A Lie Group Approach to Riemannian Batch Normalization
Authors: Ziheng Chen
Yue Song
Yunmei Liu
Nicu Sebe
Venue: ICLR 2024 poster (2024)
Keywords: Lie Groups
Riemannian Batch Normalization
SPD Neural Networks
Primary area: learning on graphs and other geometries & topologies
TL;DR: We propose a general framework for Riemannian batch normalization over Lie groups (LieBN), and showcase our LieBN on diverse Lie groups of SPD manifolds.

Abstract: Manifold-valued measurements exist in numerous applications within computer vision and machine learning. Recent studies have extended Deep Neural Networks (DNNs) to manifolds, and concomitantly, normalization techniques have also been adapted to several manifolds, referred to as Riemannian normalization. Nonetheless, most of the existing Riemannian normalization methods have been derived in an ad hoc manner and only apply to specific manifolds. This paper establishes a unified framework for Riemannian Batch Normalization (RBN) techniques on Lie groups. Our framework offers the theoretical guarantee of controlling both the Riemannian mean and variance. Empirically, we focus on Symmetric Positive Definite (SPD) manifolds, which possess three distinct types of Lie group structures. Using the deformation concept, we generalize the existing Lie groups on SPD manifolds into three families of parameterized Lie groups. Specific normalization layers induced by these Lie groups are then proposed for SPD neural networks. We demonstrate the effectiveness of our approach through three sets of experiments: radar recognition, human action recognition, and electroencephalography (EEG) classification. The code is available at https://github.com/GitZH-Chen/LieBN.git.

Published as a conference paper at ICLR 2024
A LIE GROUP APPROACH TO RIEMANNIAN BATCH NORMALIZATION
Ziheng Chen', Yue Song'; Yunmei Liu & Nicu Sebe'
I University of Trento, 2 University of Louisville wihengg.chie ch0163. com, yue. songguninnns
ABSTRACT
Manifold-valued measurements exist numerous applications within computer vision and machine learning. Recent studies have extended Deep Neural Net- works (DNNs) to manifolds, and concomitantly, normalization techniques have also been adapted several manifolds, referred to as Riemannian normaliza- tion. Nonetheless, most the existing Riemannian normalization methods have been derived in ad hoc manner and only apply to specific manifolds. This pa- per establishes a unified framework for Riemannian Batch Normalization (RBN) techniques on Lie groups. Our framework offers the theoretical guarantee of controlling both the Riemannian mean and variance. Empirically, we focus on Symmetric Positive Definite (SPD) manifolds, which possess three distinct types of Lie group structures. Using the deformation concept, we generalize the ex- isting Lie groups on SPD manifolds into three families of parameterized Lie groups. Specific normalization layers induced by these Lie groups are then pro- posed for SPD neural networks. We demonstrate the effectiveness of our ap- proach through three sets of experiments: radar recognition, human action recog- nition, and electreencephalloggapyy (EEG) classification. The code is available git
attps://githnee com Git
INTRODUCTION
Over the past decade or so, Deep Neural Networks (DNNs) have achieved remarkable progress across various scientific fields (Hochreiter & Schmidhuber 1997 Krizhevsky et al., 2012) He al., 2016 Vaswani et al. 2017). Conventionally, DNNs have been developed with the underlying as- sumption of the Fucl lidennnan geometry inherent input data. Nonetheless, there exists plethora of applications wherein the latent spaces are defined by non-Euclidean structures such as manifolds (Bronstein et al.) 2017). To address this issue, researchers have attempted extend various types of DNNs to manifolds based on the theories of Riemannian geometry (Huang & Van Gool 2017
Huang et al., 2017: 2018 Ganea et al. 2018 Chakraborty al. 2018 Brooks al. 20 9a
Brooks 2020 Brooks et al., 2020 Chen et al. 2020 Chakraborty et al. 2020 Chakraborty 2020
Chen et al. [021] Wang al. 2022b a Nguyen 2022a
Nguyen & Yang 2023
Chen et al.
2023c e b: Wang et al. 2024).
Motivated by the great success of normalization techniques within DNNs (loffe & Szegedy 2015)]
Ba et 2016 Ulyanov et al. 2016 Wu & He 2018 Chen et al. 2023a),, researchers have sought to devise normalization layers tail ored for manifold-valued data. Brooks et al. (2019) introduced Rie- mannian Batch Normalization (RBN) specifically designed for SPD manifolds, with the ability to regulate the Riemannian mean. This approach was further refined in Kobler et al. (202b)) to extend the control over the Riemannian variance. However, the above methods are constrained within the dnffine-invarian metric (AIM) SPD manifolds, limiting their applicability and generality. On the other hand, Chakraborrty (2020) proposed two distinct Riemannian normalization frameworks, one tailored for Riemannian homogeneous spaces and the other catering matrix Lie groups. Nonethe- less, the normalization designed for Riemannian homogeneous spaces cannol regulate mean nor variance, while the normalization approach for matrix Lie groups is confined to a specific type of distance (Chakraborry 2020 Sec. 3.2). A principled Riemannian normalization framework capable of controlling both Riemannian mean and variance remains unexplored.
"Corresponding author

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Published as a conference paper at ICLR 2024
Given that Batch Normalization (BN) (loffe & Szegedy 2013) serves the foundational proto- type for various types of normalization, our paper only concentrates on RBN currently and can be readily extended to other normalization techniques. As several manifold-valued measurements form Lie groups, including SPD manifolds, Special Orthogonal (SO) groups, and Special Euclidean (SH) groups, we further direct our attention towards Lie groups. We propose general framework for RBN over Lie groups, referred to as LieBN, and validate our approach in normalizing both the Riemannian mean and variance. On the empirical side, we focus on SPD manifolds, where three distinct types of Lie groups have been recognized in the literature. We generalize these existing Lie groups into parameterized forms through the deformation concept. Then, we showcase our LieBN framework on SPD manifolds under these Lie groups and propose specific normalization layers. Extensive experiments conducted on widely-used SPD benchmarks demonstrate the effectiveness of our framework. We highlight that our work is entirely theoretically different from Brooks et al.) (2019b); Kobler et al. (2022a); Lou et al. (2020), and more general than Chakraborty (2020). The previous RBN methods are either designed for specific manifold or metric (Brooks al. 2019b Kobler et al., 2022a Chakraborty) 2020), or fail to control mean and variance (Lou et al. 2020),
while our LieBN guarantees the normalization of mean and variance general Lie groups. In sum- mary. our main contributions are as follows: (a) a general Lie group batch normalization framework with controllable first- and second-order statistical moments, and (b) the specific construction of our LieBN layers on SPD manifolds based on three deformed Lie groups and their application on SPD neural networks. Due to page limites, all the proofs are placed App.
2
PRELIMINARIES
This section provides brief review of the Lie group and the geometries of SPD manifolds. For more in-depth discussions, please refer to Tu (201T); Do Carmo & Flaherty Francis (1992).
Definition 2.1 (Lie Groups). A manifold is Lie group, if it forms a group with group operation C such that m(x (z. y) + z @ y and i(zz Z are both smooth, where zg is the group inverse.
Definition 2.2 (Left-ivarrince)) if for any 2, y E G and Vı, V/2 E T&M,
A Riemannian metric over a Lie group {G, O is left-invariant,
Oy (V, V2) IL. (Tas,, Tex (V2)):
where (z) = x O y is the left translation by æ, and Lxe,y is the differential map of Lg at y.
Lie group group and also a manifold. The most natural Riemannian metric on a Lie group is the left-invariant metrid] Similarly, one can define the right-invariant metric as Def. 2.2 A bi- invariant Riemannian metric is the one with both left and right invariance. Given the analogous properties of left and right-invariant metrics, this paper focuses on left-invariant metrics.
The idea of pullback is ubiquitous in differential geometry and can be considered as a natural coun- terpart of bijection in the set theory.
Definition 2.3 (Pullback Metrics). Suppose M2 are smooth manifolds, g is Riemannian metric on M2, and f M1 - M2 is smooth. Then the pullback of by f is defined point-wisely,
(V')),(,,)) = 9r6 (F- ((),
(2)
where p M. f-p(()) is the differential map of p. and V, W T..MM If f" is positive definite, is a Riemannian metric on Mi, called the pullback metric defined by f.
The most common pullback metric obtained through the pullback ditionally, if (M2, (2) constitutes a Lie group, the diffeomorphism f can pull back the group operation 32 to @1 on M₁, which is defined as VP, Q E M1, P @1 Q = f-(((()) Oo f(Q)). Henceforth, (M, g). abbreviated as M, always signify a Lie group with left-invariant metric.
diffeomorphism ʃ. Ad-
Now, we briefly review the geometry of SPD manifolds. We denote n x n SPD matrices as S and n X n real symmetric matrices as S". As shown Arsigny et al. (0005). S forms a man- ifold, known as the SPD manifold, and S is a Euclidean space. SPD manifolds exhibit three Lie group structures, each associated with an invariant metric. These metrics include the Log-Fuclidean
Left invariant metric always exists for every Lie group Do Carmo & Flaherty Francis 1992
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Metric (LEM) (Arsigny et al. 2005
Invarinnv Metric (AIM) Pennec et al. 2006), and Log-
Cholesky Metric (LCM) (Lin 2019). In Thanwerdas & Pennec (022)) LEM and AIM are general- ized into wwo-parameter families of metrics, denoted as (a, B)- and (a, 3) AMM, respectively.
Both (a, 9)MEEM and (a, A-MAIMM are defined by the O(n) ((n)-invariant inner product on the tangent
space the identity matrix, expressed as:
(V, ( = aVV, (V + trrr())tr(W),
(3)
where (.-) is the Frobenius inner product, V.W& min(a, a + n²) > 0}.
W E TT = S", (E {(a, 3) R²
In fact. 3)-EM,, (a, 3)MMI,, and LCM are all pullback metrics. Specifically, (a, 3)-LEM is the pullback metric from St (Chen et al. (023c), while A)MM is the pullback metric from a left-invariant metric on the Cholesky manifold (Thanwerdas & Pennee 2022b). Additionally, as shown in Chen et al.] (2023d), LCM is the pullback metric from the Fuclidean space f of lower triangular matrices by the diffcomorphism defined as
(Pc(P) LL] + DIog(L),
(4)
where P LL represents the Cholesky decomposition, I.] is the strictly lower part of square matrix, and Diog(L) L) is a diagonal matrix consisting of the logarithm of the diagonal element of L.
Let {wi...}} be weights satisfying convexity constraint, i.e.,Vi, w; O and Σ, w, 1. The weighted Fréchet mean (WFM) of a set of SPD matrices (Pr...v) is defined as
WFM((w;), (P.)) = argmin
NV ;= Σ
w; dd S);
(5)
SEST_
where -) denotes the geodesic distance. When w, I/N for all i, then Eq. 5 is reduced the Fréchet mean, denoted as FM((P:)). The Fréchet variance 2 is the attained value at the minimizer of the Fréchet mean. In this paper. we will interchangeably use the terms Riemannian mean with Fréchet mean, and Riemannian variance with Fréchet variance. For (a, 3EMM and LCM, the Fréchet mean has closed-form expression. Moreover, since (a, --MM has non-positive sectional curvature (Thanwerdas & Pennec, 2023] Tab. 5), the Fréchet mean exists uniquely (Berger 2003 6.1.5) and can be computed by the Karcher flow algorithm (Karcher 1977).
Given SPD matrices P, Q E S along with tangent vectors V.W W. E TpS++4 we introduce the following notations. Specifically, we denote gp(-;)) as the Riemannian metric at P, Logp(-) as the Riemannian logarithm at P. mexp(-) and mlog(-) the matrix exponentiation and logarithm, and d(.,-) as the geodesic distance, respectively. Additionally, Chol(-) signifies the Cholesky decompo- sition. We use T. Chol(P) and K Chol(Q) denote the Cholesky factor of P and Q. K. and TL diagonal matrices with diagonal elements from K and L. mlog., and (Chol),, represent the differentials of mlog and Chol at P and L. We denote | [[a.8] and || ||e as the norm induced by (--) -) and the standard Frobenius norm. We summarize all the necessary ingredients Tab.
Table 1: Lie group structures and the associated Riemannian operators on SPD manifolds.
Metric
(a, 9)-M
(a, A)-AIM
LCM
(V. W]]
(ming, p (W))(10))
(P- V. WP Iyo
Exs, Vol E. Wo
mlog (Q $ PQ ())(
d(P. Q)
[mllo(()) mlog(Q)))
lIve Chol(P) PLo Chol
Q P
mexp(mlog (P) mlog(())
KPK
Chol "hol-'([L KL)
FM(IP))
mexp mlog Pt)
mlog
Karcher Flow
* (1 Σ, (..)
Logp Q
(mlog,,p) p) (mlog(Q) mlog(P))
pl mlog PP)QP?)) p (Chol")) ).,, [[K] - [4] + (log(L-'k)
Invariance
Bi-invariance
Left-invariance
Bi-invariance
3 REVISITING NORMALIZATION
3.1 REVISITING EUCLIDEAN NORMALIZATION
In Euclidean DNNs, normalization stands as pivotal technique for accelerating network training by mitigating the issue of internal covariate shift (loffe & Szegedy 2015). While various normalization
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methods have been introduced (loffe & Szegedy 2015 Ba et al. 2016 Ulyanov al. 2016 Wu
& He 2018), they all share a common fundamental concept: the regulation of the first and second statistical moments. In this paper, focus batch normalization only.
Given a batch of activations {II....v the core operation of batch normalization can be expressed as:
14 Veg
Vi < N, 34 7
+
where 14b is the batch mean, og is the batch variance, y is the scaling parameter, and 3 is the biasing parameter.
3.2 REVISITING EXISTING RBN
Inspired by the remarkable success of normalization techniques in traditional DNNs, endeavors have been made develop Riemannian normalization approaches tailored for manifolds. Here we recap some representative methods. However, we note that none of the existing methods effectively handle the first and second moments in a principled manner.
Brooks et al. (2019b) introduced RBN over SPD manifolds under AIM, with operations defined as:
Centering from geometric mean M : Vi < N, P; + M P Biasing towards SPD parameter B : Vi < N, P, P,B.,
(8)
where (P....) are SPD matrices, and M their Fréchet mean under AIM. Define map
[P) Expq [PTP+Q (Logp(S))),
(9)
where P.Q.S S and Exp, Log, PT are Riemannian exponential map, Riemannian logarith- mic map, and parallel transportation along the geodesic, respectively. Then under AIM, Eqs. and (8 can be more generally expressed
TI-B [T (P:)
(10)
However, Eqs. and 8 only consider the Riemannian mear and does not consider the Rieman- nian variance. To remedy this limitation, Kobler al. (2022b) further improved the RBN to enable control over the Riemannian variance. The key operation is formulated as
Vi < N, P. Dr r-
(11)
where 22 is the Fréchet variance, R is scaling factor. However, this method is still limited to SPD manifolds under AIM. In parallel. Chakraborty (2020) Algs. 1-2) proposed a general frame- work for Riemannian homogeneous spaces based on Eq. (10) by considering both first and second moments. However, as stated in Chakrabortt (2020) Sec. 3.1), Eq. (TO) does not generally guaran- tee the control over Riemannian mean, resulting in agnostic Riemannian statistics. To mitigate this limitation, Chakraborty (2020) Algs. 3-4) further proposed normalization over the matrix Lie group. However, the discussion is limited
certain distance, limiting the applicability of their method.
On the other hand, Lou et al.) (2020) proposed an RBN based on variance of Eq. (T)). Although their framework encompasses the standard Euclidean BN when the latent geometry the standard Euclidean geometry, their approach suffers from the same problem of agnostic Riemannian statistics on general manifolds.
In summary, prevailing Riemannian normalization approaches lack a principled guarantee for trolling the first and second-orde statistics. In contrast, as will be elucidated, our method can govern first and second-order statistics for all Lie groups. We summarize the above RBN methods in Tab.
RIEMANNIAN NORMALIZATION ON LIE GROUPS
In this section, the neutral element in the Lie group M is denoted as E. Notably, the neutral element E is not necessarily the identity matrix. We first clarify the essential properties of Euclidean BN and then present our normalization method, tailored for Lie groups
lthough not mentioned in the original paper, Eqs. and 8 can guarantee the mean of the resulting samples under ATM, as Eqs. and (8) actions of (2).
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Table 2: Summary of some representative RBN methods.
Involved Statistics
Controllable Mean
Controllable Variance
Methods
Applicution Scenarios
SPDBN (Brooks al.
2019b
Mean
NIA
SPD manifolds under AIM SPD manifolds under AIM
SPDBN Koole et ai.
40226
Mean+ Variance
Chakraborty 20200 Algs.
1-2
Mean+ Vuriance
Riemannian homogeneous space
Chakraboty 20201
Algs.
3-4)
Mean+ Variance
A certain Lie group structure and distance Geodesically complete manifolds General Lie groups
KBN [Lou ef a.. 2020
Alg. 2)
Mean+ Variance
X
X
Ours
Mean+ Variance
Two key points regarding Euclidean BN, as expressed by Eq. (6), are worth highlighting: (a) The standard BN (loffe Szegedy 2013) implicitly assumes a Gaussian distribution and can effectively normalize and transform the latent Gaussian distribution; (b) The centering and biasing operations control the mean, while the scaling controls the variance. Therefore, extending BN into Lie groups requires defining Gaussian distribution, centering, biasing, and scaling Lie groups.
On manifolds, several notions of Gaussian distribution have been proposed (Pennec 2004 Chakraborty & Vemuri 2019 [Barbaresco] [021]. In this work, we adopt the definition from Chakraborty & Vemuri (2019), which characterizes a Gaussian distribution on the Lie group M with mean parameter . E M and variance o², This distribution is denoted (MM, o.). and its Probability Density Function (P.D.F.) is defined as
p (X M, 0²) k(p) (- ddx, M)
(12)
20
where k(ar) is the normalizing constant, exp(-) the scalar exponentiation, and d(.; -) is the geodesic distance. On Lie groups, the natural counterparts of addition and subtraction in Eq. (6) are group op- erations. Therefore, centering and biasing on Lie groups can be defined by Lie group left translation. Additionally, scaling can be defined by scaling on the tangent space at the Riemannian mean.
Now, we can extend Eq. (6) into Lie groups M. For a batch of activation {P....N E M), we define the key operations of Lie group BN (LieBN) as:
Centering to the neutral element E: Vi < N, + L DM (P.),
(13)
Scaling the dispersion: Vi S N, P ExPE
Vu² +c Logg(())
(14)
Biasing towards parameter B M: V N, P. + P.(P))
(15)
where M is the Fréchet mean, x is the Fréchet variance, M E M is the group inverse of M, M and LG are left translations (Lg = G @ P.). and E [R is a scaling parameter. To clarify the effect of our method in controlling mean and variance, we first present the following two propositions, one related to population statistics and the other related to sample statistics.
Given random point X over M, and the Gaussian distribution ((M v²) defined in Eq. 112 we have the following properties for the population statistics:
Proposition 4.1 (Population).
1. (MLE of M) Given {P....N E M} i.i.d. sampled from (M, v²), the maximum likelihood estimator (MLE) of M is the sample Fréchet mean.
2. (Homogeneity) Given X ~ (N,,? and B E M, LB(X) N(LB(M), v?)
Proposition 4.2 (Sample).
Given N samples (P...N E M), demoting (s(P.) Expg is (P.), we have the following properties for the sample statistics:
Homogeneity of the sample mean: FM(Ln(P.)) = -LPPP(())))BBBBB
(16)
Controllable dispersion from E: Σn w: d ('. 82 D d E),
(17)
where {w....) weights satisfying a convexity constraint, i.e. w/ and Σ, 10 =
When M corresponds to R equipped with the standard Fuclidean metric, Fq. 12 reduces to the Euclidean
Gaussian distribution.
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Prop. 4.1 and Eq. (16 imply that centering and biasing in Eqs.
13 and (3) can control the sample and population mean. As the post-centering mean is E, Eq.
implies that Eq. (14) can control the dispersion. Although the population variance after Eq. (14) generally agnostic, some
cases such as SPD manifolds under LEM and LCM, Hq. (14) can normalize the population variance. Please refer App. Co technical details.
Similar to loffe & Szegedy (2015). we use moving averages to update running statistics. Now, we present our general framework of LieBN in Alg. I Importantly, when M =R, Alg. is reduced to the standard Euclidean BN. More details are exposed in App.
Remark 4.3. The MLE of the mean of the Gaussian distribution in Eq. 12 have been examined
in several previous works (Said et al., 2017 Chakraborty & Vemuri 2019 Chakraborty 2020). However, these studies primarily focus on particular manifolds or specific metries. In contrast, our contribution lies in presenting universally applicable result for all Lie groups with left-invariant metrics. While Eq. (12) briefly appeared Kobler et al. (2022b), the authors only focus on SPD manifolds under AIM. The transformation of the population under their proposed RBN remains unexplored as well. Besides, while Chakraborty (2020) analyzed the population properties for their RBN over matrix Lie groups, their results were confined within a specific distance. In contrast, our work provides more extensive examination, encompassing both population and sample properties of our LieBN in general manner. Besides, all the discussion about our LicBN can be readily transferred to right-invariant metrics. This paper focuses on LieBN based on left-invariant metrics.
Algorithm 1: Lie Group Batch Normalization (LieBN) Algorithm
Input
: A batch of activations (PI...N E M), a small positive constant c, and momentum 7 [0. running mean M, = E, running variance of 1, biasing parameter / M, scaling parameter E IR((0), Normalized activations {PI...N
Output
:
training then
Compute batch mean M, and variance vi of {PI...):
Update running statistics Mr + WFM 7, 7h (Mr, Mo)), (.))).a oZ + (1 -1--) 7)0? + Yu3; end
if training then M Mb, 22 vg:
else .M Mr, 22 + u,2
for i + 1 to N do Centering to the neutral element E: P. + Mo (P.)
Scaling the dispersion: P; + ExpE Voltee Logg((.)
Biasing towards parameter B: P. < LB(P.)
end
5 LIEBN ON THE LIE GROUPS OF SPD MANIFOLDS
This section showeases our Alg. D on SPD manifolds. Firstly, we extend the current Lie groups on SPD manifolds by the concept of deformation, resulting in three families of parameterized Lie groups. Subsequently, we construct LieBN layers based on these generalized Lie groups.
5.1 DEFORMED LIE GROUPS OF SPD MANIFOLDS
As shown in Tab. T there are three types of Lie groups on SPD manifolds, each equipped with left- invariant metric. These metrics include (a,, 3))AIM, (a, (0,,3)-EMM and LCM. For clarity, we denote the group operations w.r.t. (a, ()MMM,, (a,, L-MMMMM and LCM as GAI (yLE and (LCC respectively.
In Thanwerdas & Pennec 2019b). (a. -AMM is further extended into parameerssesers families of metrics by the pullback of matrix power function Pa(-) and scaled by s.. denoted as (0, a, AMM-I... The power function serves deformation wherein (0, a. 33)AIMM encompasses (a, AAMM when 0 = 1, and includes (a, B)-MEMM as approaches 0 (Thanwerdas Pennec 201 20)9 Inspired by the
deforming utility of the power function, define the dewer-de metries of (a, LEM and
LCM as the pullback metrics by Po and scaled by Jt. We denote these two metrics as (9, a, 3-MM and O-L.CM, respectively. We have the following results w.r.t. the deformation.

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Proposition 5.1 (Deformation). (6. a. 3)M is equal (a.3)-LEM, B-LCM interpolates between 9- g-LEM (0 = 0) and LCM (8 = 1), with g-LEM defined as
(V, W)r 0(mlog.,,p P(V), p VP E S "V,,VV &TSS+
(18)
where @(Vi, V/2) 2 (Vi, (DV(),,, D(V)) (V.)) is a diagonal matrix consisting of the diagonal elements of Vv and mlog., p is the differential map P.
As (0,a,3)-LEN, is equal to (,,M),LEM, in the following, we focus on (a, 3)-LEM (0,a, a, AMMM and 0-LCM. As a diffeomorphism, Pa also can pull back the group operation @-i and @LC, denoted as @ and @aLLC, We have the following proposition on the invariance.
Proposition 5.2 (Invariance).
(0, is left-invariant w.r.t @-AI, while 0-I.CM bi-
invariant w.E.. -0-C
5.2 LIEBN ON SPD MANIFOLDS
Now, we showcase our LieBN framework illustrated in Alg. on SPD manifolds. As discussed in Sec. [5.1] there are three families of left-invariant metries, namely (9, a, 3)MM (a, ()-EM,, and 9-LCM. Since all three metric families are pullback metrics, the LieBN based on these metrics can be simplified and calculated the co-domain. We denote Alg. [as
Li; 5, e,,, Vp, E (PI...N
(19)
Then we can obtain the following theorem.
Theorem 5.3. ( Given Lie group M1. Lie group M2 with left-invariant metric g², and a diffeomorphism f M₁ + M2. then f induces a left-invariant metric g on Mi. denoted as g¹ g² For a batch of activation {Pi...) in M1, LieBN P; B (,7) in M1 can be calculated in M2 by the following process:
Mapping data into M2: P. ((P)), B = ((B),
(20)
Calculating LieBN in (M2. g²} P: = (P;
(21)
Mapping the normalized data back to Mi P: (P.).
(22)
Given metric g + the power-deformed metric Ø P g is equal to P () Therefore, the LieBN under ğ can be calculated by the LieBN under 9. Besides, as the Christoffel symbols remain the same under constant scaling, the LieBNs under g and y only differ in the variance. We denote (a.. and g! (f,o.))-A as the metric tensors of (a, AMM and (0, a, B))-IM,, respectively Based on the above discussions, the computations of the LieBN under g (8)A are reduced to the LieBN under g (B.)))) Furthermore, as shown in Chen et al. (2023c). (a, --MM is a pullback metric from the Euclidean space S" of symmetric matrices, while the proof of Prop. 5.2 indicates that 0-LCM is a pullback metric from the Euclidean space _ of lower triangular matrices. Note that in the Euclidean space S" or L" shown in App. D Eq. (21) simplifies to the standard Euclidean BN. We denote P, 2, Pi and P2 as points in the codomain (ST. ++ with (ar, -))-IM for (0, a, -AMM, S" for (o., 3).LEM, and _ for O-LCM, respectively). We summarize all the necessary ingredients in Tab. 3 for calculating LieBN on SPD manifolds. Note that for (0, a, 3AAM., our scaling oper-
ation dehned in Eq. (14) encompasses the scaling operation in Kobler et al. 2022b Eq. (9))
special case, when (0, a, (1,
6 EXPERIMENTS
In this section, we implement our three families of LieBN to SPD neural networks. Following the previous work (Huang & Van Gool, 2017 Brooks et al., 2019b Kobler et al. (022a)). we adopt three different applications: radar recognition recognition on the HDM05 (Müller et al. 2007) and FPHA (Garcia-Hernando et al, 2018)) datasets, and EEG classification the Hinss2021 dataset (Hinss et al. 2021). More details on datasets and
the Radar dataset (Brooks et al. 2019b), human action
rrper-parameters are exposed in App. G Besides SPD neural networks, we also implement LieBN on special orthogonal groups and present some preliminary experiments (see App. H).
Implementation details: Note that our LieBN layers are architecture-agnosse and can be applied to any existing SPD neural network. In this paper, we focus on two network architectures: SPDNet
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Table 3: Key operators in calculating LieBN on SPD manifolds.
Metric
(0, a. 9)-MMM
(0,
0-LCM
Pullback Map
Pa
mlog
P. PooYLO
Codomain
18T 4. GAA1, 329 (
{8",
{C", 11
LP(P)
KPK
P+Q
P P+C Q
Rieman- nian and Lie group operators in the codomain
K PPK-
P-Q
P-C Q
Expg Logg
P
sP
aP
Arithmetic average
Arithmetic average
FM
Karcher Flow
(PM((1,1-)),
pd (r 1
Arithmetic
Arithmetic
weighted average
weighted average
Table 4: 10-fold average results of SPDNet with and without SPDBN LieBN on the Radar, HDM05, and FPHA datasets. For simplicity, WeeBN-Metric-(PP) is abbreviated as Metric-(#). For the LieBN under each metrie, if the LieBN induced by the standard metric (0 1) is saturated, report the LieBN under the deformed metric the rightmost columns of the table.
(a) Radar dataset.
Method
SPDNet
SPDNetBN
AIM-(1)
LEM-(I)
LCM-(I)
LCM-(-0.5)
Fit Time (s)
0.98
1.56
1.62
1.28
1.11
1.43
Mcan+STD
93.25=1.10
94.85=0.99
95,47=0.90 96.27
94.89=1.04
93.52+1.07
94.80+0.71
Max
94.4
96.13
96.8
95.2
95,73
(b) HDMOS and FPHA datascts.
Method
SPDNet
SPDNeIBN
AIM-I)
LEM-(I)
LCM-(1)
AIM-(1.5)
LCM-(0.5)
Fit Time (8)
0.57
0.97
1.14
0.87
0.66
1.46
1.01
HDMO5
Mcan:STD
19.1370.67
66.72+0.52
67.7920.65
65.0520.63
66.6840.71
68. 18.1620.68
70.8420.92
Max
60.34
67.66
68.75
66.05
68.52
69.25
72.27
Fit Time (s)
0.32
0.62
0.80
0.55
0.39
1.03
0.65
FPHA
Mcan#STD
85.59=0.72
89.33+0.49
99.70:0.51
86.5620.79 79
77.64 1.00
90.39:0.66
86.33=0.43
Max
86
90.17
90.5
87,83
79
92.17
87
Huang & Van Gool 2017) for the Radar, HDM05, and FPHA datasets; and TSMNet (Kobler ctt al. 2022a) for the Hinss2021 dataset. For the SPDNet architecture, we compare our LieBN with SPDNetBN (Brooks et al. 2019b), which applies the SPDBN (Eqs. ( and (8)) to SPDNet. Consistent with SPDNetBN, we apply our LieBN after each transformation layer (BiMap layer in App. [ In the EEG application, the state-of-the-an Riemannian method is TSMNet with SPD domain-specific momentum batch normalization (TSMNNr-SPDDDSMBN (Kobler et al. 2022a), which is domain adaptation version of Kobler et al. (022b)). For fair comparison, implement a domain-specitee momentum LieBN, referred to as DSMLieBN (detailed in App. E Following Kobler et al. (2022a). we apply our DSMLieBN before the LogEig layer (detailed Ap- pendix B in TSMNet. We use the standard cross-entropy loss as the training objective and optimize the parameters with the Riemannian AMSGGad optimizer (Bécigneul & Ganea 2018). The network architectures are represented as (do, di...., ... d, }, where the dimension of the parameter in the i-th BiMap layer d, x d The experiments are conducted with a learning rate of 5e batch size of 30, and training epoch of 200 on the Radar, HDM05, and FPHA datasets. For the Hinss2021 dataset, following Kobler et al. (2022a). we use a learning rate of 1e 3 with a weight decay of 1e -4_ a batch size of 50, and training epoch of 50. All experiments
also
an Intel Core 19-7960X CPU with 32GB RAM and an NVIDIA GeForce RTX 2080 Ti GPU. Eyaluation methods are explained App. G.3
6.1 EXPERIMENTAL RESULTS
For each family of LieBN DSMMLiBNN, we report representatives: the standard one induced from the standard metric (0 = 1), and the one induced from the deformed metric with selected 0. If the standard one is already saturated, only report the resudts of the standard ones.
Application to SPDNet: As SPDNet is the most classic SPD network, apply our LieBN to SPDNet on the Radar, HDM05, and FPHA datasets. Additionally, we compare our method with SPDNetBN, which applies the SPDBN in Eqs. and ( to SPDNet. Following Brooks et al.
8

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Table 5: Cross Cross-validation results of TSMNet with SPDDSMBN and DSMLieBN on the Hinss dataset. For simplicity, DSML.eBN-MMerii-000 is abbreviated as Metric-(0). For the DSMLieBN under each metric, if the DSMLieBN induced by the standard metric (0 1) is not saturated, we report the DSMLieBN under the deformed metric at the bottom rows of the table.
(a) Inter-session classification
(b) Inter-subject classification
Method
Fit Time (s)
McantSTD
Method
Fit Time (8)
Mcan=STD>
SPDDSMBN
0.16
12+19.9.87
SPDDSMBN
7.74
50.10+8.08
AIM-(1)
0.16
55.10:7.61
AIM-(1)
6.94
50.04+8.01
LEM-(1)
0.13
54.95=10.09
LEM-(1)
4.71
50.95#6.40
LCM-(1)
0.10
51.5426.85 88
1.CM-(1)
3.59
51.86++453
LCM-(0.5)
0.15
53.11
AIM-(-0.5)
8.71
53,97+8.78
(002d)) we use the architectures of {20, 16, 8). {93, 30), and {63, 33) for the Radar, HDM05 and FPHA datasets, respectively. The 10-fold average results, including the aver- age training time (slepcoh), summarized in Tab. 4 We have three key observations regarding the choice of metrics, deformation, and training efficiency. The choice of metrics: The metric that yields the most effective LieBN layer differs for each dataset. Specifically, the optimal LieBN layers on these three datasets are the ones induced by AIM-(1), LCM-(0,5), and AIM-(1.5), respectively, which improves the performance of SPDNet by 2.22% %, 11.71 % and 4.8% Additionally, al- though the LCM-based LieBN performs worse than other LieBN variants on the Radar and FPHA datasets, exhibits the best performance on the HDM05 dataset. These observations highlight the advantage of the generality of our LieBN approach. The effect of deformation: Deformation pat- terns also vary across datasets. Firstly, the standard AIM is already saturated on the Radar dataset. Secondly, indicated in Tab. 4 appropriate deformation factor 0 can further enhance the per- formance of LieBN, Notably, even though the LicBN induced by LCM-(I) impedes the learning of SPDNet the FPHA datasets, it can improve performance when appropriate deformation factor 0 is applied. These findings highlight the utility of the deforming geometry of the SPD manifold. Efficiency: Our LieBN achieves comparable or even better efficiency than SPDNetBN, although compared with SPDNetBN, our LieBN places additional consideration on variance. Particularly, the LieBN induced by standard LEM or LCM exhibits better efficiency than SPDNetBN, Even with deformation, the LCM-based LieBN is still comparable with SPDNetBN in terms of efficiency. This phenomenon could be attributed to the fast and simple computation of LCM and LEM.
2019b): Chen et al.
Application EEG classification: We evaluate our method on the architecture of TSMNet for two tasks, inter-session and inter-subject EEG classification. Following Kobler et al. (2022a), we adopt the architecture of { (40, 20). Compared the SPDDSMBN, TSMNet+DSMLiett -AM obtains the highest average scores of 55. 10% and 53.97% in inter-session and -sujject transfer learning, improv- ing the SPDDSMBN by 0.98% and 3.87% respectively. In the inter-subject scenario, the advantage of the efficiency of our LieBN over SPDDSMBN is more obvious. Specifically, both the LEM- and LCM-based DSMLieBN achieve similar or better performance compared to SPDDSMBN, while requiring considerably less training time. For example, DSMLLeBN-LCM--0 achieves better sults with only half the training time of SPDDSMBN on inter-subject tasks. Interestingly, under the standard AIM, the sole difference between SPDDSMBN and our DSMLieBN is the way of center- ing and biasing. SPDDSMBN applies the inverse square root and square root to fulfill centering and biasing, while AlM-induced LieBN uses more efficient Cholesky decomposition. As such, the DSMLieBN induced by the standard AIM is more efficient than SPDDSMBN, particularly on the inter-subject task.
7 CONCLUSIONS
This paper proposes novel framework called LieBN, enabling batch normalization over Lie groups. Our LieBN can effectively normalize both the sample and population statistics. Besides, we gener- alize the existing Lie groups SPD manifolds and showease framework on the parameterized Lie groups of SPD manifolds. Extensive experiments demonstrate the advantage of our LieBN.
There are several other types of Lie groups in machine learning, such as special Euclidean groups As future avenue, we shall extend our LieBN to other Lie groups

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Published as a conference paper at ICLR 2024
ACKNOWLEDGMENTS
This work was partly supported by the MUR PNRR project FAIR (PE00000013) funded by the NextGenerationEU, by the EU Horizon project ELIAS (No. 101120237), and by gift donation from Cisco. The authors also gratefully acknowledge financial support from the China Scholarship Council (CSC).
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APPENDIX CONTENTS
Notations
16
Basic Layers in SPDNet and TSMNet
16
Statistical Results of Scaling in the LieBN
16
LieBN as Natural Generalization of Euclidean BN
19
Domain-specific Momentum LieBN for EEG Classification
19
Backpropagation of Matrix Functions
20
Additional Details and Experiments of LieBN on SPD manifolds
21
Datasets and Preprocessing
G.2
Hyper- Paremeerrssterss
21
Evaluation Methods
G.4
Empirical Insights on the paraae-arsmesers in LicBN on SPD Manifolds
21
G.4.1
The Effect of B in SPD LieBN
22
H
Prelininary Experiments on Rotation Matrices
22
Geometry on Rotation Matrices
22
H.2
Datasets and Preprocessing
22
H.3
Implementation Details
23
4
Results
23
Proofs of the Lemmas and Theories in the Main Paper
23
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A NOTATIONS
For better clarity, we summarize all the notations used in this paper in Tab. 6
Table 6: Summary of notations.
Notation
Explanation
{M, O,g or abbreviated as MM
Lic group with group operation @ and left-invariant Riemannian metric
Pol
Group inverse of P E .M
TpM
The tangent space at P E .M
9p(:.) or (:))
The Riemannian metric at P E .M
: ||P
The norm induced by ( on Tr.M
d(
Geodesic distance
FM
Fréchet mean
WFM
Weighted Fréchet mean
Logp
The Riemanniin logarithm at P
PTr-Q
The Riemannian parallel transportation along the geodesic connecting / and Q
Lp
The Lie group left translation by P E EM
The differential map of the smooth map f at E E
["g
The pullback metric by f from
St
The SPD manifold
The Euclidean space of symmetric matrices
L"
The Euclidean space of lower triangular matrices
(
The standard Frobenius inner product
The norm induced by the standard Frobenius inner product
ST
ST
{(x, 3) R min(a, + nı3) > 0}
The O(n)- iinariiat Euclidean inner product
The norn induced by ((n)-invariant Euclidean inner product
9 glan)j
The Riemannian metric tensor of 3)-AM
The Riemannian metric tensor of (6, o. 5)-MM
@AI
The group operation w.r.t. AIM
GLE
The group operation w.r.t LEM
@LC
The group operation w.r.t. LCM
Riemannian Gaussian distribution
exp
Scalar exponentiation
mlog
Matrix logarithum
mexp
Matrix exponentiation
Chol
Cholesky decomposition
Dlog
The diagonal element-wise logarithm
tie
Dlog Chol
The strictly lower triangular part of a square matrix
DO))
A diagonal matrix with diagonal elements from square matrix Matrix power function
Pa(-) or (-)
B BASIC LAYERS IN SPDNET AND TSMNET
SPDNet (Huang & Van Gool 2017) is the most classic SPD neural network. SPDNet mimics the conventional densely connected feedforward network, consisting of three basic building blocks
BiMap layer: 5 = W* 1rrksk-11p - with W semi-orthogonal,
(23)
ReEig layer: 5 = U max ke.))k-11, with sk Uk -
(24)
LogEig layer: 5* = log(
(25)
where max(.) is element-wise maximization. BiMap and ReEig mimic transformation and non- linear activation, while LogEig maps SPD matrices into the tangent space at the identity matrix for classification.
TSMNet (Kobler et al. 2022a) can be illustrate as Ste + fec + JBIM + SReEig + LooggEgg where fie and Jac denote temporal and spatial convolution, respectively.
C STATISTICAL RESULTS OF SCALING IN THE LIEBN
In this section, we will show the effect of our scaling (Eq. (14) on the population. We will see that while the resulting population variance generally agnostic, becomes analytic under certain
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cireumstances, such SPD manifolds under LEM or LCM. As result, Eq. 14 can normalize and transform the latent Gaussian distribution.
To simplify, let (PP)) Expy (s (P)] Similar to the main paper, M denotes Lie group with a left-invariant metric. First, we present a lemma on the resulting P.D.F. of a random point transformed by d..
Lemma C.1. Given a random point X distributed over M with P.D.E. px, the of = (x is given by:
py (Q) Apx(o!-)
(26)
where A
Here
|enotes the determinant, and o and Ls, (2)'0-)--1 are the
16 (0)00011
differentials.
Proof. For the sake of simplicity, we will denote o, as throughout this proof. The volume element w.r.t. left-invariant metric Haar measure (Pennec & Ayache 1998 Sec. 3.2):
dP
de M(P = [.p,,
(27)
where [Lp,,El is the determinan of the differential of Lp the neutral element E. Then have
(Q)
d, M()
(L0-(Q)).E!
(L0-111)00-
(28)
1074
dr. M(Q) [1-110)00-111
M(Q).
The probability of =(P) in a set M
F(O(P) E y) F(P a- (Y))
-f 1-1
(29)
px (()
/ A
Therefore, the density of Y = o(X) is
py (Q) Apx (px()(-)(()))
(30)
The above lemma implies that when A is constant, Y also follows Gaussian distribution.
Corollary C.2. Following the notations in Lem. C. if c is constant and X N(E,, then Y also follows a Gaussian distribution, i.e.y (E, 202)
'This should be more precisely understood as the determinant of the matrix representation of T.p., under local coordinate
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Proof.
d(o,+(Q): E)3
pr = ck(o) exp
20
d(Expg 1/s LoLog (2), E))
(6) exp
20²
(31)
Logg (Q)
(6) exp
2s20²
d(Q, F))
(6) exp
2s2
where || || is the norm of the tangent space the neutral element E.
Cor. C.2 implies that when A c, ts can scale the population variance and further transform the Gaussian distribution. Simple computations show that in the standard Euclidean space R A = 1/s. Therefore, natural to expect that the pullback of I² also enjoys constant A.
Proposition C.3. Consider n-dimenssonal Lie group M pulled back from the standard Euclidean spuce R" by the diffeomorphism v/ : . - R". In other words, the group operations and Rieman- nian metric M are defined by vb from R Then A remains constant M.
Proof. To simplify notation, we denote &, as Φ. Under the given assumption, the group addition and Riemannian metric on M are defined follows:
VP + (v(P) (Q1))
(32)
ge
where g the standard Euclidean metric. Therefore, & can be simplified
s(P) = ExpE [s (P)]
+/ EExpo ei (s (
(33)
+ (sxb((
where Exp and Lög the Riemannian exponential and logarithmic maps in R", which reduced to vector addition and subtraction, respectively. Therefore, the inverse of øi is
(P) - (V/a(P))-
(34)
Besides, L=-'((Q)QQ-1 can also be further simplified:
L (P)) = -- (4/ab((Q) +((2)+((P)
(35)
The differentials of Eqs.
and (35) Q are
$
1- (1) o P4.0.
(36)
8
(Q=110)00-+ 1) -
1, %/..0-
(37)
Therefore, A 1/6 for all Q M.
By Prop. we can directly obtain the following corollary.
Corollary C.4. Given Lie group M pulled back from the Euclidean space, and random point X K~N *~(EE, o²) over M, Y = (X) ~ NN,, 202)
In machine learning, several Lie groups derived by the pullback from the standard Euclidean space. As shown Chen et al. (023c) and the proof of Prop. [.2] (o, -LMM and #-LCM are pullback metrics from the Euclidean metric. Therefore, for the Lie groups of SPD manifolds w.r.t. (a, 3EMM and 0-LCM, Eq. (14) can transform the Gaussian distribution. Specifically, given
random point X N(M, o²) Eqs. 13) to 15 transform the Gaussian distribution as:
N(M, ((E, - N(F,
N(B
(38)
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where M and employed to normalize X, and in Eq. [4) is omitted. The above process exactly mirrors the transformation of Gaussian distributions within the framework of standard BN (loffe & Szegedy 2015).
Remark C.5. A similar result to our Cor. C.4 was also presented Chakraborty 2020 Prop. 3). However, in his proof, the author did not account for the Haar measure and only considered the P.D.F., casting doubt on the validity of their results. Additionally, their discussion is limited to matrix Lie groups, specifically under the distance d) - (nlog(P-1Q)1 In contrast, we rectify their proof and consider general Lie groups.
D LIEBN AS NATURAL GENERALIZATION OF EUCLIDEAN BN
The centering and biasing in Euclidean BN correspond to the group action of R. From a geometric perspective, the standard Euclidean metric is invariant under this group operation. Consequently, not surprising that our LieBN algorithm formulated in Alg. serves standard Euclidean batch normalization. We formalize this fact in the following proposition.
natural generalization of
Proposition D.1. The LieBN algorithum presented in Alg. ii equivalent the standard Euclidean BN when M = R", both during the training and testing phases.
Proof. The core of this proof lies in the fact that RR, (1) the Fréchet mean and variance reduced to the familiar Euclidean statistics, (2) the calculation of the running mean becomes the weighted arithmetic mean. (3) Eqs. (13) to (15) become Eq. (6): We prove these three points one by one.
As stated in Lou et al. 20000, Prop. G.1 and Cor. G.2), from the view of the product manifold, the element- wisee Fréchet mean and variance on R" are equivalent to the vector-valued Euclidean variance and mean.
Besides, by similar proof as in Lou et al. 2020 Prop. G.I), the weighted Fréchet mean R is simplified as the weighted arithmetic average. Therefore, on R.,, the calculation of running statistics in our Alg. becomes the familiar moving average.
Thirdly, on R we know that (y) x + y, Exp, v = x Log, y/ - x, and the neutral element is O. Since statistics, as well the Euclidean BN, are calculated element-wisely, we can safely consider a single element, f.e.R" = R. For a batch of activations { E R}, where the batch mean and batch variance denoted as μh and vf. then Eqs. (13 15) rewrote
1)
Logo(L- (a:))) Lo
4b
LB
ExPo
+ + +
(39)
0g +e
Vog
The above equation is the exact computation of the standard Euclidean BN.
E DOMAIN-SPECIFIC MOMENTUM LIEBN FOR EEG CLASSIFICATION
Kobler et al. (2022a) proposed SPD domain-specifit momentum batch normalization (SPDDSMBN) as a domain adaptation approach for EEG classification. SPDDSMBN, based on Eq. (1). per- formed normalization of mean and variance on SPD manifolds under the specific AIM. Addition- ally. SPDDSMBN utilized separate momentums for updating training and testing running statistics, inspired by the work of Yong et al. (2020). Following Kobler et al. [002a Alg. 1), we also present momentum LieBN (MLieBN) Alg. Here is fixed and Yraai defined as
Arain
- max (x,00) p, where p domains.per batch
(40)
Furthermore, in line with Kobler al. (2022a). adopt multi-channel mechanisms for domain- specific MLieBN (DSMLieBN), where each domain has its own MLieBN layer. Similar to Kobler ct al. (2022a), we the biasing parameter equal the neutral element, and the scaling factor is shared across all domains. We denote Alg. 2 as (P s, ).rain. Then our DSM-
LieBN follows
DSMLieBN MLieBN, (P3 E,, 8,6,, Nin YP, 3 (PI..N),
(41)
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Algorithm 2: Momentum LieBN (MLieBN) Algorithm
Input
: A batch of activations (PI...N & M), and a small positive constant running mean M, E, rumning variance 52 for training running mean M, E, rumning variance oiद for testing biasing parameter B E M, scaling parameter E IR/ /O], momentum for training and testing "train:? 7 E [0, 1]
Output :
Normalized activations {PI...n)
training then
Compute batch mean Mr and variance o of (Pi...N):
M, + WFM((1 Itrain Terain b. (M,, Mb};;
of < (1 Ytrain )0² Mtrai 1/6
M, < WFM( [1 - --, ?,, Mc)):
o2 < (1 70 7
end
if training then M <- Mr, x2 s- i?:
else M <- M, x² i
for i + 1 to N do
Centering to the neutral element E: P. + Mo¹ (P.)
Scaling the dispersion: P; + ExpE VoR+e
Logg( (P.)
Biasing towards parameter B: P. <- (B()
end
where i is the index of the domain. We follow the official code of SPDDSMEN to implement our DSMLieBN. In word, the only difference between DSMLieBN and SPDDSMBN is the different way of normalization.
Analogous Thm. 5.3 computations for DSMLieBN under pullback metrics can also be performed by mapping, calculating, and then remapping.
F BACKPROPAGATION OF MATRIX FUNCTIONS
Our implementation of LieBN on SPD manifolds involves several matrix functions. Thus, we em- ploy matrix backpropagation (BP) (Ionescu et al.) 2015) for gradient computation. These matrix operations can be divided into Cholesky decomposition and the functions based on Eigendecompo- sition.
The differentiation of the Cholesky decomposition can be found Murray 2016 Eq. 8) or Lin
(2019 Props. 4). Besides, our homemade BP of the Cholesky decomposition yields similar gradient to the one generated by autograd of torch. coorch.iinalle www.llin....llssteee....... experiments, we use torch. www.ll.......................... cholesky.
Therefore, during the
The second type of matrix functions is based on Eigendeconpossiion, such as matrix exponential, logarithm, and power. Although torch (Paszke et al. 2019) supports autograd of Eigendecomposi- tion, it requires the computation of 8,-8, (Ionescu et al. 2015 Props. 1), where ò; and o, denote
eigenvalues. This might trigger numerical instability when d; approximates 85. Following Brooks al. 2019b),, we use the Daleckii- -reiin formula (Bhatia 2013 Thm. V.3.3) to calculate the BP of Eigen-based matrix functions. In detail, for a matrix function defined as X = f(S) = U
with S USUT as the eigendecomposition of an SPD matrix, its BP expressed as
v = UKK @ LUU)UU
(42)
where Vx L is the Euclidean gradient of the loss function w.r.t. X. Matrix K is defined as
((i)) d;-d;
if if + 0j otherwise
Kij
(43)
(o.)
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where Σ = diag(01, 02, ,o)). Eq. 43 demonstrates the numerical stability of Daleckii-Kri Kreein formula.
G
ADDITIONAL DETAILS AND EXPERIMENTS OF LIEBN ON SPD MANIFOLDS
We use the official code of SPDNetBN (Brooks et al. 2019b and TSMNe Kobler et al. 2022a) implement our experiments the SPDNet and TSMNet backbones.
G.1 DATASETS AND PREPROCESSING
Radar dataset (Brooks et al. 2019b) contains 3,000 synthetic radar signals. Following the pro- tocol in Brooks et al. (2019b), 9b,, each signal is split into windows of length 20, resulting in 3,000 covariance matrices of the size 20 x 20 equally distributed in 3 classes. HDM05 dataset (Müller et al. (007) consists of 2,273 Axeleton-basee motion capture sequences executed by different actors. Each frame consists of 3D coordinates of 31 joints, allowing the representation of each sequence a 93 x 93 covariance matrix. In line with Brooks et al. (2019b), trim the dataset down to 2086 instances scattered throughout 117 classes by removing some under-represeeneed clips. FPHA Hermado et al., 2018) includes 1,175 skeleton- based first-person hand gesture videos of 45 different categories with 600 clips for training and 575 for testing. Following Wang et al. 202 we represent each sequence as 63 x 63 covariance matrix. Hinss2021 dataset (Hinss et 2021
is recently released competition dataset containing EEG signals for mental workload estimation. The dataset employed for two tasks, namely inter-session and inter-subject, which are treated as domain adaptation problems. Geometry-awart methods (Yair et al. 2019 Kobler et al. 2002a) have demonstrated promising performance in EEG classification. We follow Kobler et al. (2022a) for data preprocessing. In detail, the python package MOABB (Jayaram & Barachant 20188) and MNE Gramlort 2013) are used to preprocess the datasets. The applied steps include resampling the EEG signals 250/256 Hz, applying temporal filters to extract oscillatory EEG activity in the 4 to 36 Hz range, extracting short segments < 3s) associated with a class label, and finally obtaining 40 x 40 SPD covariance matrices.
G.2 HYPER-PARAMETERS
We implement the SPD LieBN and DSMLieBN induced by three standard left-invariant met- rics, namely AIM, LEM, and LCM, along with their parameterized metrics. Therefore, our three hyper-parameters, i.e., ax B). where 0 controls deformation. In our LieBN, (a. only affects variance caleulation. Therefore, set (a, B) (1, and only tune the deformation factor 9 from the candidate values of #0.5, =1, and 115.. We de- note [Baseline]+[BN (Mtric(Metri as the baseline endowed with specific LieBN, such as LPDNee+LieBN-AIM-(() and TSMNet-DSMLiBBN..CM..
method has a maximum
G.3 EVALUATION METHODS
In line with the previous work (Brooks et al. 2019b Kobler et al. 2022a), we use accuracy as the scoring metric for the Radar, HDM05, and FPHA datasets, and balanced accuracy (i.e., the average recall across classes) for the Hinss2021 dataset. Ten-fold experiments FPHA datasets are carried out with randomized initialization and split (split is officially fixed for the FPHA dataset), while the Hinss2021 dataset, models are fit and evaluated with randomized leave 5% of the sessions (inter-session) or subjects (inter-subject) out cross- volidavaliotioon scheme.
the Radar, HDM05, and
G.4 EMPIRICAL INSIGHTS ON THE PAPMMETRR IN LIEBN ON SPD MANIFOLDS
Our SPD LieBN has at most three types of hyper-parameters: Riemannian metric, deformation factor 0, and (nn)-invarian) parameters (a, 3). The general order of importance should be Riemannian metric 0 > (a,3) 3).
The most significant parameter is the choice of Riemannian metric, as all the geometric properties are sourced from a metric. safe choice would start with AIM, and then decide whether to explore other metrics further. The most important reason is the property of affine invariance of AIM, which
Supplemental.zip bbag:://gthhu.comkoooee////MMSS
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is natural characteristic of covariance matrices. In experiments, the LieBN-AIM generally achieves the best performance. However, AIM is not always the best metric. As shown in Tab. 4b) the best result the HDM05 dataset is achieved by LCM-based LieBN, which improves the vanilla SPDNet by 11.71 %. Therefore, when choosing Riemannian metrics on SPD manifolds, a safe choice would start with AIM and extend other metries. Besides, if efficiency is an important factor, one should first consider LCM, as it is the most efficient one.
The second one is the deformation factor 0. As we discussed in Sec. 5.1 interpolates between different types of metrics (# = 1 and 0 + 0). Inspired by this, we select H around its deformation boundaries (1 and 0). In this paper we roughly select 0 from (=0.5, =1, +155
The less important parameters are 3). Recalling Alg. | and Tab. | (a, 3) only affects the calculation of variance, which should have less effects compared with the above two parameters. Therefore, we simply set (a, B) (1, during experiments.
G.4.1 THE EFFECT OF 3 IN SPD LIEBN
Recalling Eq. (3), 3 controls the relative importance of the trace part against the inner product. Therefore, we set the candidate values
as {1, 1n, 1/x², 0, c, -./n2), where is the input dimension of LieBN, and is small positive scalar to ensure (mn)-invariance, i.e., (a, 3) E ST. 1/2² and 1/1, means averaging the trace in Eq. 3 while the sign of B denotes suppressing (-), enhancing (+), neutralizing (0) the trace.
We focus on AlM-based LieBN on the HDM05 dataset. We set 1.5, as it is the best deformation factor under this scenario. Other network settings remain the same the main paper. The 10-fold average results are presented Tab. 7 Note that on this setting, n = 30. As expected, B has minor effects on our LieBN.
Table 7: The effect of different ,3 for AlM-based LieBN on the HDM05 dataset.
Beta
-1/302
-0.03
1/302
1/30
SeD
18.08.88
68. 12+0.74
68.2040.85
68. 18+0.85
68. 16+0.80 68.1 16#0.68
H PRELIMINARY EXPERIMENTS ON ROTATION MATRICES
This section implements our LieBN in Alg. on the special orthogonal groups, i.e., (e)),, also known as rotation matrices. We apply our LieBN to the classic LieNet (Huang & Van Gool (017). where the latent space is the special orthogonal group.
H.1 GEOMETRY ON ROTATION MATRICES
Table The associated Riemannian operators on Rotation matrices.
Operators
dR(RS)
Log, Expy(A)
(t..s)(t)
FM
Expression
||mlog (RT |] mlog(R) mexp(A) R (log S)) Manton 2004 Alg. 1)
We denote R, e SO(n), and si (t) the geodesic connecting R and S. The neutral ele-
ments rotation matrices is the identity matrix. Tab, summarizes all the necessary Riemannian ingredients of the invariant metric S0(n).
For the specific S3(3), the matrix logarithm and exponentiation can be calculated without decom- position (Murray al. 2017 Exs. A. 11 and A.14).
H.2 DATASETS AND PREPROCESSING
Following LieNet. validate our LieBN the G3D dataset (Bloom et al. 2012). This dataset
Bloom et al. 2012) consists of 663 sequences of 20 different gaming actions. Each sequence is recorded by 3D locations of 20 joints (i.e., 19 bones). Following Huang & Van Gool (017))
the code of Vemulapalli et (2014) to represent each skeleton sequence as point on the Lie group
SO NxT (3), where N and T denote spatial and temporal dimensions. As preprocessed in Huang & Van Gool 2017), set 7 as 100 for each sequence the G3D.
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H.3 IMPLEMENTATION DETAILS
LieNet: The LieNet consists of three basic layers: RotMap, RotPooling, and LogMap layers. The RotMap mimics the convolutional layer, while the RotPooling extends the pooling layers to rotation matrices. The logMap layer maps the rotation matrix into the tangent space at the identity for classification. Note that the official code of LieNe is developed by Matlab. We follow the open- sourced Pytorch code to implement our experiments. To reproduce LieNet more faithfully, we made the following modifications to this Pytorch code. We re-code the Mop and RotPooling layers to make them consistent with the official Matlab implementation. In addition, existing Riemannian optimization package geoopt [éécigneul & Ganea (2018) into SO(3) to allow for Riemannian version of SGD, ADAM, and AMSGrad on SO(3), which is missing in the current package. However, we find that SGD is the best optimizer for LieNet. Therefore, we adopt SGD during the experiments. We apply our LieBN before the LogMap layer and refer to this network as LieNetLieBN, Note that the dimension of features in LieNet is B x T x 3 x 3, we calculate Lie group statistics along the batch and spatial dimensions (B 7), resulting in an N 3 x running mean.
also extend the
Training Details: Following Huang al. (2017). we focus on the suggested 3Blocks architecture for the G3D dataset. The learning rate is le with weight decay of le-s. Following LieNet, we adopt 10-fold cross-subject test setting, where half of the subjects are used for training and the other half are employed for testing.
H.4 RESULTS
The 10-fold results are shown in Tab. 9 Due to different software, our reimplemented LieNet is slightly worse than the performance reported Huaag et al. (2017). However, we still can observe a clear improvement of LieNetLieBN over LieNet.
Table 9: Results of LieNet with or without LieBN on the G3D dataset.
G3D
Methods
STD
Max
LieNet LieNetLieBN
87.91=0.90 88.8
89.73
90.67
I PROOFS OF THE LEMMAS AND THEORIES IN THE MAIN PAPER
Proof of Prop. 4.7
Property
The MLE of M is
N
d),
MMLE = argmax log(K(>))
2x
(44)
argmin d),
Property 2
We denote Y Lx(X), and px and py as the density of X and respectively. The density of Y
(Q) = P (Q))
(Q),
kk(a))e))))))) exp
202
(45)
d(Q, LB,Ln(()))
(((o))xp
20
http:///gghnb....m/iiwwwuuuuu//r hhtpp://github.cco////1997771/1 icNet
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The first equation is obtained by (Pennec 2004 Thm. 7), while the last equation is obtained by the isometry of the left translation.
Proof of Prop. 4.2
The isometry of Lg can directly obtain the homogeneity of the sample mean.
Now let us focus on Eq. (17). We have the following:
we (P)
Σ
Wi ||s Logg Prlle
Σ
N
Wi Logg P: 1
(46)
1=1
2
(P.P.. E),
i=1
where || I|E is the norm TEM.
Proof of Prop. 5.1 We first prove the case of (0, B)MM,M, and then proceed to the case of 0-LCM.
8L,M; For clarity, we denote the metric tensor of (0, a, 3)-LEMMMMMM as
g (ODDLLE
82 P (
(47)
where (a..)).LE is the metric tensor of (a, B3-LEMM. Let P E S and V.W. E TpS++ then we have
9p
()...))LLE
(V,W)
p2PP)(P)
1 (L, (Po,.p(V), ((W)
@2 ((mlog Po) (V), (mlog Po). (W))(a.8)
1
(48)
(mlog,, p (V), mlog, p (a))la.3)
(a.9)-LB (V, W). gp
0-LCM: Let first review well-known fact of deformed metrics (hhanwerds & Pennec 2022a 1
Let = ga P7 g be the pofer-deformed metric on SPD Then when 0 tends to 0, for all P E S
and all V TpS* we have
(P(V.V) V) - gr (V), pV), (V)).
(49)
By Eq. we can readily obtain the results.
Proof of Prop. 5.2 (a, 3)-AIM is left-invariant (Thanwerdas & Pennec 2022b As the pullback of (a, AAM, (0, a, 3MMAIM is left-invariant as well, Besides, Chen et al. (202dd) shows that LCM is the pullback metrie from the Euclidean space of L". Therefore, #-LCM bi-invariant.
Besides,[Chen
Proof of Thm. 5.3
We denote Eqs. 13 to on Mr, i = 1,2 2 as the mapping 5'(- B)
Let B = [PI...N and f(B) {f P N).
The core of this proof lies in three points:
1. The Fréchet mean and variance of M₁ correspond to the counterparts of f(B) in M2.
2. (P,),M u2, B, Mi equal (- v², ((B), s)).
3. The updates of running statistics in MM correspond the counterparts in M2-
We denote M as the Fréchet mean of B, and v² as the Fréchet variance of B. Then, by the isometry of f. the Fréchet mean and variance of ((3) are f(M) and v², respectively.
On M,, i = 2, we denote L' O², Exp" Log" as the Lie group and Riemannian operators, E as the neutral element, and Eq. (14)
oi (.). With the isometry and Lie group isomorphism of
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have the following equations:
L1 M-1 = =f L2
((M)) of.
(50)
6¹
0 Expe [s Logg (+)
f (Exple [s Logg (f()))
/ oo f,,
(51)
1 f-1 (m of.
(52)
Then we have
= (UPA)OO,
(53)
Lastly, we show the correspondence between running statistics. Since the Fréchet variance is the same for both B and f(B), we focus on the running mean. Let M, and fM,) denote the initial values of the running means in M and Mr respectively, and WFM represent the weighted Fréchet mean in M, Then the updated running mean in M1 is
WFM ((1 7), (M, M)) f 'WWMM
({1 ({1 1,7,,,), (fM)))
(54)
we can further simply the above equation as
WFM = S-1 WFM
(55)
Denoting LieBN' as the LieBN algorithm on M,, Eq.
53 and Eq. 55
imply that:
LieBN' s, = f-1
(56)
25


Reviews:
[{"review_id": "Ftv4p0eMPg", "reviewer": "Reviewer_Urfr", "summary": "The authors describe a new kind of Batch Normalization layer for\nRiemaniann neural networks. The proposed technique is a theoretical\nimprovement over existing batch-norm layers by being a generalized and\nunified view on all previously proposed technique, using the Lie-group\nstructure of the manifold.", "questions": "- What is the interest of the (alpha, beta) generalization ? In\n  particular in context of neural networks ? And what are the value used\n  in the experiments ?\n- In the article about RBN, Brooks et al discuss about the amount of\n  data need to achieve good performance. Any insight about this for your\n  layer ?\n- Is there a link between Frechet variance and the variance of the\n  Gaussian used for normalization ?\n\n- I guess it should be \"neutral element\" instead of identity in Eq 13 ?", "limitations": "", "rating": 8, "confidence": 5, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "- A very pleasant to read recap on all batch norm-like layers for\n  Riemannian networks.\n- Theoretical guaranties on the control provided by the layer.\n- Convincing experimental evaluation (not SOTA obviously, but an\n  improvement over other manifold methods)\n- LieBN reduces to the classical BN for Euclidean manifold.", "weaknesses": "- Nearly all tables are barely readables.\n- Novelty of the work should emphasized more. LieBN provides more\n  guaranties and a more sound approach. But in the writing, it is not\n  completely clear of what is a full novelty over previous methods and\n  what is a generalization.\n- A broad zoo of choices (AIM, LEM, LCM and alpha beta variants), but no\n  clue on choosing. It's an usual question with this type of methods,\n  and it always a little bit disapointing to simply benchmark over all\n  the possible choices.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pJPAT2ChMv", "reviewer": "Reviewer_ZRPd", "summary": "This paper proposes a batch normalization layer for neural networks on Lie groups. The authors then focus on SPD neural networks to showcase their approach. The proposed method is validated on radar recognition, action recognition, and electroencephalography (EEG) classification.", "questions": "The proposed technique is a simple tweak of those from Kobler et al. (2022b), Lou et al. (2020), Chakraborty (2020). \nNo new concepts or ideas have been developped w.r.t. these works. While the authors state that the proposed technique works for Lie groups and is able to control mean and variance in contrast to these works, extensions of these works to Lie groups, as done in the paper, are trivial.\n\nThe experimental results are not convincing since the proposed method is only compared with some SPD neural networks. For example, on human action recognition, the proposed method is outperformed by the method of Laraba et al. (2017) on HDM05 dataset by a large margin (72.27\\% vs. 83.33\\%). This shows that the proposed technique is probably not effective compared to other learning techniques designed in Euclidean space. \n\n*Question*\n\nHow does the proposed method perform on another Lie groups, e.g. when being used in LieNet (Huang et al., 2017) ?\n\n*References*\n\n1. Sohaib Laraba, Mohammed Brahimi, Joëlle Tilmanne, Thierry Dutoit: 3D skeleton-based action recognition by representing motion capture sequences as 2D-RGB images. Comput. Animat. Virtual Worlds 28(3-4) (2017)\n\n2. Zhiwu Huang, Chengde Wan, Thomas Probst, Luc Van Gool: Deep Learning on Lie Groups for Skeleton-Based Action Recognition. CVPR 2017: 1243-1252.", "limitations": "", "rating": 1, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 1, "strengths": "* Proofs are given in the supplementary material (I did not thoroughly check them)\n* Experiment results show improvements over some state-of-the-art SPD neural networks", "weaknesses": "* The paper lacks of novelty\n* Experimental results are not convincing\n* No discussion about the limitations of the proposed approach", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "0rmf6062pD", "reviewer": "Reviewer_B3vJ", "summary": "Study of Deep Neural Networks (DNNs) on manifolds, associated with normalization techniques, with a unified framework for Riemannian Batch Normalization (RBN) techniques on Lie groups. Theoretical guarantee are provided to caracterize the stability of the process. \nApproach is illustrated for Symmetric Positive Definite (SPD) manifolds, with three families of parameterized Lie groups, in a SPD neural networks.  Experiments have been done for radar recognition, human action recognition, and electroencephalography (EEG) classification.", "questions": "Add the following references on batch normalization\n[B] Daniel Brooks. Deep Learning and Information Geometry for Time-Series Classification. Machine Learning [cs.LG]. Sorbonne Université, 2020. English. ⟨NNT : 2020SORUS276⟩. ⟨tel-03984879⟩; https://theses.hal.science/tel-03984879\n[C] D. Brooks, O. Schwander, F. Barbaresco, J. . -Y. Schneider and M. Cord, \"Deep Learning and Information Geometry for Drone Micro-Doppler Radar Classification,\" 2020 IEEE Radar Conference (RadarConf20), Florence, Italy, 2020, pp. 1-6, doi: 10.1109/RadarConf2043947.2020.9266689.\n[D] D. Brooks, O. Schwander, F. Barbaresco, J. -Y. Schneider and M. Cord, \"A Hermitian Positive Definite neural network for micro-Doppler complex covariance processing,\" 2019 International Radar Conference (RADAR), Toulon, France, 2019, pp. 1-6, doi: 10.1109/RADAR41533.2019.171277.\n[E] D. A. Brooks, O. Schwander, F. Barbaresco, J. -Y. Schneider and M. Cord, \"Complex-valued neural networks for fully-temporal micro-Doppler classification,\" 2019 20th International Radar Symposium (IRS), Ulm, Germany, 2019, pp. 1-10, doi: 10.23919/IRS.2019.8768161.\n[F] D. A. Brooks, O. Schwander, F. Barbaresco, J. -Y. Schneider and M. Cord, \"Exploring Complex Time-series Representations for Riemannian Machine Learning of Radar Data,\" ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Brighton, UK, 2019, pp. 3672-3676, doi: 10.1109/ICASSP.2019.8683056.\n[G] Brooks, D., Schwander, O., Barbaresco, F., Schneider, JY., Cord, M. (2019). Second-Order Networks in PyTorch. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2019. Lecture Notes in Computer Science(), vol 11712. Springer, Cham. https://doi.org/10.1007/978-3-030-26980-7_78", "limitations": "", "rating": 8, "confidence": 5, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "Interesting algorithm LieBN, which enables batch normalization over Lie groups, to normalize both the sample and population statistics.and apply to SPD manifolds.", "weaknesses": "Density of probability on SPD matrix could be only defined as invariant to all the automorphisms of SPD manifold. To assess which density verify this property, you have to consider \"Lie Groups Thermodynamics\" developped by Jean-Marie Souriau. Consider upper-half space of Siegel (pure imaginary axis is the space of SPD matrix) where the Lie group SU(n,n) acts transitivelly. With Souriau method, you are able to compute the Gibbs density of maximum entropy that is covariant to SU(n,n). If you restrict to the imaginary axis, you find the density for SPD matrices. See the following reference and put it in your references:\n[A] Barbaresco, F. (2021). Gaussian Distributions on the Space of Symmetric Positive Definite Matrices from Souriau’s Gibbs State for Siegel Domains by Coadjoint Orbit and Moment Map. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science(), vol 12829. Springer, Cham. https://doi.org/10.1007/978-3-030-80209-7_28", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "l3ksoigxku", "reviewer": "Reviewer_LaGM", "summary": "The paper concerns batch normalization for Lie group valued data. The authors propose a unified framework for batch normalization that they claim offers theoretical guarantees.", "questions": "I believe the authors need to argue convincingly what is the contribution of the paper, and why the paper presents a significant contribution relative to the previously methods.", "limitations": "", "rating": 3, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 1, "strengths": "I have a hard time finding strengths that were not already presented in previous papers. I hope the authors can argue to the opposite, but as of now I am not sure of what is the actual contribution of the paper.", "weaknesses": "- I am unsure what is the contribution of the paper. As the authors state, the normalization scheme they propose has been used in previous work. There are some claims like \"In contrast, our work provides a more extensive examination, encompassing both population and sample properties of our LieBN in a general manner. Besides, all the discussion about our LieBN can be readily transferred to right-invariant metrics. \" but I was not able to find out what specifically these differences are. The approach seems to be almost exactly the same when I look up in the cited papers where it is applied to Lie groups as well.\n- using the Riemannian or Lie group exp and log maps for batch normalization was a good idea the first time it was presented, but I don't see the value added with the current paper", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "etNNQWrrDG", "reviewer": "Reviewer_R6Kt", "summary": "This paper proposes a batch-normalization (BN) method for manifold-valued features in neural networks. While in prior works several BN techniques are proposed for specific types of manifold, the authors present a general manifold-based BN formulation from a viewpoint of Lie-group. Besides, especially for SPD manifolds, practical BN methods are derived from the general formulation based on three types of pull-back metrics [Chen+23].\nThe experimental results using SPDnet and TSMnet demonstrate that the propose methods exhibit competitive performance to SOTAs.", "questions": "The above-mentioned concerns should be addressed especially in the following points.\n- Analysis about the SPD-BN methods in Table 3 from computational viewpoint in comparison to the other RNs.\n- Empirical comparison regarding computation cost.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "+ A general formulation of manifold-based BN is presented through reviewing/summarizing several Riemannian-normalization (RN) approaches.\n+ Practical BNs for SPD matrices are derived in an efficient form from the general formulation.", "weaknesses": "Novelty of LieBN in Sec.4 is limited as it is rather straightforward from the prior works [Kobler+22a] and [Chakraborty+20].\n\nOn the other hand, the pull-back metrics [Chen+23ab] are effectively applied to the general formulation to instantiate practical BN methods for SPD manifolds in an interesting way.\nThough especially pull-back Euclidean metrics seem to be efficient as shown in Table 3, this paper lacks in-depth analysis about the methods from qualitative and/or computational viewpoint.\nIt is demanded to clarify computational details such as by showing back-props through comparison to the other RN approaches based on complicated manifold-based computation, which would significantly improve reproducibility.\n\nConsidering \\theta-parameterization does not work so well as shown in Sec.6, such a parametric extension might be redundant, rather complicating the discussion in Sec.4. In stead of that extension, it may be better to focus on analyzing the practical BN methods shown in Table 3.\n\nAs to empirical performance results reported in the experiments, superiority of the method is less clear since the performance improvement is not significant due to large stds of performance scores.\nTo clarify the efficacy of the proposed method, it should be compared with the other RN methods in terms of computation cost not only the classification performance.\n\nBased on the experimental results, one cannot identify the best SPD-BN method that outperforms the others consistently. Although the authors insist such an inconsistency shows generality of the approach, it is less understandable and unfavorable from a practical viewpoint. In this case, provide some discussion and/or analysis about connection between types of metrics and tasks (or network architectures) for rendering insights into the SPD metrics.\n\nMinor comments:\nIn Eq.5: $\\frac{1}{N} \\sum$ -> $\\sum$", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)