (gy osum in e sion law),
=∥⊖Q⊕P∥g (gy oisome y o he au omo phism),
= dg y(Q, P).
(64)
Gy oisome y o he gy oin e se:
dg y(⊖P, ⊖Q)
=∥P⊖Q∥g ,
=∥⊖Q⊕P∥g ( gy ocommu a i i y and gy oisome y o he au omo phism),
= dg y(Q, P),
= dg y(P, Q)(Symme y o he gy odis ance).
(65)
G.4 PROOF OF PROP.3.6
P oo o P op. 3.6.
We i s show he exp essions o he gy odis ance on
M
. Then we p oceed o
show he gy odis ance and gy oisome ies on
MK
. We ollow all he no a ions in Tab. 2and deno e
he geodesic dis ance unde a speci ic geome y as dgeo.
Exp essions o gy odis ance on M:
Fo {Sn
++,⊕AI}, we ha e he ollowing:
dg y(P, Q) = P−1
2QP−1
2g ,
=mlog(P−1
2QP−1
2)(LogI= mlog),
= dgeo(P, Q).
(66)
Fo {Sn
++,⊕LE}, we ha e he ollowing:
dg y(P, Q) = ∥mexp(−mlog(P) + mlog(Q))∥g ,
=mlog−1
∗,I(−mlog(P) + mlog(Q))I,(LogI(P) = mlog−1
∗,I(mlog(P))),
=∥mlog(P)−mlog(Q)∥,( he pullback o LEM (Chen e al.,2024e))
= dgeo(P, Q),
(67)
whe e
mlog−1
∗,I
is he in e se o he di e en ial map o
mlog
, and
∥·∥I
is he no m induced by he
LEM a I.
32

Published as a con e ence pape a ICLR 2025
As LCM is also a pullback me ic (Chen e al.,2024e), {Sn
++,⊕LC} ollow he same logic:
dg y(P, Q) = ψ−1
LC(−ψLC(P) + ψLC(Q))g ,
=(ψLC)−1
∗,I(−ψLC(P) + ψLC(Q))I(LogI(P)=(ψLC)−1
∗,IψLC(P)),
=∥ψLC(P)−ψLC(Q)∥,( he pullback o LCM (Chen e al.,2024e))
= dgeo(P, Q).
(68)
Fo {
G (p, n),e
⊕G }, we ha e he ollowing:
dg y(P, Q) = mexp(−[P , e
Ip,n])Qmexp([P, e
Ip,n])g (⊖P=−P),
=e
P⊤Qe
Pg (e
P= mexp([P, e
Ip,n]) ∈SO(n)),
=1
2Loge
Ip,n e
P⊤Qe
P(∥·∥I=1
2∥·∥),
=1
2[Ω,e
Ip,n],
(69)
whe e [Ω,e
Ip,n] = Loge
Ip,n e
P⊤Qe
P. Fo Ω, we ha e he ollowing:
Ω = 1
2mlog In−2e
P⊤Qe
PIn−2e
Ip,n((Bendoka e al.,2024, P op. 5.6)),
=1
2mlog e
P⊤(In−2Q)e
Pe
P⊤In−2e
Pe
Ip,n e
P⊤e
P
=1
2mlog e
P⊤(In−2Q)In−2e
Pe
Ip,n e
P⊤e
P,
(1)
=e
P⊤1
2mlog (In−2Q)In−2e
Pe
Ip,n e
P⊤e
P,
(2)
=e
P⊤1
2mlog ((In−2Q) (In−2P))e
P,
(3)
=e
P⊤Ωe
P,
(70)
Eq. (70) ollows om he ollowing ac s:
(1)
When
mlog(B)
is well-de ined and
A
is non-singula ,
A−1(log B)A= log A−1BA
(Ho n & Johnson,2012).
(2) P= mexp([P, e
Ip,n])e
Ip,n mexp(−[P, e
Ip,n]) (Nguyen,2022a, Eq. (36)).
(3) Le Ω = 1
2mlog ((In−2Q) (In−2P))
Combining Eqs. (69) and (70), we ha e
dg y(P, Q) = 1
2[Ω,e
Ip,n],
=1
2[e
P⊤Ωe
P, e
Ip,n],
(1)
=1
2e
P⊤[Ω,e
Pe
Ip,n e
P⊤]e
P,
(2)
=1
2[Ω,e
Pe
Ip,n e
P⊤],
(3)
=1
2[Ω, P],
(4)
=1
2∥LogP(Q)∥,
(5)
=∥LogP(Q)∥P,
(6)
= dgeo(P, Q).
(71)
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Published as a con e ence pape a ICLR 2025
Eq. (71) ollows om he ollowing ac s:
(1)
[e
P⊤Ωe
P, e
Ip,n] = e
P⊤Ωe
Pe
Ip,n −e
Ip,n e
P⊤Ωe
P,
=e
P⊤Ωe
Pe
Ip,n e
P⊤−e
Pe
Ip,n e
P⊤Ωe
P,
=e
P⊤[Ω,e
Pe
Ip,n e
P⊤]e
P.
(72)
(2) Euclidean no m is in a ian unde he ac ion A7→ OAO⊤,∀A∈Rn×n, O ∈O(n).
(3) P= mexp([P, e
Ip,n])e
Ip,n mexp(−[P, e
Ip,n]).
(4) LogP(Q) = [Ω, P ].
(5) ∥·∥P=1
2∥·∥,∀P∈
G (p, n).
(6) dgeo(P, Q) = ∥LogP(Q)∥P o any P, Q ∈
G (p, n)no in each o he ’s cu locus.
Fo
{G (p, n),⊕G }
, we i s make he ollowing no a ions: We deno e he geodesic dis ance,
gy odis ance, Riemannian loga i hm a
Ip,n
, and Riemannian me ic a
Ip,n
on
G (p, n)
as
dgeo
,
dg y
,
LogIp,n
, and
∥·∥ONB
Ip,n
. The coun e pa s on
G (p, n)
a e
g
dgeo
,
g
dg y
,
g
Loge
Ip,n
, and
∥·∥PP
e
Ip,n
. As shown
by Nguyen e al. (2024, App. N),
π: G (p, n)→
G (p, n)
is a Riemannian isome y. Then, o any
U, V ∈G (p, n)wi h π(U) = Pand π(V) = Q, we ha e he ollowing:
dg y(U, V ) = LogIp,n ⊖G U⊕G V
ONB
Ip,n
(1)
=LogIp,n π−1(e
⊖G Pe
⊕G Q)
ONB
Ip,n
(2)
=π−1
∗,Ip,n g
Loge
Ip,n e
⊖G Pe
⊕G Q
ONB
Ip,n
(3)
=g
Loge
Ip,n e
⊖G Pe
⊕G Q
PP
e
Ip,n
=g
dg y(P, Q)
=g
dgeo(P, Q)
(4)
= dgeo(U, V )
(73)
The abo e comes om he ollowing.
(1) Gy o addi ions unde he Riemannian isome y (Nguyen & Yang,2023, Lem. 2.1).
(2,3,4) Riemannian ope a o s unde he Riemannian isome y (Gallie & Quain ance,2020).
Cons an Cu a u e Spaces:
When
MK=Rn(K= 0)
, he gy o s uc u es de ined in Eq. (3)-Eq. (9) a e educed o he ec o
s uc u es. The claim can be di ec ly p o ed. In he ollowing, we p esen he p oo o
K= 0
. We
i s show he exp ession o he gy odis ance unde
MK
, hen he isome y o gy a ion, and inally
he esul s on he gy oisome y o gy o ansla ion and gy oin e se. In he ollowing,
a, b, x, y
a e
a bi a y poin s in MK.
Gy odis ance and geodesic dis ance: The Riemannian me ic, loga i hm, and geodesic dis ance on
he CCS (Skopek e al.,2019) a e
gK
x=λK
x2gE,(74)
LogK
x(y) = 2
p|K|λK
x
an−1
Kp|K|∥−x⊕Ky∥−x⊕Ky
∥−x⊕Ky∥,(75)
dgeo(x, y) = 2
p|K| an−1
Kp|K|∥−x⊕y∥,(76)
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Published as a con e ence pape a ICLR 2025
whe e
λK
x=2
1+K∥x∥2
,
gE
is he s anda d Euclidean inne p oduc , and
⊕K
is he gy o addi ion in
Tab. 2. Especially, when x= 0, we ha e
gK
0= 22gE,(77)
LogK
0(y) = 1
p|K| an−1
Kp|K|∥y∥y
∥y∥(78)
Fo he gy odis ance, we ha e he ollowing:
dg y(x, y) = ∥−x⊕y∥g ,
=∥Log0(−x⊕y)∥0,
= 2 
1
p|K| an−1
Kp|K|∥−x⊕y∥−x⊕y
∥−x⊕y∥,
=2
p|K| an−1
Kp|K|∥−x⊕y∥(∀s > 0, an−1
K(s)>0),
(79)
whe e ∥·∥0is he no m induced by gK
0.
No m in a iance unde gy a ion: As
MK
o ms a eal inne p oduc gy o ec o spaces (Unga ,
2009, De . 3.2), any gy a ion on CCS p ese es he no m induced by s anda d inne p oduc :
∥gy [a, b](x)∥=∥x∥,∀x. (80)
Fo he gy ono m, we ha e he ollowing:
∥gy [a, b]x∥g = 2∥Log0(gy [a, b]x)∥0,
=2
p|K| an−1
Kp|K|∥gy [a, b]x∥,
=2
p|K| an−1
Kp|K|∥x∥,
=∥x∥g .
(81)
Isome y o le gy o ansla ion and gy oin e se: No e ha
MK
o ms a gy ocommu a i e gy og oup.
Acco ding o Thm. 3.5, we can ob ain he esul s.
G.5 PROOF OF THM.4.1
P oo o Thm. 4.1.
Acco ding o Thm. 3.3 and Thm. 3.5, any le gy o ansla ion is a gy oisome y.
The e o e, o any Q∈ M, we ha e he ollowing:
dg y (B⊕Pi, Q)(1)
= dg y (⊖B⊕(B⊕Pi),⊖B⊕Q)
(2)
= dg y ((⊖B⊕B)⊕gy [⊖B, B](Pi)),⊖B⊕Q)
(3)
= dg y (Pi,⊖B⊕Q).
(82)
The abo e comes om he ollowing.
(1) Any le gy o ansla ion is a gy oisome y.
(2) Le gy oassocia i e law.
(3) ⊖B⊕B=Eand pseudo- educ ion.
Deno ing he gy omean o {Pi}and {B⊕Pi}as Mand
M, we ha e he ollowing:
B⊕M(1)
=B⊕(⊖B⊕
M)
(2)
= gy [B, ⊖B](
M)
(3)
=
M.
(83)
The abo e comes om he ollowing.
35
Published as a con e ence pape a ICLR 2025
(1) Eq. (82) indica es ha M=⊖B⊕
M.
(2) Le gy oassocia i e law.
(3) Pseudo- educ ion.
Now, we p oceed o deal wi h he second p ope y. We ha e he ollowing:
dg y( ⊙Pi, E)(1)
=∥⊖E⊕( ⊙Pi)∥g
(2)
=∥ ⊙Pi∥g
=∥ LogEPi∥E
=| |∥LogEPi∥E
=| |∥Pi∥g
(3)
=| |∥⊖E⊕Pi∥g
=| |dg y(E, Pi)
(4)
=| |dg y(Pi, E)
(84)
The abo e ollows om he ollowing.
(1) Symme y o gy odis ance (Thm. 3.5).
(2) ⊖E=E.
(3) Pi=⊖E⊕Pi.
(4) Symme y o gy odis ance.
The las equa ion in Eq. (84) indica es he homogenei y o dispe sion om E.
G.6 PROOF OF PROP.5.1
P oo o P op. 5.1.
We only need o p o e he equi alence o gy odis ance and geodesic dis ance.
No e ha e e y Lie g oup is au oma ically a gy og oup wi h each gy a ion as he iden i y map.
We deno e
{M,⊕, g}
as a Lie g oup wi h le -in a ian me ic
g
. Fo any
P
and
Q
in
M
, we ha e
he ollowing:
dg y(P, Q)(1)
=∥⊖P⊕Q∥g
(2)
=∥LogE(⊖P⊕Q)∥E
= dgeo(E, ⊖P⊕Q)
(3)
= dgeo(P, P ⊕(⊖P⊕Q))
(4)
= dgeo(P, Q).
(85)
The abo e de i a ion comes om he ollowing.
1. De ini ion o gy odis ance Eq. (9).
2. De ini ion o gy ono m Eq. (8).
3. Unde a le -in a ian me ic, any le Lie g oup ansla ion is a Riemannian isome y.
4. P⊕(⊖P⊕Q)=(P⊖P)⊕Q( he associa i e o g oup addi ion)
=E⊕Q
=Q.
(86)
The e o e, he gy omean and gy o a iance a e exac ly he F éche mean and a iance unde he
geodesic dis ance, while he unning mean upda es a e also iden ical unde gy odis ance and geodesic
dis ance.
36
Published as a con e ence pape a ICLR 2025
G.7 PROOF OF PROP.6.1
We i s e iew a as and s able algo i hm o he ONB G assmannian loga i hm (Bendoka e al.,
2024, Alg. 5.3), and he calcula ion o G assmannian loga i hm unde he p ojec o pe spec i e by
he ONB G assmannian loga i hm (Nguyen e al.,2024, P op. 3.12).
Algo i hm 2: G assmann loga i hm unde he ONB pe spec i e (Bendoka e al.,2024, Alg. 5.3)
Inpu : U, Y ∈G (p, n)a e S ie le ep esen a i es unde ONB pe spec i e.
1QSRTSVD
:= YTUwi h Sin ascending o de , and Qand Rcolumn-wisely lipped acco dingly;
2ˆ
S=√In−S2;
3∆=(In−UU⊤)Y Qa csin( ˆ
S)
ˆ
SRT;
Ou pu : LogU(Y)=∆
Alg. 2 e iews a as and s able algo i hm o he G assmannian Riemannian loga i hm unde he
ONB pe spec i e
G (p, n)
. The anilla Riemannian loga i hm in Tab. 8 equi es an
n×p
SVD and a
p×p
ma ix in e se, while Alg. 2only equi es an
p×p
SVD. The e o e, Alg. 2is mo e e icien
han he anilla loga i hm. Besides, Alg. 2can also e u n a unique angen ec o when
Y
is in he
cu locus o U. Fo mo e de ails, please e e o Bendoka e al. (2024, Sec. 5.2).
As he p ojec o pe spec i e is isome ic o he ONB pe spec i e, he G assmannian loga i hm unde
he p ojec o pe spec i e can be calcula ed by he ONB G assmannian loga i hm (Nguyen e al.,2024,
P op. 3.12).
P oposi ion G.2 ((Nguyen e al.,2024)).Gi en any
P, Q ∈
G (p, n)
wi h
U=π−1(P)
and
V=π−1(Q), he Riemannian loga i hm g
LogP(Q)on
G (p, n)is gi en as
g
LogP(Q) = π∗,U (LogUV),(87)
whe e
Log
is he Riemannian loga i hm unde he ONB pe spec i e,
π∗,U :TUG (p, n)→
TP
G (p, n)is he di e en ial map o πa U, which is de ined as
π∗,U (∆) = ∆U⊤+U∆⊤,∀∆∈TUG (p, n).(88)
Now, we begin o p esen he p oo .
P oo o P op. 6.1.We i s show he exp ession o LogIp,n and g
Loge
Ip,n .
Fi s no e he ollowing:
(In−Ip,nI⊤
p,n) = 0 0
0In−p,(89)
U⊤Ip,n =U⊤
1, U⊤
2Ip
0
=U⊤
1,
(90)
By he abo e wo equa ions, he ONB G assmannian loga i hm a Ip,n is
LogIp,n (U) = 0 0
0In−pU1
U2Qa csin( ˆ
S)
ˆ
SRT(Alg. 2)
= 0
U2Qa csin( ˆ
S)
ˆ
SRT!
=0
e
U2,
(91)
whe e
QSRTSVD
:= U⊤
1
wi h
S
in ascending o de , and
Q
and
R
column-wisely lipped acco dingly,
and ˆ
S=√In−S2.
37

Published as a con e ence pape a ICLR 2025
Fo g
Loge
Ip,n , we ha e
g
Loge
Ip,n (UU⊤)(1)
=π∗,Ip,n LogIp,n (U)
(2)
=π∗,Ip,n  0
e
U2
(3)
=0e
U⊤
2
e
U20.
(92)
The abo e de i a ion comes om he ollowing.
(1) P op. G.2
(2) Eq. (91)
(3) Fo any ∆ = (∆⊤
1,∆⊤
2)⊤∈TIp,n G (p, n), whe e ∆1is p×p, we ha e he ollowing
π∗,Ip,n ∆1
∆2=∆1
∆2(Ip,0) + Ip
0∆⊤
1,∆⊤
2
=∆10
∆20+∆⊤
1∆⊤
2
0 0 
=∆1+ ∆⊤
1∆⊤
2
∆20.
(93)
Combining all he abo e esul s oge he , we ha e he ollowing:
[UU⊤,e
Ip,n] = hg
Loge
Ip,n (UU⊤),e
Ip,ni
= 0e
U⊤
2
e
U20,e
Ip,n
=0e
U⊤
2
e
U20Ip0
0 0 −Ip0
0 0  0e
U⊤
2
e
U20
=0−e
UT
2
e
UT
20.
(94)
38