Lie Group Decompositions for Equivariant Neural Networks
Authors: Mircea Mironenco
Patrick Forré
Venue: ICLR 2024 poster (2024)
Keywords: equivariant neural networks
lie groups
group convolution
geometric deep learning
Primary area: representation learning for computer vision, audio, language, and other modalities

Abstract: Invariance and equivariance to geometrical transformations have proven to be very useful inductive biases when training (convolutional) neural network models, especially in the low-data regime.
Much work has focused on the case where the symmetry group employed is compact or abelian, or both.
Recent work has explored enlarging the class of transformations used to the case of Lie groups, principally through the use of their Lie algebra, as well as the group exponential and logarithm maps.
The applicability of such methods to larger transformation groups is limited by the fact that depending on the group of interest $G$, the exponential map may not be surjective.
Further limitations are encountered when $G$ is neither compact nor abelian.
Using the structure and geometry of Lie groups and their homogeneous spaces, we present a framework by which it is possible to work with such groups primarily focusing on the Lie groups $G = \textnormal{GL}^{+}(n, \mathbb{R})$ and $G = \textnormal{SL}(n, \mathbb{R})$, as well as their representation as affine transformations $\mathbb{R}^{n} \rtimes G$.
Invariant integration as well as a global parametrization is realized by decomposing the "larger" groups into subgroups and submanifolds which can be handled individually.
Under this framework, we show how convolution kernels can be parametrized to build models equivariant with respect to affine transformations.
We evaluate the robustness and out-of-distribution generalisation capability of our model on the standard affine-invariant benchmark classification task, where we outperform all previous equivariant models as well as all Capsule Network proposals.

Published as a conference paper at ICLR 2024
LIE GROUP DECOMPOSITIONS FOR EQUIVARIANT NEURAL NETWORKS
Mircea Mironenco
Patrick Forré
Al4Science Lab, AMI.ab Informatics Institute
A14Science Lab, AMLab Informatics Institute
University of Amsterdam
University of Amsterdam
mircea.mironenco @ gmail.com
p.d.forre@uval @va..nl
ABSTRACT
Invariance and equivariance to geometrical transformations have proven to be very useful inductive biases when training (convolutional) neural network models, es- pecially in the low-data regime. Much work has focused on the case where the symmetry group employed is compact or abelian, or both. Recent work has ex- plored enlarging the class of transformations used the case of Lie groups, prin- cipally through the use of their Lie algebra, as well as the group exponential and logarithm maps. The applicability of such methods is limited by the fact that de- pending on the group of interest G, the exponential map may not be surjective. Further limitations are encountered when G' is neither compact ing the structure and geometry of Lie groups and their homogeneous spaces, we present a framework by which it is possible
abelian. Us-
work with such groups primarily focusing on the groups G = GL" + (n, R) and G = SL(n, I), as well as their repre- sentation affine transformations IR" x. Invariant integration as well as global parametrization is realized by decomposition into subgroups and submanifolds which can be handled individually. Under this framework, we show how convolu- tion kernels can be parametrized to build models equivariant with respect to affine transformations]] We evaluate the robustness and outoo-diistibution generalisa- tion capability of our model on the benchmark affine-invariant classification task, outperforming previous proposals.
INTRODUCTION
Symmetry constraints in the form of invariance equivariance to geometric transformations have shown to be widely applicable inductive biases in the context of deep learning (Bronstein et al.) 2021]. Group-theoretic methods for imposing such constraints have led numerous breakthroughs across variety of data modalities. CNNs (LeCun et al. 1995) which make use of translation cquivariance while operating on image data have been generalized in several directions. Group- equivariant convolutional neural networks (GCNNs) represent one such generalization Cohen & Welling (016). GCNNs make use of group convolution operators to construct layers that produce representations which transform in predictable manner whenever the input signal is transformed by an a-priori chosen symmetry group G. These models have been shown to exhibit increased gen- eralization capabilities, while being less sensitive to G-perturbations of the input data. For these reasons, equivariant architectures have been proposed for signals in a variety of domains such as graphs (Han et al. )022), sets (Zaheer al. 2017) or point clouds data (Thomas et al. (018). Con- structing equivariant networks entails first choosing a group G, a representation for the signal space in which our data lives and description of the way this space transforms when the group it. Choosing a particular group G entails making a modelling assumption about the underlying (ge- ometrical) structure of the data that should be preserved. Early work has focused on the case where G is finite, with later work largely concentrated on the Euclidean group E(n), and its subgroups SE(n, IK) or (O()). Working with continuous groups much more challenging, and the vast ma- jority of equivariant models focus on the case where the group G has set of desirable topological and structural properties, namely G is either compact or abelian, or both.
on
Our code is publicly available at http::://gittbb.coom/// www.mi wew.iiiiinnnonooggggnnn

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Published as a conference paper at ICLR 2024
Recent work (Bekkers) 2019; Finzi et al., 2020) explores the possibility of building equivariant networks for Lie groups continuous groups with smooth structure. This research direction is promising since it allows for the modelling of symmetry groups beyond Euclidean geometry. Affine and projective geometry, respectively affine and homography transformations are ubiquitous within computer vision, robotics and computer graphics (Zacur et al. (2014). Accounting for larger degree of geometric variation has the promise of making (vision) architectures more robust to real-world data shifts. When working with non-compact and non-abelian Lie groups, for which the group exponential is not surjective, standard harmonic analysis tools cannot be employed directly. Our contribution is a framework making it possible work with such groups.
Contributions We present procedure by which invariant integration with respect to the Haar measure can be done in principled manner, allowing for efficient numerical integration scheme to be realized. We then construct global parametrization maps which allow us to map elements back and forth between the Lie algebra and the group, addressing the non-surjectivity of the group exponential. We apply our framework to the groups G = GL GL*(n, R) and G = SL(n, RR, and more broadly the family of affine matrix Lie groups Aff(G) == Rn x G GL(n, R). The methodology and tools are generally applicable to any Lie group with finitely many connected components, and we explain how our approach can be seen as a generalization of previous proposals for constructing equviariant layers when working with the regular representation of a topological group.
2 RELATED WORK
Recent proposals for Lie group equivariance (Bekkers, 2019, Finzi et al., 2020) focus on the infinite- dimensional regular representation of group and rely the group exponential map allow con- volution kernels to be defined analytically in the Lie algebra of the group. Working with the regular representation entails dealing with an intractable convolution integral over the group, and (Monte Carlo) numerical integration procedure approximating the integral needs to be employed, which requires sampling group elements with respect to the Haar measure of the group. Unfortunately, the applicability of these methods is limited to Lie groups for which the group exponential map is surjective, which is not the case for the affine group Aff R)). These methods also rely on the fact that for compact and abelian groups sampling with respect to the Haar measure of the group is straightforward. which is not the case for the affine groups of interest. Manaonald et al. (2022) pro- pose a framework which while still relying on the group exponential. This proposal is more closely reviewed in Sec. A.3 together with other related equivariant models.
be applied to arbitrary Lie groups, aiming to address such limitations
3 BACKGROUND
Continuous group equivariance A Lie group G' is group as well as smooth manifold, such that Vg. h E G the group operation (g, h) r> gh and the inversion map g +> g- are smooth. GL, IR) denotes the Lie group consisting of all invertible n X matrices. A linear matrix Lie group refers to a Lie subgroup of GL(n, R). GL(n, IR), the translation group (R", +) and the family of affine groups Af) ,, G S GL IR) are our primary interest, with G usually being one of GL (n. IR), SL(n, KK) < GL (n, R) or SO(n). Equivariance with respect to the action of a locally compact operators. We recall that in the continuous setting we model our signals as functions f X + R defined on some underlying domain X. For example, images and feature maps can be defined as K K-channel functions f E (R22 k which are square-integrable (with respect to the measure μ)), and which have bounded support in practice, e.g. f [-1, C R² + RRK Cg denotes the left-regular representation of G. encoding the action of G on function spaces. For any continuous f E C(X):
group G can be realized by constructing layers using the ponss-coorllaiioo/oooouuiiii
(1) Every locally compact group G has a left (right) invariant Radon measure μa called the left (right) Haar measure of G. The Haar measure allows for G-invariant integration to be realized, and for the group convolution to be defined. To state the invariance property of μG, define the functional Apci
[x)f(x)= = f')
G
Aue= L'(G) + IR, (fo [ flg)dpo(o), e ('))'G
(2)
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Then, left Haar measure respects Aya (Cgf) Apa (f) for any g G and f (G).. Additional details on convolution operators are provided in Sec. Sec. A.I with Lie groups reviewed in Sec. B.1
Convolution operators For a group G acting transitively on locally compact spaces X and Y we then seek construct an operator K (X) + (Y) satisfying the equivariance constraint o = Ko Cg. We formalize two scenarios, when X is a homogeneous space of G (not necessarily group) and Y = G, and the case where X Y G. Focusing on the second case, if du.x dug is the Haar measure on G, the integral operator IC can be defined was the standard convolution/cross- IR be kernel that is invariant to the left action of G in both arguments,
correlation. Let k: Y such that k(gx, 9y) k(x,, y) for any (x, v) E Y x X and E G. Let μxx be G-invariant Radon measure on X, and define K. Cs (X) - (P(G) (p E (1,23) such that Vʃ E LP():
Ck: Cvf(v) = [ (w))ka,,))uxx)
Vy EY
CR is G-equivariant: L. o C2 = Cn o Cn. Vg E G (A.2). Since X = Y = G are homogeneous
spaces of G we can easily define a bi-invariant kernel by projection k(x, y) = k(g)'z) (k : G + I) for any (x, y) E Y x X, where 9230 for some fixed you The kernel is bi-invariant:
k(her, byy = k((hg,) -1/) ko, h har) = (0, x) k(x,,), v), E
(4)
For the case Y = and 9y = y (yo = e, the identity of G) this corresponds to a cross-correlation. For a convolution operator, we would analogously define k(x, y) = k( where x = Ja for To E X. In this case the essential component needed for equivariance of the operator Cs is the G- invariant measure d which is the Haar measure when X = Y = G. When X is a homogeneous space of G. but not necessarily G itself, we have to work with an operator which takes in a signal in (X( and produces a signal (G' on the group. This encompasses the case of the lifting layers, which are commonly employed when working with the regular representation
group (Cohen) & WWllingg 20166; Kondor & Trivedi 2018). The kernel k(-) in this case can be derived through an equivariance constraint as in [Bekkers (2019); Cohen et al. (2019). can also be shown (A.2) that an equivariant lifting cross-correlation can be defined as an operator Cº such that for any J E IP((X):
O% ff cffig /
Vg E G
(5)
where 8 G + R records the change of variables by the action of G (see A.2
Group cross-
correlation C = C₁ and convolution C% operators will be defined for any f E (G))):
OffigH [ (1))(o-'))daa E
f Caf,
(6)
CffigH [( ((0)) (99))(())))dc(()) E
: Cif
Lie algebra parametrization The tangent space at the identity of Lie group G is denoted by g and called the Lie algebra of G. A Lie algebra is vector space equipped with a bilinear map [- : g x g - g called the Lie bracket. To construct an equivariant layer using the Lie algebra of the group, one defines the kernels k(-) in (6) or (7) as functions which take in Lie algebra elements. This requires a map & + G which is (at least locally) diffeomorphism, with an inverse that can be easily calculated, preferably in closed-form. This allows us rewrite the kernel k : G R as:
k(g g) k(E(E-1(g (g-"))) Ro(E (g "g)
(8)
R with learnable parameters f. Using the inverse map - (g- g), he maps the Lie algebra coordinates of the 'offset' group element g 9 (for cross-oorrlltioons) to real values corresponding to the evaluation k(g -')). Our kernels are now maps hg of- G - IR, requiring the implementation of f----- (-) and a particular choice for the Lie algebra kernel he. This deseription encompusses recent proposals for Lie group cquivariant layers. In Bekkers (2019) the kernels are implemented by modelling Ko via B-splines, while Finzi et al. (2020) choose to parametrize he as small MLPs. Once hi and & are chosen, we can approximate e.g. the cross-correlation using Monte Carlo integration:
ko(+) is effectively an approximation of k(-) of the form Ko o
/ f()) (oT1))) ~ GG(
ko ('))()))))),, g1 PG
(9)
N
3

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where (G( denotes the volume of the integration space G and 9; MG indicates that is sampled (uniformly) with respect to the Haar measure. This allows one to obtain equivariance (in expectation) with respect to G. For compact groups, Me can be normalized such that (G 1. To summarize, we record the components of the framework which are needed for D to realize an equivariant operator. Namely,
require (1) a parametrization map <-1 G g. as well as (2) the implementation of an efficient sampling scheme with respect to the Haar measure μc such that numerical integration is feasible in practice.
4 LIE GROUP DECOMPOSITIONS FOR CONTINUOUS EQUIVARIANCE
Limitations of the group exponential For every Lie group we can define the Lie group exponen-
tial map expm : g G, which is a diffcomorphism locally around 0 g. Since we are interested in GL,,, RR and its subgroups, we can make things more concrete as follows. Man (R) = M, (R) (the vector space of
real matrices) is the Lie algebra of GL, R) (Sec. B. The notation gl(n, IR) = (R, is used for this identification. For G = (n. IR) with g gl(n, R), the group exponential is the matrix exponential expm _g((n, R) + GL with the power series expression X + ex = Lk=o क X The map & in (8) is most commonly implemented as the group exponential { == expm. Given a subgroup G S GL,,, RR) for which expm surjective, every ele- ment g E G' can be expressed as g = expm
e for X E g, and fast routines for calculating expm(-) are available. In this case, the inverse map { is given by the matrix logarithm, giving us:
( = logm(g -
logm G +
(10)
In general, we need to consider if both & and g- need to be implemented, and whether these maps are available in closed form. Assuming there exist X and Y such that ex = =g and ex = g. logm(g 9) = logm(e xxe A key optimization underlying this frame- work is enabled by employing the BCH formula (..I. which tells us that for abelian Lie groups X + Y. This simplifies calculations considerably and allows one to work primarily at the level of the Lie algebra, bypassing the need to calculate and sample the kernel inputs "ğ at the group level. Considering the affine Lie groups Aff G < Gn R). this simplification can be used for example for the abelian groups G S0(2) and G= R (2) x SO(2), consisting of rota- tions and scaling. |Bekkers] |cakeng.BBekkers (2019): Finzi et al. (2020) primarily work with these groups, and choose & and <-1 to be the matrix exponential and logarithm, respectively. If the group is non-abelian but the exponential remains surjective (such as with compact groups like SO(3)). expm(.) remains a generally valid choice for & as long as 5-1 can be accurately calculated in closed-form. For the non-abelian, non non-compact groups Sn, IR) or GL (n, R) the non- surjecccctt of the exponential map limits the applicability of the matrix logarithm outside of neighborhood around the identity (Prop. B.2). The class of equivariant networks that can be implemented with this framework is then firstly limited by the parametrization maps & and g-1, motivating the search for an alternative.
(10) can be rewritten
logm(eXX
Another key limitation is that for ( to realize an equivariant estimator when numerically approx- integral, sampling needs be realised with respect to the Haar measure of the group G. Techniques for sampling with respect to the Haar measure on the groups SO(n) or R x (n) x SO(n) are known, and generally reduce to working with uniform measures on Euclidean spaces or unit quaternions in the case of SO(3). We aim to address these limitations, allowing the previously described framework to be generalized to arbitrary Lie groups G < Gl, IR). We further seek a solution that places minimal limitations on the class of 'Lie algebra kernels' ki : g -> IR that can be used, and one should be able to employ any kg that uses the coordinates of tangent vectors in g expressed in some basis. In the following we present set of generally applicable tools while considering SL IR) and GL+(n. (n, R) as working examples, since these groups require more consideration and represent our primary application.
imating the convolution/cros--coreeeaa
4.1 LIE GROUP DECOMPOSITION THEORY
We exploit the fact that the groups GL + (n, R) and SL(n,, IR) have an underlying product structure that allows them to be decomposed into subgroups and submanifolds which are easier to work with individually. More precisely, G {GL* (n, R), SL R)) can be decomposed as product P x H, where H v is the maximal compact subgroup of and P S G is a submanifold which is diffeomorphic to R for some k > 0. and we have diffeomorphism y P H - G.
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Similar decompositions available for a larger class of groups G < GL, (Abbaspour & Moskowitz [007] Ch. 6). It can be shown that the map yp is chosen correctly the Haar measure JlG can be expressed as the pushforward measure ya ( where μ is a G- measure on P and par is the Haar measure on H. In some cases the group decomposition presents a corresponding Lie algehra decomposition, which we can leverage to build the parametrization map 5-1 - g-
Factorizing the Haar measure Let G be a locally compact group of interest (e.g. GL (n, R)), with (left) Haar measure μaG. Assume there exist set of subspaces or subgroups P S G, K S G. such that G PK, and a homeomorphism 4p : P x K + G. Further assume that pp and μuK are (left) G-invariant Radon measures the corresponding spaces. We look to express (up to multiplicative coefficients) the Haar measure μa as the pushforward of the product measure μP under the map sp. This allows for the following change of variables for any (G):
[1o)dko(()) =
-f SEEK F(p(p, k)) @ pk) = S f((((, (k)dp(()) (11)
In the context of Monte Carlo simulation this will enable us to produce random samples distributed according the measure Jl by sampling on the independent factor spaces P and K and constructing a sample on P x K and respectively on G using the map y. The space P will either be another closed subgroup. or a measurable subset P C G that is homeomorphic to the quotient space G/K. In particular, if P is not a subgroup, we will focus on the case where P' is homogeneous space of G with stabilizer K such that P = G/K. K. When the left and right Haar measure of a group coincide, the group is called unimodular. The groups GL (n, RR, (n,,, I) are unimodular, however this is not true for all affine groups Aff For groups which are of an issue in practice. However, GL. (n. IR) is not polume-preserving and we also desire that our framework be general enough to deal with the non non-unimodular case as well. G is not unimodular and μG is its left Haar measure, there exists continuous group homomorphism Ag - 18 called the modular function of G. which records the degree to which H4G fails to be right-invariant. We now have the tools necessary to record two possible integral decomposition methods.
this is not as much
Theorem 4.1. (1) Let G be a locally compact group, H < G a closed subgroup, with left Haar measures μG and pa respectively. There is a G-invariant Radon measure PGjH on G/H if and only if AGl = An. The measure PG/H is unique up to a scalar factor and if suitably normalized:
1. [(o)duo(s) = -Sai I (b)) Af E
(12)
(2) Let P S G, K SC G closed subgroups such that G = PK. Assume that Pn K is compact, and Zo denotes the stabilizer of the transitive left action of P x K on G given by (p, k) g pgk- for any (p.k) E P x K and g E G. Let G. P and K be o-compact (which holds for matrix Lie groups), PG, pP and Pk left Haar measures the modular function of G restricted to K. Then μa is given by PC = n. (up @ A-1 where
G, P, and K respectively and A A is
T P K - (P x K) is the canonical projection. In integral form we have:
1. f()d)a(())
Jp f(pk)
Da (k)
duk (k)du()))))) (G)
(13)
Ax x
Proof. Folland 2016 Theorem 2.51) and Wijsman 1990 Proposition 7.6.1).
The existence and range of the convolution operators (for arbitrary Lie groups G) are described in Sec. [.2.I] with the non-unimodular case being covered by Prop. A..] When going to the Lie group setting, we can already deal with semi-direct products of groups of the form x G. The modular function on N x G is ANxa(n,9 g) = A (n) Aagg)(())- (Kaniuth & Taylor 2013). The term 5 5: G - R records the effect of the action of G on N, and it coincides with the term 5(.) used in the lifting layer definition (27). Concretely, take the affine groups Aff(G) R" x G, G < Gn, R), defined under the semi-direct product structure:
Aft R G ((x, A) R", A GEG]
(14)
G acts on R.T by matrix multiplication and for (zr, A), (y, B) E Aff(G), the product and inverse are: (x, A)(y, B)= (x + Ay, AB), (z, -
(15)
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Elements of IR are concretely represented as column vectors. Viewing (R", +) the additive group, we have 8: G - R given by 8(A) = [det(A) for any A E G. Applying Thm. 4.1
gives:
SANOO) f()) ( = [ L [((x, A)) dxd
AA E CC((f(G)))
(16)
JR.
|dct(A)
Expressing the cross-correlation Ca of (7) in this product space we have for / E LA(()G)))
C&f: (z, A) [/ / f(ī, A)k((((, ' ( " JR"
(17)
For the affine groups Aff(G) = R" X G a parametrization map EAT(G) IK" $ g - Aff(G) will simply be the identity on the first factor, since the Lie algebra of Aff(GL(nn, IR)) decomposes as I" @ gl(n, IR) when represented
Lie subalgebra of gl(n 1, 1,). We then left with the parametrization and invariant integration of the G-factor of Aff We provide a solution for the cases G = SL(n, R) and G = GL + R), while remarking that a solution for Gn,, R) can be immediately extended to (n,,, IR) (Sec. B.4). Our approach is based on generalized Polar decomposition of matrices, which is applicable in the case of reductive (GL + (n. R)) semi-simple (SI,(1, R)) Lie groups. An alternative decomposition is discussed in Sec. B.6.2
Manifold splitting via Cartan/Polar decomposition Let Sym(n, I) be the vector space of of symmetric positive definite (SPD)
real symmetric matrices and (os I) the subset of Sym(n, R) matrices. Denote by SPos(n, R) the subset of Pos(n, IR) consisting of SPD matrices with unit deter- minant, and by Sym (n, R) the subspace of Sym(n, R) of traceless real symmetric matrices. Any matrix A Gn, R) can be uniquely decomposed via the left polar decomposition A PR
where P E Pos(n, I) and R E O(n) B. B.4b. The factors of this decomposition are uniquely deter- mined and we have a bijection GL [1, lR] Pos(n, R) x O(n) given by:
(18) For the reader unfamiliar with Lie group structure theory, the following results can simply be under- stood in terms of matrix factorizations commonly used in numerical linear algebra. The polar de- composition splits the manifold GL (n, R) into the product Pos(n, R) x SO(n), and SL(n, IR) into (Pos(n, IR) x SO()). We use the notation G - M x H to cover both cases. This decomposition can be generalized, as the spaces Pos(n, R) = GL + (n, R))SO(n) and SPos(n, R) = SL.n, R))((()) are actually symmettrcc spaces, and Cartan decomposition is available in this case (B.4). The Car- tan decomposition tells us how to decompose not only at the level of the Lie group, but also at the level of the Lie algebra. In fact, using this decomposition we can also obtain factorization of the measure on these groups. Let (G/H, H, M, m) define our 'Lie group data', corresponding to (GL+ (n, RR)SO(()), (n, R), (ymm(, or (SL(n, R)),())))))) SPos(n, R,,, Sym, (R))). Theorem 4.2. Let (G/H, M, m) be as above, and denote by g. the Lie algebras of G and H.
A (VAAT, VAAT A); A GL, R)
1. The matrix exponential and logarithm are diffeomorphisms between m and M, respectively. For any P E M and a E R. the power map P + p is smooth and can be expressed as: pa expm(alogm(P) P)), vP E Pos(n, R))
(19)
2. G and G = m x H. We have group-level diffeomorphisms: M
X: X:M x H + G, x(P, H PR
(00)
4
+ (tm
(21)
3. The above maps can be inverted in closed-form:
x x : G
M H, x A AAT VAAT A)
(22)
1 logm( (AA"))
:
Φ- A logm(AA")) expm(
(23)
2
A proof and further references for Theorem 4.2 can be found in Sec. B.5 At the level of the Lie
algebra, have the decomposition gl(n, RR so(n) @ Sym(n, R). The Lie algebra of SL R) is
si(n, R) = {X E gi(n, R) tr(X)= = 0}. It decomposes similarly sl(n,, IR) so(n) @ Sym, (n, IR). The Cartan decomposition of g is therefore expressed as g = h @ m where h so(n) with
Sym(n, R) if G GL+ (n, RR) and m (nmo(n, IR) if G = SL, R).

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4.2 A PARAMETRIZATION BASED ON THE CARTAN DECOMPOSITION
Consider again the notation H, M, m) as in Theorem 4.2 (G = (n,
SL(n, R)).
Concrete integral decompositions From Theorem 4.2 and the fact that symmetric matrices have a unique square root, we actually have equivalent decompositions for A E G as A = -PR or A = =51/2 R for S, P E M, R E H and P = S For GL(n, R). the decomposition A = 5 5 R. has a factorization of the Haar measure of GL(n, R) as product of invariant measures on (os RR) (shortened Pos(n) and O(n). Let (Poa(n)) denote the GL, R) invariant measure on (os(n).
Theorem 4.3. Denote G = GL(n, R), H = O(),, and let μG be the Haar measure on G and Hn the Haar measure H normalized by Vol(H) 1. For A E G. under the decomposition A S1/2R, S E Pos(n). R E H, the measure on G splits as dua (A) B (addprosm) (S) (R), where Bn = Vol(O(w)) is normalizing constant. Restricting to G = GL (n,, IR and H = SO(n)
2
and ignoring constants, we have:
f f [(A)dno(A)- (pon(n) J -
RRdp: (S), Vf Co(G)
(24)
(A) = |det(A)]="dA, with dA the Lebesgue measure on R=² We now describe how to sample on the individual factors to obtain GL(n, IR) samples.
The Haar measure of GL,n, R) is dH
Theorem 4.4. Ifa random matrix A E Gn,,, IR) has a left-O(n)) invariant density function relative IAAT I then ( (AAT)1/2 = 51/22 and R = (AAT)-11/2 A are independent random matrices and R has a uniform probability distribution on O(n). The uniform distribution on O(n) will be the normalized Haar measure po(n). Conversely, if S E Pos(n) has density function f Pos(n) + R2o relative (Ppo(n) and R O(n) is uniformly distributed with respect the Haar measure Mo(a), then A == S²/2 has a density function 2,2 ((AA"))det((
relative to dA.
Theorems and are known results that appear in the random matrix theory literature, but have not seen recent application in the context of deep learning. In (B.6) we provide more de- tails and references. Using the decomposition A 51/2 R invariant integration problems on G can be transferred to the product space M x H, and we can express up to normalization the in- variant measure Nc as (-(M @ pr)). To construct samples (A,,..,,A., A..) ~ PG one produces samples (RI,.... R (M...... Mp. ~ Pas. Then #c-distributed random values are obtained by {A {s((M, R.),..., (MM, ). where again & M x H + G is given by p: (S, Rt) + S1/2
JA where fr will be the uniform distribution on H, and samples
=
Mapping to the Lie algebra and back Any A E G can be expressed uniquely as A = e xR for X E m and R E H. Since H = SO(n) in both cases, the fact that expm s0(m) + SO(n) is surjective, allows using --1 22). Then by taking the principal branch of the matrix logarithm on H = =OO(m), Y
to write it A ex ex, Y E sa(n). The factors X and R e² are obtained
logm( R). A map g- G g as described in
g and (9 is constructed g-1 = (idm x logm)
Φ-1 More precisely, for any A = exee E G, using
we obtain the tangent vectors (Y,X) X) so(n) x m and since g = so $ we have unique Z X + E g. Details are given in BB.7
Define Kg = kg a c- G + R as our Lie algebra kernel. A Monte Carlo approximation of cross-correlation operator C (G) + L²()) as in 99 will be of the form:
1 Σ Ro. 9), d ~ pG. Ag E
Ckf gg 9 + N
(25)
For affine groups, every element (a,, A) of R x G. can be uniquely decomposed as (z, (0, with / the n x n identity matrix. One can use the fact that ((x.A)) = ((x,1) write:
(x, AE, A)) (( A) L Lo,,), Á)] L.
A)]
(26)
An efficient implementation of convolutional layer can be realised in practice for n E {2, 3} by first obtaining the transformed kernel ((AA 2 A Á) and then applying the translation C, using an efficient convolution routine, done for example in Cohen & Welling (2016): Bekkers 2019).
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In practice, the exact discretization of the translation factor E" will depend the support of the input data. For example, if our input signals are defined compactly on a grid (e.g. 2D images), we can approximate continuous convolution (Finzi et al., 2020) by sampling the translation factor in uniform grid of coordinates ā ~ [-1, 1)"h C Ik" as the parametrization (AK(,) R gl(n, IR) + G is the identity map for the first factor. We then approximate lifting cross-correlation layer by:
/ (0)
ch A)
(27)
R"
I, ~ 1--1,11"
(28)
For the non-lifting layers, starting from
17 denoting dÁ = (Ac(A) and applying 26 we have:
[Ckf](z, A) = / f(t A) A A) )däd
(29)
Using Theorem 4.3 denote the invariant measures JM and μu by d$ and dR, we obtain:
[Cxf]<z, Bn
dSaRdz
(30)
H
The kernel in (5 is now of the form Ko R² x G - R. giving us:
[Z, A)
Ma, R [oo (. (L))((55)
(31)
where o; ~ 11 S, R3 (pa & &HH) and R. sampled uniformly with respect μи. V records both the volume of the integration space from the MC approximation well as the constant 8n.
5 EXPERIMENTS
For all experiments correlations that are equivariant (in expectation) with respect to the groups R2 x GL x (2, (2,1) IR and R² S1, R). Details regarding the network architecture and training are given in Appendix D
ResNet-style architecture, replacing convolutional layers with cross-
Aline-transformation invariance We evaluate our model on a benchmark affine-invariant image classification task employing the afNIST datase
2 The main works compare with the affine
equivariant model of Macaond et al. (2022) and the Capsule Networks Ribeiro et al. (2020a b) which are of the art for this task. The experimental setup involves training on the standard set of 50000 non-transformed MNIST images (padded to 40 x 40), and evaluating on the afNNIST test set, which consists of 320000 transfreandf MNIST images. The model never sees the transformed a images during training, and we do not use any data augmentation techniques. In this case, robustness with respect to the larger groups of the affine family of transformations needed. For fair comparison we roughly equalize the number of parameters with the referenced models.
Table I: afNNIST classification accuracy, after training on MNIST.
Model
afNNSST Acc.
MNIST Acc.
Parameters
MC. Samples
R² SL(2, R)
98.2(+0.1)
99.55(+0.1)
370K
10
VB CapsNet [Ribeiro al.
2020b]
98.1
99.7
175K
RU CapsNet RRbciro et al.
2020a
97.69
99.72
>> 580K
RR2 GL (2, R)
97.4(+0..2)
99.5(+0.1)
395K
affConv [MacDonald et al.
2022
95.08
98.7
374K
100
aftine CapsNet Gu & Tresp 2020
93.21
99.23
Equivariant CapsNet (Lenssen
2018
98.42
235K
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Table I reports the average test performance of our model the final epoch, over five training with different initialisations. We observe that our equivariant models are robust and generalize well, with the R² SL(2, IR model outperforming all previous equivariant models and Capsule Networks. Note that, compared to MacDonnad et al. (2022) our sampling scheme requires 10 times less sam- ples to realize accurate Monte Carlo approximation of the convolution. The TR2 x GL. IR) model performs slightly worse than the volume-preservee affine group IR² x SL(2, IR). This can be explained by considering that the afINIST dataset contains only a small degree of scaling.
Homography transformations We further evaluate and report in Table 2 the performance of the same model evaluate the homNIST dataset of MacHonald et al. (2022). The setup is identical to the afINIST case, with the images now being transformed by random homographies. We observe a similar degree of robustness in this case, again outperforming previous methods applied to this task.
Table 2: homNIST classification.
Model
homNIST Acc.
MC. Samples
R² S2 R)
98.3(10..1)
10
B² x GL. (2,
97.71(+0.1)
10
affConv (MacDonald al. 2022)
95.71
100
As our models are only equivariant in expectation, we analyze numerically in Sec. the degree to which the equivariance error is dependent the number of Monte Carlo samples used approxi- mate the convolationlcros-ccoeelaaoo integral.
CONCLUSION
We have built a framework for constructing equivariant networks when working with matrix Lie groups that are not necessarily compact or abelian. Using the structure theory of semisim- ple/reductive Lie groups we have shown one possible avenue for constructing invariant/euuivariant (convolutional) layers primarily relying tools which allow to decompose larger groups into smaller ones. In our preliminary experiments, the robustness and -ittt---dittiittionn capabilities of the equivariant models were shown to outperform previous proposals tasks where the symmetry group of relevance is one of GL+ (nz R) or SL(n, R).
Our contribution is largely theoretical, providing framework by which equivariance/Iivariacce to complex symmetry groups can be obtained. Further experiments will look to validate the applica- bility of our method to other data modalities, such as point clouds molecules, in Finzi et al. 2020).
While we have primarily focused on convolution operators, we remark that the tools explored here are immediately applicable to closely-related machine learning models which employ Lie groups and their regular representation for invariance/equiiaarince For example, the "LieTransformer" architecture proposed in Hutchinson et al. (2021) opts to replace convolutional layers with self- attention layers, while still using the Lie algebra of the group as mechanism for incorporating positional information. They face the same challenge in that their parametrization is dependent on mapping elements back and forth between a chosen Lie group and its Lie algebra, and they require mechanism for sampling on the desired group. The methods presented here directly applicable in this case.
Future work will explore expanding the class of Lie groups employed by such models using the tools presented here. Another potential avenue to explore is the applicability of the presented tools to the problem of 'partial' and 'learned invariance/equiivrianne (Benton et al.; 2020). The sampling mechanism of the product decomposition allows one to specify a probability distribution for the non- orthogonal factor, which could be learned from data.

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ACKNOWLEDGMENTS
The presentation of this paper at the conference was financially supported by the Amsterdam ELLIS Unit and Qualcomm.
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Appendix
Table of Contents
Additional background
15
A.1 Topological groups and the Haar measure
15
A.2 Equivariant convolutional operators
16
A.3
Related work
18
Lie group decompositions
Lie groups
B.2
Primer on Riemannian Geometry
................................
21
B.3
Group actions Homogencous spaces
................................
23
B,4
The Cartan/Polar decomposition
28
B
5
Proof of theorem|4.2
30
B
6
Integral factorizations for the Cartan/ Polar decomposition
31
B.7
More details on the Lie algehra parametrization
33
Architecture & training details
34
Equivariance error analysis
34
A ADDITIONAL BACKGROUND
A symmetry group refers to set of transformations which preserve some underlying structure present in the data. Formally, a group G set together with an associative binary operation G x G G which tells us how group elements be composed to form another. Every element E G has an inverse g- E G. and the group has an identity element e E G such that gg e.
A.1 TOPOLOGICAL GROUPS AND THE HAAR MEASURE
Hausdorff topological toosoorfttopollggcall space M is locally compact if each point m M has a compact neigh- borhood. A topological group G' is group as well as a Hausdorff topological space such that the group operation (g. h) + gh and inversion map + g are continuous. For the following and more details on Radon measures see Folland (1999): Lang (2012).
Radon measures Let (X, Bx, 11) be a measure space with Hausdorff topological space X, Bx the o-algebra of Borel sets and μ : Bx > J0, oo] any measure on Bx (referred to as Borel measure). The measure y is called locally finite if every point æ E X has an open neighborhood U for which p(U) < co. Borel set X (E E Bx) is called inner regular if:
p(E)= sup( K E, K compact)
(32)
Respectively, a Borel set E X is called outer regular if:
M [E)== inf (v(V) V E, V open]
(33)
The measure j is called inner (outer) regular if all Borel sets are inner (outer) regular, It is called regular if it is both inner and outer regular. μ Radon measure if is locally finite, inner regular on open sets and outer regular. is o-finite if there exists countable family of Borel sets (Bn (nn,, where μu( B.,) < oo, Vn E and UnEN B, = X. If μ a Borel measure on a Hausdorff topological space X, local finiteness will imply that μ is finite on compact subsets of X.
AA topological space where any distinct points have disjoint neighbourhoods is called Hausdorff.
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Theorem A.1. Every locally compact group G has a left (right) nonzero Radon measure Pa: such that (G(g = (c(B) (respectively Pa bc(B) = Pc(B)) for any Borel subset C and any g G. The measure PG is called the left (right) Haar measure of G and if vG is another Haar measure on G, then PaG = c. vG for some c E I
When integrating with respect to the left Haar measure Pa we have for any f E Cc(G):
I [(xr)duG = [ [(x))ua, Vy EG
(34)
All of the groups and topological spaces in the main text will be o-compact. A locally compact space X is o-compact (or countable at infinity) if a countable union of compact subsets. We will use the notation fa to refer the left Haar measure and when needed the notation Jaz. fzz()) (+) and
(n(+) will be used to differentiate left and right Haar measures.
Remark A.2. If X is a homogeneous space of G (but not G itself), then a G-invariant Radon mea- sure dux on X (if it exists) respects the same invariance property presented in Thm. A.7 and (3 and we simply refer to dux as a G-invariant measure. For review of such measures, see (Folland 2016 Chapter 2.6).
Function spaces Suppose (X, Bx,,xx1 (4) is a measure space where is locally compact Hausdorff space and μ Radon measure X. 1 (X, ['(X) for <= p denotes the space of equivalence classes of functions {ʃ X R Borel measurable and fxlfpd oo) that agree y-almost everywhere. Equipped with the norm |l/ = (xIS(Pd)) LP(X) Banach space. C(X) == C(X,R) denotes the space of continuous real-valued functions on X. The support of f E C(X) is defined as supp(f) = fo E X f(x) * 07. We state that a function f has compact support whenever supp(f) is compact. (x( denotes the subspace of C(X) of continuous functions with compact support.
A.2 EQUIVARIANT CONVOLUTIONAL OPERATORS
In this section show that the convolution/croos-correee operators 6 7 as well as the lifting cross-correlation (5) are equivariant. We then clarify the existence and range of these operators. For non-lifting potpuutiioncrss-ccorelaaaae operator, recall that X Y G and k Y x X + R is a bi-invariant kernel:
k(gr, gu) k(z, v), V G
(35)
J4 is a G-invariant Radon measure on X. The operator Cs was defined for any f E "x (X):
C f - / (()) Ay
(36)
Cn f Ck maps element of L' L'(X to L (G), respecting C, o Cs Cg for any g E G:
(,((C))))) Cul(g-'v) / f(x)k(z, yddxx()
(37)
x
(Haar invariance) = / f(0 t)ddxx((
(38)
x
(Kernel bi-invariance) / fgg (ox
(39)
lifting cross-correlation operator Cf was defined for f E (+(X))b:
cf ctt:g [ (z)) ddxx(=) Vg ege G
(40)
*See Proposition 2.9 and Theorems 2.10 & 2.20 of Folland 2016
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where X is a homogeneous G-space with Radon measure duu and ó G + R2 records the change of variables produced by the action of G:
S [(x)dux(x)= = /s ((ox))(g)dxx(1)
(41)
In the case where X RRn and Y Aff for Aff(G) the semi-direct product Aff(G) = IR" x G. G' S GL(nn RR), we have Aff(G) acting transitively on X, and we can make the identification X Aff(G)/G Each element g E Gf) can be represented as g = (2.h) with x E R", h E G. In this case, we have 6(g) = (ee(g)) [de(()). Note that the form of the kernel can equivalently be derived through an equivariance constraint relation as in [Bekkers] (2019) Theorem 1). The lifting layer is then more concretely of the form S = ʃ * k where:
(f k)) { f()) (1)
dr
(42)
The lifting layer is equivariant (f,lf * k] = |Cg * k), since for any 9 = (ō, E Aff(G) we have:
Cglf * (g) (f (g) 9) {
dr
3 f(z))(g (allg-''aa -gxx) (h-
(43)
dr
(x)
5 ((g'ko))(g" "o)k(g-a))dees (x))lett(h-)) |det(h (h) 'h)l
(44)
J Cal(T) 1x)
dr
[de)(h)] ([Cg.f] [] * (g)
(45)
A similar derivation appears in Mac et al. 2022 Theorem 4.2). The lifting layer equivari-
ance here was derived for the case where X the homogeneous space R" Aff(G) however a similar process is available for more general homogencous spaces. One only has to identify the appropriate relatively invariant measure 5(-) (1-))--p4. dux which appears in (5). For more details on rela-
tively invariant measures see Hewitt & Ross (012).
A.2.1 EXISTENCE AND RANGE OF CONVOLUTION OPERATORS
If G is a locally compact Hausdorff group. C.(G) is dense in L'( for 1 x therefore approximate functions in LP(G) using functions in (G). While some results hold in a more general setting, we assume that all topological groups are (locally) compact Hausdorff and second countable (and therefore o-compact), as the Lie groups of interest satisfy these properties.
We can
Proposition A.3. 6 We record the following results concerning the existence and range of the con- volution operators for locally compact group G.
1. If f E (G) and for any k E IP(G) (1 <= <= co) then ( converges absolutely, f * k E LP(G) and If * kllp I ||
2. If G is not unimodular, f E LP(G) and k E (') n (G) then f f*ke k E LP(G).
3. If f,k E L2) then f * E Co(G), see also (Hewitt & Ross 2012 Theorem 20.16).
For the correscorreltionn operator, if we define the involution f" f)) we can write f * k as f * k- and reuse the results of Proposition A.3 In the case of nonunimodular semi-direct product group G N x H. where Ii is unimodular, the Haar measure is of the form Pa = Do ux 0 " Since AG : + (0, 5o) is unbounded, we therefore always make the assumption that the support of fl is compact set where A is bounded.
Assumption. We assume f.k E (G) n L²) and additionally k E C.G).. When working with image data, one can also take f C.(G) directly. Going forward we therefore establish for p E {1,2}, of : LX + LP(G) defined by (5) and C or Op are similarly operators LP(G) + (G) given by 66 and 77.
Folland 1999
Proposition 7.9).
Folland 2016
Propositions 2.40 & 2.41).
ooreepoons to Thm. 4.1 in integral form. A more precise reference
Wijsman 1990 Comollary 7.6.3).
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The restriction to compact subset of G can also be motivated if we wish to employ Monte Carlo approximation of the integral using a uniform distribution, since the Haar measure is finite on compact subsets. However, the restriction will not tied to the injectivity/surjecciitty radius of the group exponential map, as we will use a different parametrization as shown the main text.
A.3 RELATED WORK
The theory behind constructing equivariant convolutional layers when our input space is homo- geneous space some locally compact topologiaal group is covered in Cohen & Welling (2016); Kondor & Trivedi 2018); : Cohen et al. (2019); Bekkers (2019). A differential geometric formulation
not necessarily limited to homogeneous spaces given in Weiler et al. 2021 2023 and a review focused on the application of induced representations in the context of neural networks can be found
in Kondor& & Trivedi (2018); Cohen et al. 2019)..
Employing Monte Carlo integration to approximate convolution integrals has also been proposed
e.g. in Hermmsslla et al. 2018); Finzi et al. (2020); Romero et al. (2022); Knigge et al. (2022).
Hutchhinnon al. [2021 the Lie algebra parametrization employed with convolution operators
being replaced with self-attention layers. Steerable CNNs make use of group representation theory to
parametrise convolution kernels by solving a kernel steerability Consee & Welling (201
Weiler & Cesa (2019); Lang & Weiler 22211; ; Cesa et al. (2022).
In the context of dimen--imensional group representation Finzi et al. (2021) present general solution for constructing equivariant MLPs. They present a framework for sol solving the equivariance constraint by making use the generators of the Lie algebra of group. The resulting linear system is solved for finite dimensional representations by using the singular value decomposition. More general solutions are developed in [ogaasskyy et al. (2020); Batatia et al. (2023) for wider class of Lie groups and representations.
MacDonald al. (2022) The limitations of previous Lie algebra methods (as reviewed Sec. Dp also discussed in MacDonald et al. (2022), which proposes a possible solution while still work- ing with the group exponential. To overcome its lack of surjectivity and be able to sample with respect the Haar measure of Aff(GL R))= R x GL,nn, R), the domain of integration itself is restricted and the convolution integral reformulated, ensuring that the group elements ğ 9 (using the convolution operator notation) within the injectivity radius of the exponential map. This is done by first changing the convolution operator after the lifting layer to work with the right Haar measure
(1 (f+ - 100
= (kG
(46)
For non-lifting layers instead of the kernel k(.), the feature map f(-) is now evaluated at fgg
Proposition A.4. F Let G be Lie group with Lie algebra g and suppose U S g is neighborhood of E and expm(U) U) neighborhood in e E G such that the group exponential expm :g -> G is diffeomorphism of onto expm(U). For f CG) with support in expm(U) we have:
[ ((9)dpoll) - f f(expm(X)) det -a-adx
)dX
(47)
ad_
where det (~-e (1-e- - is the Jacobian determinant of the differential of expm and dX is the ad Lebesgue measure on g.
The change of variables of Proposition A.4 and the expression (46) allow MacDonald et al. (2022)
define the non-lifting convolutional layers to be of the form:
- f(9ø 1k( -
/ f(ge )ko det 1 e ad
* k))
)XX
(48)
ad
*See MacDonald et al. 2022 Theorem 4.3) or (Folland 2016 (2.32) (2.36)). Here, we used the convo- lution operator to he consistent with the derivation of Mac Donald et al. (2022).
(Helgason
1984 Thenrem 1.14), (MacDonald et al. 2022 Theorem 4.4).
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kg is again learnable map that takes in Lie algebra elements approximating A expp gg R. The difference here lies in the fact that one does not need to use the inverse map < to map back to g. By treating det (-=--->>>)
(proportional to) density function, sampling is realised directly on the Lie algebra using standard MCMC methods. One starts the sampling process from the outer-most convolution integral and rejects samples that lie outside of the support of the exponential map.
x
While the approach can be applied to any matrix Lie group, in practice because every possible --'g element must be precalculated and kept in memory before the forward pass, the scalability of the method is greatly limited due to its memory requirements. Further numerical errors are introduced due to the fact that det 1-e
is approximated with limited precision by the power
Ek= (K+1J] (adx)*. Additionally, while discretization of the integral is always necessary, this approach is limited in that the domain of integration must still be restricted the injectivity/surjectiitty radius of the group exponential for the change of variables to apply. In this paper, we also employ a change of variables in the context of invariant integration with respect to the Haar measure. However, rather than working with the tangent space of the group as in Prop. AA.4] we make use of group-level decompositions into independent factor spaces, and show that the Haar measure decomposes a product of invariant measures, allowing us to construct a sampling scheme on the lower-dimensional subcomponents, as explained in Sections 4.1 and B.6
series expression xd_x
(-1)
B LIE GROUP DECOMPOSITIONS
B.1 LIE GROUPS
A Lie group G is a group as well as a smooth manifold, such that both the group operation and the inversion map are smooth. Lie groups are therefore second countable Hausdorff topological spaces. An abelian Lie group G is a Lie group and an abelian group, i.c. group for which the order of the group operation does not matter gh = hg. Vg, h E G. Mnn (RR) = M,, (TR) denotes the vector space of n n matrices. It is canonically isomorphic to R=² which is locally compact.
Closed or open subsets of M, (R) will be locally compact with respect to the induced topology TO One such open subset is GL, IR), the Lie group of invertible matrices:
GL R) {X (R) det(X)
+ 0}
(49)
The notation H S G (H <, G) used to indicate that H is a (proper) subgroup of G, rather than just a (proper) subset H C G (H C G). We are only interested in closed Lie subgroups of GL R).
A (closed) Lie subgroup II H of a Lie group G will refer of G (with the induced topology). A linear or matrix Lie group is defined to be Lie subgroup of (n, IR), and will therefore be locally compact and second countable. GL(n,, R). the translation group (R" +) and the family of affine groups RD x H. H S GL, RR) are our primary interest, with H being one of the groups:
closed subgroup and submanifold
GL. R)- = {X E GL.nn det(X) > 0). the identity component of Gl, R); It it also referred to as the positive general linear group;
R) {X (L,,,R) R) det(X)= = 1}, the special linear group;
. 0(n) = (X E GL(n, X = I,), the orthogonal group:
(O(m) = {X E (()) det(X)= = 1}. the special orthogonal group.
G is smooth with d(cxpm)) id, making expm a diffcomorphism expm|u : + V of some neighborhood U of E g onto a neighborhood V of G.
Proposition B.1. The group exponential map expm
The notation expm : :g + for the Lie group exponential is used to differentiate it from the Riemannian exponential, which is mentioned later Unless the group G can be equipped with
10
Lec 2010
Proposition 4.66).
An abstract subgroup of G is submanifold iff H is closed with the induced topology (Gallier & Theorem 19.18) 18)).
Quaintance 2020
2013
Proposition 20.8).
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bi-invariant Riemannian metric, the exponentials do not coincide (see Proposition B.10 M,, (I) equipped with the matrix commutator [X, = XY -Y--XX YX for X, Y E M, (R) is a Lie algebra. and more precisely it is (canonically isomorphic to) the Lie algebra of GL(n, R We use the notation gl(n, IR) = (RR) when working with this identification. For G = GL with g gl(n, IR), the group exponential is given by the matrix exponential:
expm gl(n, R) + GL(n, XHE
X
(50)
can define the inverse of the group exponential (expmlu) ) V - U which is a diffcomorphism of V onto U. For matrix Lie groups this map is the matrix logarithm which we denote by logm(-). Its power series expression is:
From Prop. B.1
1 1I- AE GL(n,
logm( A) =
(51)
1=1 k
The existence of the inverse of the matrix exponential is characterized as follows.
Proposition B.2. 14 Let B(I, {X M,, ||X - I|| 1} where || || is a norm on
(RR (e.g. Frobenius norm) and the identity matrix. Note that B(1,1) C GL(n, R). Then
for every 9 E B(I,) we have expm(logm(g)) g and for every X logg)) we have logm(expm(XX) =.
Recall from Section that if the parametrization map ç: g - G is chosen to be the group exponen- tial { = expm, then g-1 is given by the matrix logarithm:
5) = logm(g-'
logm G +
(52)
For the following, see Hall 2015 Chapter 5). Assuming there exist X and Y such that ex و
and e² = g. 10 can be rewritten as logm (ex e e²). The Salpee--CCaeellaaatttt
(BCH) formula
states that there exists a sufficiently small open subset E U C g so that exex e and one has:
logm(e x x = x + Y
| [x,
X
[X,
(53)
12
12
For abelian Lie groups this reduces to:
logm( x =
(54)
For V a finite-dimensional vector space over I. we denote by GL(V) the group of invertible linear maps of V and by gi(V, IR) = End(V) ( the space of linear maps V - V. GL(V) admits a Lie group structure as is isomorphic to GL(n, R) once basis is chosen. The space gi(V R) can be made into a Lie algebra under the commutator bracket and it is isomorphic to M,, (IR) gl(n, R).
Definition B.3. Let V be a vector space. A ffiite-dimensional real) representation of a Lie group G is Lie group homomorphism p: G + GV(.). For g a (real) Lie algebra, representation of g is a Lie algebra homomorphism Φ g + gl(V,R).
The conjugation (inner automorphism) map C G + G is defined such that C, Lg o Rg-
C G - G, C h + ghg
(55)
The adjoint representation G, is given by the homomorphism!
15
Ad : G + GL( + Ad,
(56)
where Ad, : g + g. Ad, = d(C,). (dI.g) o
The differential of Ad is used to
define the adjoint representation of g. denoted by ad:
ad : + gl(g), adx = (x)
(57)
Lee
2013
Example 8.36 & Proposition 8.41).
Faraut
2008
Theorem 2.2.1)
T.ce
2013
Proposition 20.24).
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B.2 PRIMER ON RIEMANNIAN GEOMETRY
For more details on smooth and Riemannian manifolds see Lee 2013
Riemannian manifolds A Riemannian metric (tensor) g smooth manifold M is covariant 2-tensor field smoothly assigning to each p E M, an inner product 9p T x T,M - IR at its tangent space ,,M
p (-(-)-) = (s
(58)
smooth manifold M with a Riemannian metric g is a Riemannian manifold (M,9). Definition B.4. Let (M,9) and (N, h) be Riemannian manifolds. A map & : M + N is an isometry if & is a diffeomorphism and g h. Equivalently, 6 is bijective, smooth and Vp E M,
dØp: T.M + TTip) N is linear isometry:
go(u, v) = hacp)) (d))(u), do)( Vu, E T,M
(59)
An affine connection is bilinear map V that maps a pair of vector fields X,Y another vector field VxY, which is the covariant derivative of Y with respect to X. The affine connection allows us define the notion of parallel vector field. If M smooth manifold and V connection on M, then for I - M smooth curve, any vector field X along 7 is called parallel if V+(1) X = 0
for any I, where 7 (to) drta ( l..) A smooth curve y
M is geodesic (with respect to V) iff (() is parallel along y, that is (() = 0, vL E J. For every point p M and every tangent vector u E TyM there exists some interval 1 = (-n) m). > 0 around 0 and a unique
geodesic y 1 + M satisfying:
(0) p. and (0) =
(60)
There exists a unique geodesie y satisfying these conditions and for which the domain cannot be extended. In this case, 7 is the unique maximal geodesic satisfying the initial conditions (60). We denote it by Tp.u and say that "p is a geodesic through with initial velocity v.
Definition B.5 (Exponential map of connection). Let M be manifold and V connection on M. Define for a point p E M the set D(p) = (u TrM 7nx((1) defined).
The exponential map exp," D(p) + M is given by:
expp :
(61)
Proposition B.6. The differential of the exponential map d( d(exp,)) at 0 is the identity on T M. For every E M, The exponential exp, is diffeomorphism from an open subset U C T.M centered at 0 such that exp, S M is open.
is therefore possible to build a local chart (exp, (U), exp, around every point E M using the inverse of the exponential map exp," 1 The Levi-Civita connection is the unique affine connection that is metric-compatible and torsion free.
Definition B.7. 17 nian exponential. For any p E M. a normal neighborhood of p an open neighborhood UP = exp, (B(O,e) e)) where exp, is a diffeomorphism from the open ball ( e) C T,M onto UP The injectivity radius inj (P) at is the least upper bound value e > such that expp is diffeomorphism on B(0,c). c). The chart (exp,, (B(O, (n))))))), exp, ) is called normal chart and the inverse of the exponential exp,,' is the Riemannian logarithm.
The exponential map of the
connection will be called the Rieman-
B.2.1 LIE GROUPS AS RIEMANNIAN MANIFOLDS
A Riemannian metric on a Lie group G is called left-invarint iff the left-translation map is an isometry:
(u, v) = ((dLa)zu, u,
(d) (dLL))))))L))) Va, G. Yu, Vu T,G
(62)
-ighttiivvriant metrics are defined analogously. bi-invariint metric on G is Riemannian metric that is both left and right invariant. To specify an invariant metric we can use the following.
16 Gallier & Quaintance
2020
Propositions 16.4 & 16.5).
Gallier & Quaintance
2020
Definitions 16.8-16.10).
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Proposition B.8. 18 For G a Lie group with Lie algebra there is a one-to-one correspondence between inner products on g and left (right) invariant metrics on G.
Left (right) invariant metrics can therefore be determined uniquely by specifying an inner product on g. This can be seen since for any G and u, E T.G:
(u, v)... = (d(Lz->)= d( vn
(63)
The analogue result for bi-invariant metrics is the following.
Proposition B.9. 19 There is one-to-one correspondence between Ad-invariant inner products on g and bi- ii-invariant metrics on G. An Ad-invariant inner product on g defined such that for any g E G. Adg is a linear isometry:
(u, v) (u) Vg G, Vu
(64)
Lie groups with bi- bi-innariant metrics are convenient work with as the group and Riemannian exponential maps coincide at the identity.
Proposition B.10. 20 If Lie group G is compuct, then it has bi-invariant metric. If G has bi-invariant metric. then the Riemannian exponential at the identity exp. TeG - G coincides with the Lie group exponential expm g G.
The Lie groups of interest SL(n, R), GL R) SE(n, R) do not admit bi-invariant Riemannian metries |T| When only left (right) invariant metric is available, it is still possible in some cases to obtain closed-form expressions for geodesics such the Riemannian exponential. Suppose the group GL(n, I)) is endowed with the canonical left-invariant metric determined by the inner product (X,Y) == tr X Y) for X, Y E gl(n, R). Then for any A E GL R):
g4 (x -
= (A-*X, X2), VX,, E GLGLL,
(65)
closed-form expression for the Riemannian exponential map at the identity is given by
22
exp =ex" exxxxxXx" x-xx AX g(),
(66)
On the right-hand side we have used the Lie group exponential, given by the matrix exponential. The same expression holds for SL(n, R 23 and can be used to define the exponential at any point.
we equip Gn. IR) or SL(n, IR) with their canonical left-invariant metric, the Riemannian exponential is available in closed form given by (66). As opposed to the Lie group exponential, the Riemannian exponential is surjective, and one could see it as a possible choice for the parametrization map &
Remark B.11.
G.
However, explained in Section 4 if one cannot work only at the level of Lie algebra, and group elements have to be mapped from the group G to g. the map 5- would also need to be available, We are not aware of a closed-form expression for the Riemannian logarithm on the groups GL * (n, R.,, (L, (n,,TK) TR), corresponding to the canonical left-invarian metric.
One could employ a shooting or relaxation method compute the Riemannian logarithun as done for example in (Rentmeesters et al. 2013] Chapter 6.2). However, since --1 is used at every (lifting) cross-correlation layer, this would add a large computational cost during the forward pass. This issue molivated the search for an alternative solution, as the one proposed in the main text.
Gallier Quaintance
2020
Propositions 21.1 1. & 21.2)
19 Gallier & Quaintance
2020
Proposition 21.3).
Gallier Quaintance
2020
Propositions 21.6 21.20).
A connected Lie group G has
bi-invariant metric iff it isomorphic to the Cartesian product of compact
group and an additive vector group (R",+) for n > 0 (GGallirr & Quaintance 2020 Theorem 21.9)).
SSee Andruchow et al. (2014); Martin & Neff 2016 Note that in this case, the metric will be left invariant
and right-O(n) -iinvarrannn
"See 7acur et al.
2014
Section 8.9) which references Wang (969
Helgason (1979)
2'See 7acur al.
2014
Section 9) for a general discussion obtaining the Riemannian logarithm in the
context of matrix Iic groups
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B.3 GROUP ACTIONS & HOMOGENEOUS SPACES
Let G be group and M a set. The left action of G on M is map A G M + M satisfying for any m M and group elements h,g g E G:
X(e,
and A(h, A(g. m)) = X(g.,
(67)
Right actions are defined analogously. Where is clear we are referring to a left action we use the notation g or gm for A(g, m.. Using the action
define the map Ag M - M given by Ag i + g x. Since Lie groups are locally compact topological groups some results will be useful if stated more generally. If G is a topological group and M topological space, then À is continuous and Ag. will be a homeomorphism. Whereas when G is a Lie group and M a smooth manifold the action is smooth and > will be diffeomorphism. The action of G on M is transitive if:
Vx,y y E M 3y E G: x -
(68)
If a group G' acts transitively on a set M, then M is called a homogeneous space of G. For any point x E M, the set of group elements that fix form subgroup of G called the isotropy group or stabilizer of x, and denoted by Gg. The orbit of point E M is denoted by Oz:
G₂ = E g. g-x==]
(69)
O2 G x (g ge C
(70)
Let G also act on a set N. A map f M + N is equivariant if it commutes with the action of G:
fig m) = g f(m), Vm M, Vg e G
(71)
Proposition B.12. : M be a transitive left action of a group G on set M, and denote by H G. the stabilizer of M. The map by π G - G H denotes the canonical
projection + gH on the left cosets for any E G. For any such E M the projection (or orbit) map n. + M is a surjective map defined by:
Tz Tz: G + M, n2 : 9 1 X(g, x) = .
(72)
Since nx is surjective and we have ( (9H)= = gH x = g H g = (o) any G, it induces a bijection dx G/H + M by passing to the quotient:
Tx 02 TT, o, ( -
(73)
Theorem B.13. Let G be locally compact Hausdorff group that also o-compaet. Suppose G acts transitively and continuously on locally compact Hausdorff space M. For any E M, the stabilizer G2 a closed subgroup of and the quotient space G/Gz z is Hausdorff. The projection G G/ H is a continuous open map, and the orbit map Tx G - M is also continuous. Furthermore, the map 62 : G M is a homeonorphism, and it is G-equivariant with the action of G G/H defined in 74
If M and N are smooth manifolds, differential p M. π is smooth submersion if dip surjective for every p E M. The subset I, and it is properly embedded submanifold The 'analogue' of Theorem B.13 in the Lie group setting are the following results.
M + N a smooth map, and dip, : T,M + TT(p)NN N its
(z) for x N is referred to as the fiber (of T)) 28
Theorem B.14. Suppose is a closed Lie subgroup of a Lie group G. There exists a unique smooth structure the set of left cosets G H so that the canonical projection л G - G/ H is smooth submersion. Furthermore, the left action of G on the cosets:
G G/Hl + G/H, (g1,92lt) + 919211
(74)
is transitive and smooth, i.e. G/H is homogeneous G-space.
2020 Proposition 5.2). 26 Gallier & Quaintance 2020 Proposition 3.7 & Theorems 5.13-5.14).
See for example (Gallier& & Quaintance
Abbaspour & Moskowitz 2007 Theorem 0.4.5).
Lec
2013
Corollaries 5.13 & 5.14..
T.ce
2013
Theorem 21.17).
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G/H is also referred to coser manifold. Note that π G - G/H is Geequivariant, and we have a diffeomorphism T G/H + G/H, Th : gg + hgH such that:
H
Tg o T.g. Tgh = Tg The Vg, E
(75)
Theorem B.15. 3D
Let G M- - M be smooth transitive left action of a Lie group G on
smooth manifold M. For any x E M, let G2 = - denote its stabilizen. The stabilizer is a closed Lie subgroup of G. &z G/H + M defined as in 7
Ox : G/H + M, gll + X(g - g g.
(76)
is a diffeomorphism. Furthermore, ØR is equivariant with respect to the action of G on G/ G/H and the action of G on M. The projection map a : - M, g + Mg, x) = g x (which can be expressed as n. 2z o π is a smooth submersion.
In the main text, we employ the decomposition of a group G as G/H x H for < G a closed subgroup. The following sections describe the specific class of homogeneous spaces G/H for which these decompositions are realised. Our starting point is the following general result.
Proposition B.16. Let G be a Lie group, H closed Lie subgroup, and denote M G/ If the projection π. G + M has smooth cross section a : M - G (T o = idM) then:
9 M 11 G, s0m, h) (m))
(77)
defines a diffeomorphism from the product space M x H onto G.
Proof. The proof is given in O'Neill, 1983 Lemma 11.16). Here we give a more verbose descrip- tion of the construction since the inverse of this map is mentioned in the following sections. Let H Gz be the stabilizer of a point x E M, where M G/H. To show that 4 : M x H + G diffeomorphism we define the inverse map v G + M x H such that:
& g
(o((())) 9). E
(78)
vb is smooth as it is a composition of smooth maps. To show that vb is well-defined one shows (m(g)) E H = Gg. Recall from Proposition B.12 the projection T (g) g for any g E G
and that we' ve assumed π ooo- = idMi
o ( x = (o (g) 9. x
(79)
Then (O(T -9. x (o(T(1)) therefore have (g)) E M x H, and it is clear that yo o = idg and 1b o 4 = id , the result follows.
= such that a (x())) E H = Gz. We
Lemma 11.27 of O' N (183) gives a method for constructing such map for a class of homo- geneous spaces M G H called naturally reductive. Before reviewing these spaces, we need to define Riemannian submersions. These class of submersions will allow us to describe the geometry of G/H using the geometry of G. For comprehensive description one can consult O O'ilell (1983 Chapter 7) or (Gallier& & Quaintance 2020 Section 18.3), which serve as our main references.
Suppose M and N are smooth manifolds, and π M N submersion. For any x E (MM, the fiber above given by π (xx) is a submanifold of M. For any p E --((x) then -,--1(x) (xx) = ker drpp Any complement of ker dip, = Tp -xx)-((x))) in T,M will isomorphic to N. N. In the Riemannian case for (M, g) and (N, h) smooth manifolds endowed with metrics, the fibers -=) will be Riemannian submanifolds of M, and we can define orthogonal decompo- sition with respect to the metric. The orthogonal subspaces are referred to as horizontal and vertical subspaces. More precisely, for each x (M) C N and p π (x), the tangent space T,M can be decomposed into orthogonal subspaces T, = dr, @ (ker dr,,)- Vn, @ Hp. Tangent vectors v E M,M, can be written uniquely using horizontal and vertical components:
UH + vv.
UH Hp Uv Vp
(80)
SSee [Lee 2013 Theorem 18) and Neill 1983 Proposition 11.13).
The same result is obtained in the case of topological groups when G + M is continuous cross section of n,, in which case yp M H G is then a homcomorphism.
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If Hp (Vp), then is called horizontal (vertical) tangent vector. The differential dxp of a submersion being surjective for any E M allows us to construct a vector space isomorphism drplu, H, + Nx between horizontal spaces H2 and N. x is Riemannian submer- sion if for all p E M, the differential drp restricted to Hp is a linear isometry onto Ta(p) N:
[p(u, (m)))) (dTp(()) drp(()),
Vu Hp
(81)
The main utility of Riemannian submersions in our case comes from the next theorem which de- scribes how to express geodesics in N as projections horizontal geodesics in M.
Theorem B.17. Let M + (M.g) and (N, h) equipped with the Levi-Civita connection. If - M is geodesic that starts horizontally, i.e. (0) is a horizontal vector, then 7 is a horizontal geodesic (5' is horizontal for all t. E T). Furthermore, the projection o "y = "Y is a geodesic in N of the same length as 7 Conversely, for any E M, if y is a geodesic in with 7(0) = π {p), there exists a unique local horizontal lift 7 of such that 7(0) = p and र is geodesic M.
a Riemunnian submersion between Riemannian manifolds
Theorem B.17 states that if Riemannian submersion M - N is available, horizontal geodesics in M are mapped to geodesics in N. As dr is an isomorphism when restricted to horizontal spaces, if we have smooth cross-section using the Riemannian exponential on M:
N -> M express the Riemannian exponential on N
(x) = (x)) NUE
(82)
B.3.1 NATURALLY REDUCTIVE & SYMMETRIC SPACES
Let G be a Lie group, H S G closed subgroup Ad G - GL be the adjoint representation of G. A homogeneous space G/H is reductive if there a subspace m of where:
Definition B.18. 33
g= + m, and (m) S m, wh E
(83)
That is, G/H is reductive if we can find an -ivarrat subspace complementary to h in g.
The following property gives recipe for constructing a G-invariant metric on G/H and extending to a left-invariant metric on that right H-invariant such that क G - G//l a Riemannian submersion, with h and m being the vertical and horizontal subspaces at G.
Proposition B.19. 34 Let G be Lie group, H a closed subgroup and G/ reductive homogeneous spuce with reductive decomposition g = h $ m.
1. There a one-to-one correspondence between G-invariant metrics on G/H and Ad(H)- invariant inner products m. The correspondence can be established by making drelm= m - (G H) into a linear isometry, where = (T)) eH. G-invariant metric on G/H exists iff the closure of (m) is compact. H is compact then (m(H)(m) is compact, so there exists a G-invariant metric on G/H. H.
2. Let m have an Ad( H iinvariant inner product. If we extend it to an inner product on g = h @ m such that h- m. and endow G with the corresponding left-invariant metric then the canonical map π G + G/ G is a Riemannian submersion.
The reductive homogeneous spaces of interest are the following.
Definition B.20. 15 Let G be a Lie group and H a closed subgroup of G. The homogeneous space G/H is naturally reductive it is reductive with decomposition g h @ m, has G-invariant metric and satisfies:
{Xx 2 m. Y) = (x, (m) VX, Y, Z E m
(84)
"Seee Gallier & Quaintance
2020 Proposition 18.8) or O'Neill 1983 Lemma 7.45 & Corollary 7.46).
Gallier & Quaintance 2020 Definition 23.8).
Corresponds to (Gallier & Quaintance 2020 Propositions 23.23).
Gallier Quaintance 2020 Defimition 23.9).
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In this case, is possible express geodesics in G/ H with respect to the Levi-Civita connection as orbits of one ane-mmraeteer subgroups generated by the tangent vectors in m.
Proposition B.21. Suppose GjH H is naturally reductive homogeneous space, and have = h @ m. Using the G-invariant metric of Hj a left-invariant metric is constructed on G, such that its restriction to on is Ad(H) ad(H)-invriian and we have m -h+,, and recall that in this case π G + GjH is Riemannian submersion. For every X E m the geodesic starting at ((E) eH with initial velocity dre (X) given by:
To,da, (o,d=,(x)))) = expm(tX) o = TO cxpm(tx)) Vt
(85)
Since the one- mne-parameter subgroups + expm((X) are defined for any IR, by the preceding proposition so are maximal geodesics through o and therefore through any point since we are work- ing with homogeneous space. Naturally reductive homogeneous spaces are therefore complete
Definition B.22. A connected Riemannian manifold (M,g) is (Riemannian) symmetric space
for every point E M there exists a unique isometry 8p M + M such that 8p(P) = and (dsp); = idg. Equivalently, for every E M, the map 8p is an involutive isometry (sp. = id) having p as its only fixed point.
The isometry 8p : M - M called the global symmetry of M at the p. Symmetric spaces can be constructed from 'Lie group data'. The connection can be made clear with few more definitions.
An involutive automorphism of a Lie group G is an automorphism o G + G such that a L id and o² = id. For a an involutive automorphism G. GO {g E G ( = 9} will denote the closed subgroup of fixed points of a and G6 its identity component.
Definition B.23. A symmetric pair is a triplet (G, H. σ)) where G a connected Lie group, H closed Lie subgroup of G. and a G - G an involutive automorphism of G such that G S H G If udditionally Ad(H) C GL(g) is compuct (where Ad G + GL((g is the adjoint representation of G), then (G. H, r) is a Riemannian symmetric pair.
The differential do. + g of an involutive automorphism a G G G defines the 1 eigenspaces:
h {X E dx =X]; m = {X E g dx,(X)= -X}
(86)
Theorem B.24.
Suppose that (G, o) a symmetric pair. Then the following properties hold, 1-3 make G/H into reductive homogeneous space.
Note that items
1. h = (X g do, (x) = X] is the Lie algebra of H.
2. g = h + m. where m = (X E do, (X) = -X}. The decomposition follows from do. g also being involution do? = id and the identity:
X (x + dx)( ] doo(X)), W
(87)
3. Adg (m) C, V K.
4. b.. b] C b, bb C,, [m, m C.
The map do, g - associated to a symmetric pair (G, H) is referred to as a Cartan involution, with the automorphism given by dm, as in Thm. B.24 is called Cartan Decomposition of g. If one further assumes that Gg and H are compact, then we obtain symmetric space.
G + G' being globbal Cartan involution. The decomposition h
Theorem B.25. Suppose that (G, H, a)) a Riemannian symmetric pair with Gg and H compact. Denote by g and the Lie algebras of G and H respectively.
36 "Gallie & Quaintance 2020 Propositions 23.25-23.27).
O''Neill [983] pp. 313).
(ellaaoon 2001 Chapter IV, 53).
Gallier Quaintance 2020 Theorem 23.33).
See (Ziller) 2010, Proposition 6.25) or (Helgason 2001
Proposition 3.4). Corresponds to Gallier &
Quaintance 2020
Theorem 23.34), (O'Neill 1983 Theorem 11.29).
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1. Since H is compact, G| H. admits G-invariant metric from Proposition (1) From the previous theorem, G H has a reductive decomposition g = h t m where h and m are the #1 eigenspaces of do Using the identity [m. m C 6 and assuming G-invariant metric on G/ H the natural reductivity condition of B.20 holds trivially (since h O m = {0))
2. For every p E G/H, there exists isometry 8p G/H + G/H such that sy(p) p and d( d(sp)p = -idp. making G/ H a Riemannian symmetric space. For the projection π G + G/H and (e)) eH, the symmetry at is defined such that so gll ++ (g))
8a 0
(88)
For an arbitrary p = gHl E G/H, the geodesic synmetry is given by:
Sp Tg So Tg-
(89)
By the preceding theorem symmetric spaces can be given a naturally reductive structure. We can now use Proposition 1.16 to construct global cross section, under which the Lie group can be identified with the product space x H or expm(m) x H. Recall from Proposition B.21 that geodesics starting at = (e) = eH = with initial velocity (X) for X E m are of the form To, (4) expm(LXX)o r(expm(LX)) In particular, we can obtain the following expression for the Riemannian exponential exp, T.M + M:
expo m (X)) (expm(XX), AX e
(90)
That is, the following diagram commutes:
d m
m
ToM
(91)
expm
exPo
G G
Proposition B.26. Let M = G/H be naturally reductive homogeneous space and π G - G/H
the canonical projection. If the Riemannian exponential exp, at the point o = T(e) = =H M is diffeomorphism, we can construct diffeomorphism of m x H onto G given by:
m x m H H (X,4) + expm(X)h
(92)
Proof. O'Neill 1983 Lemma 11.27). Again, we reproduce the proof as the maps defined refer- enced in later sections. The map is built by constructing a cross-section π : - G/H H using the relation 900) of the Riemannian exponential such that one first defines:
Exp,- == exp, d = π expm m + M
(93)
By hypothesis exp, : (M) - is diffeomorphism, and so is dr making Exp, a diffeomor- phism. We can define the cross-section by M - G by:
σ = expm Exp_"
(94)
Then π o expm Exp_1 Exp, Exp=" idM, and by Proposition B.16 we have diffcomorphism yp M x + G given by 777::
:: : (m, h) (m)h expm (X))h
(95)
Composing this map with the map Exp, x id we obtain the desired map &5 = yp (Exp, id
@:mx x H + G. Φ: (X, - (EEp. = cxpm
(96)
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B.4
THE CARTAN/POLAR DECOMPOSITION
Define the following subsets of M, (R):
Sym(n, R) = {P M., (T)
P pT]
(97)
(os R) = {P E Sym(n n, R) Yyu E R2, , 0, Pr > 0}
(98)
(n, R) {P E Pos(n, R) det(P) =1
(99)
Sym (n, R)- = {P E Sym(n. n, R) tr(P) = =0}
(100)
Sym(n, R) is the vector space of n n real symmetric matrices and Pos R) the subset of Sym(n, R) of symmetric positive definite (SPD) matrices. SPos(n, IR) denotes the subset of
Pos(n, (n, RR) consisting of SPD matrices with unit determinant, and Sym, (n. IR) the subspace of
Sym(n, m, R) of traceless real symmetric matrices. Every SPD matrix S E Pos(n, R) has a unique square roo 41 g1/2 P. P E Pos(n, R). which shows the uniqueness of the polar decomposition. Proposition B.27 (Polar decomposition). as A = PR or A RP, where P. P E (os(n, IR) and R. R E O(n). We refer to the factorization A = PR as the left polar decomposition and to A = RP as the right polar decomposition. We choose to work with the left polar decomposition. The factors of this decomposition are uniquely determined and we have bijection GL(n, IR) + Pos(n, R) x O(n) given by:
42 Any matrix E GL(n, I) can be uniquely decomposed
A (VAAT, VAAT A), E E GL(n, R)
(101)
As mentioned in the main text, this decomposition can be generalized using the fact that the spaces Pos(n, R) (n, ()))((n) and SPos(n, R) SL(n, R))SO(n) are symmetric spaces, and Cartan decomposition is available in this case. We first state some useful properties of Pos(n, I)) and then review its symmetric space and naturally reductive structure.
Proposition B.28. 43 Every real symmetric matrix X E Sym(n, R) has a spectral decomposition X = ODO where O SO(n) and D diag diaggdi,.... (.i....,.,d, dn). di R is a diagonal matrix consisting of the eigenvalues of X, which are positive iff X is posifive-defmite. Using this decomposition we have simplisied expressions for the matrix exponential expm : Sym(n, RR + Pos(n, IR) and logaritium logm Pos(n, R)) + Sym(n, R):
expm(X) = (diag(exp(di),-..... expm(dn))OT, AX E SSm(mn, logm(P) =
(102)
VP E IR)
(103)
Pos(n, R) is an open subset of Sym(n, IR) and a smooth manifold of dimension n(n + 1)/2, with the tangent space T TpPos(n, R.) at any P E Tppos(n, IR) naturally isomorphic (by translation) to Sym(n, R). The matrix exponential and logarithm maps are diffeomorphisms between Sym(n, R)
and (os R), and the power map P + pa is smooth for any E I. since it can be expressed as:
po = expm((llogm(?)
vP E Pos(n,
(104)
As a reference for the following results on Pos(n, IR) and (Pos(n, IR) see Förstner & Moonen (2003) Pennee 22020); Stegemeyer & Hüper (202T). The presentation here also follows (Rentmeesters et al, 2013 Section 3.5) and JezzcnnoCassade 2021 Section 3.5.3).
Pos(n, R) is a homogeneous space of the positive general linear group GL (n, R). More precisely, GL (n, R) has a smooth transitive action on Pos(n, IR) given by:
A: GL +n,, R) x Pos(n, R) - (os(n, IR), (A, APA
(105)
+ Note that every SPD matrix P Pos IR) can be written as P= AAT for some A E GL (n, RR. The isotropy group of the identity matrix Pos(n, IR corresponding to this action is the special orthogonal group (O(n) since RIR = RR for R E SO(n).
41/
(Hall
2015 Lemma 2. 281 or (Hom & Johnson 2012 Theorem 7.2.6).
42
See (Jost & Jost
2008
Lemma 7.5.1) (Hom & Johnson 2012 Theorem 7.3.1).
43
(Arsigny et al.
2007
Theorems 2.6, 2.8 & Corollary 2.9) more specifically for the matrix exp/log and
power map. For review of algehraic properties and the Riemannian manifold structure of Pos(n, IR) see Ar-
signy al. 2007)
Pennec 2020).
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Applying Theorems B.14 & B.15 we have that GL (n,"n,R)/SO (o, R). That is, have diffeomorphism:
Φ GL (n, ()))((())))))))))))))))n + Pos(n, RR,, A 4-50(m) AAT
(106)
And smooth submersion of GL + (n, R) onto Pos(n, given by:
π OT: GL Gn + Pos(n, RR),
+ AA"
(107)
Let g = gl(n, R) denote the Lie algebra of GL [k) R the space so(n) = {x E (I)
The Lie algebra of SO(n)
X = -X-} x=-x-
of skew-symmetric matrices.
Pos(n, R) = GL (n, R
(n)()) a reductive homogeneous space (see Definition B.18 since
(n(SO(n))SSmll) R)) C Sym(n, (R) I)) and we have the decomposition g @ given by:
gl(n, IR) = so(n) @ Sym(n, R)
(108)
Then h = so and m Sym(n, R), and the bracket relations [h. C b, (b, m] C m, [m, m] C hold. We now choose an inner product on gl(n, such that its restriction to (n, R) is Ad(SO(n))-invarinn) mannian metric on GL GL+(n, R). We work with scaled version of the canonical inner product (X,Y)= == rx(x))
SO(n)*
Sym(n, R) and we can use it to define left-invariant Rie-
B(X = ((X,Y) 4tr(X"),, X,Y gl(n,,)
(109)
The inner product respects the decomposition (108) into symmetric and skew-symmetric matrices, and the left-invariant metric GL (n, (which is also (ight-SO())-inarriin is:
GL (m) (X, B(- 'Y) AA GL (n, R), VX,Y Y TAGG R)
(110)
To define a GL (n, R)-invariant metric on Pos(n, R GL + *(, R))SO()), note that the differen- tial of the projection (107 at is dX) = X + x for any X E gl(n, IR), with ker(dr,) = so(n) and its restriction to Sym(n, R) gives the isomorphism:
dm m + (o,((n, R), X
(111)
We have X(P1/2,1 1)- = P. and the differential with respect to the second argument at identity is X p1/2x11/2 The linear isomorphism d( L Lp1/))) : m + TpPos((, R) then given by:
d(T o Lpss=a; X + 2p1 x1/22 VX E
(112)
We denote its inverse by 1/ Tppoos(n R) - m, such that !j X + 1 1-11/2 The in- duced (quotient) metric on Pos(n, R m is defined for any P E Pos(n, R) and X, Y E TpPos(n, R): Pon(n, R) (X,Y) (X) 9P
(P- 1P-1/22p-11 p- xp- 1/2) tr x
(113)
Endowed with this metric the action of GL.+(n, R) is by isometries and ,, is a Riemannian submer- sion. (Pos(n, Pos(n.))) Po)(n.R) is also a Riemannian symmetric space and (GL+ (n, R), SO(n), θ) is Riemannian symmetric pair, with θ the global Cartan involution:
θ GL. (nl RR) + GL (n, (n,R),
θ A (AT)-1
(114)
In this case we have O = (O(n) = 6 and we have corresponding Lie algebra involution:
= dO. gl(n, R + gl(n, R),
(115)
Analog results hold for (Pos(n, IR) SL (n. R)/SO(n))))))) such that (SL(n, RR), SO(n), θ) is Rie- mannian symmetric pair. The group SL n, R. has Lie algebra:
sl(n, R (X E gl(n, tr(X) = 0}
(116)
SL(n, H (n))))))) is example of non-compact symmetric space 110 and (113), restricting them to (L IR) and SPos(n, R), respectively.
45
We reuse the metrics
Known as the affine-invariant metric, see Thanwerdas & Pennec 2023
Proposition 3.1).
45 For classification of symmetric spaces for example (Ziller 2010
Chapter 6).
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Proposition B.29.
46 GL (n,R) can be represented product SL(n, R) x BR50 by the Lie group
isomorphism:
GL (n, SL( R50- A
dct(A)))
(117)
det(
Reusing the previously defined metrics, SL(u, IR) and SPos(n, I) are totally geodesic submani- fold 47
of (n, R) and (os(n, R),, respectively. The decomposition (17 restricted to Pos(n, IR) can shown to induce a Riemannian isometry Pos(n, R) = SPos(n, R) Ryo. The tangent space decomposition is Sym(n, I)- Sym,(n, IR) + D, where d(n, IR) are scalar diagonal matrices.
B.5 PROOF OF THEOREM
As in the main text, we let (G/H,M, m) define our 'Lie group data', corresponding to (GL+ (n, R))SO(n), Pos(n, R), Sym(n, R)) or (SL(n, R)/SO(n). SPos(n, R), (n, R)). Theorem 4.2. Let (G/HL H, M, m) be as above, and denote by the Lie algebras of G and H.
1. The matrix exponential and logarithm are diffeomorphisms between m and M, respectively. For any P M and a E the power map - pa is smooth and can be expressed as:
p = =epmmaalogmi P)), vP E Pos(n, R.
(19)
2. = H and G = m x H. We have group-level diffeomorphisms:
X: x H
x(P, R) PR
(20)
& R) (xpm)
(21)
3. The above maps can be inverted in closed-form:
x G + M HI x- AA A (V AAT, VAA A)
(22)
1 Φ : G 1-1 : (2 logm (AA"), expm( - logm( (AA7))
(23)
Proof. The theorem collection of results related to the Cartan decomposition and the structure theory of Lie groups, which can be found for example in (Bridson & Haefliger] 2013 Chapter I0)
or (Abbaspour & Moskowitz 2007 Chapter 6). Similar results apply to algebraic subgroups of Gn,, R) that are closed and stable under transposition (see (Abbaspour & Moskowitz 2007 Prop.
6.3.3 & Definition 6.3.4) or (Bridson & Haefliger 2013 Definition 10.56)).
1. The first result holds due to Proposition B.28 and the fact that for any X E we have expm(lX) M for all t R, see (Bridson & Haefliger 2013 Lemma 10.52). The Riemannian exponential on M is also diffeomorphism at any point.
2. For the group-level Cartan/Polar decomposition see (Abbaspour & Moskowitz 2007 The- orem 6.2.5 & 6.3.5). Given a tangent vector in m, the Riemannian exponential on M and the matrix exponential are related by the diffeomorphism:
(eppm(XXexxplle Exp, + M, X + expm(X) I = expm X Texpm(X)) (X) = expm(2X) (118)
Exp. is obtained from applying Proposition B.21
The map of (2I) can be obtained
from (20) and the fact that the matrix exponential is a diffeomorphism on M, or using Proposition B.26 such that (2T) corresponds to (96).
3. The map x- is simply the polar decomposition. To obtain { we use the fact that AAT E M for A E G and the identities:
logm [p2/2=
- logm(P), p-1/22 (oppm--)ogm(P)))))
VP E M
(119)
2
The identities 119 be obtained from (19).
4AA detailed treatment can he found in Dolcetti & Pertici 2015
2019).
N is a totally geodesic submanifold of Riemannian manifold M, if for any two points of N and a geodesic
7 in M that joins them, y is entirely contained in N.
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B.6 INTEGRAL FACTORIZATION FOR THE CARTAN/POLAR DECOMPOSITION
Consider again the notation (G/ H, M, m) as in Theorem 4.2 Recall that we have G GL (n, IR)
(M Pos(n, R)) G (n, IR) (M (n, (n, with H SO(n). From the proof of
Thm. 4.2 the cross-section expm Expe M G. of Prop. B.26
(S) = expml (Sgm AS E M
(120)
which is smooth (Prop. B.28) and reduces simply to the square root a (S) 5512 Since symmetric positive definite matrices have a unique square root, a M + G is a diffeomorphism. We obtain decomposition equivalent to the Polar/Cartan decomposition given by the map vp M x +
defined as in Propositions B. 16 B.26
Φ P x H G, ( 8
(121)
With the inverse defined by:
sb: G x xI,, HI v/ A (A(T) A) - (AA", (AA7) 11/2
(122)
We have equivalent decompositions which allow us to represent A G A PR or A S R for S, M, R E H and therefore P = S1/ The motivation behind presenting both decompositions is that for Gn(n, IR), the decomposition A = $1/2 R. has factorization of the Haar measure (GL(n.R) as a product of invariant measures on Pos(n, R) and (n)). The Haar measure on
Gn, R) is given for any A = (Atz) E GL R) by 4K
dPCL(M,,)(A) = [del(A)] "dA = [A)
II dAi 1,==1
(123)
where d4 is the Lebesgue measure on TR= and d'A.j is the Lebesgue measure on R. GL R) has two homeomorphic connected components consisting of the group of invertible matrices with posi- tive determinant GL (lL+((r, (n. R) and with negative determinant GL (n, R Integrating the full group GL (n, R.) can be done by integrating each component separately, and we focus on constructing solution for the identity component GL* (n, IR.. We use the shorter notation (os(n) and SPos(n) going forward to denote (os(n, R) and SPos n, R). Using similar notation scheme as in 123). the
unique (up to scaling) GL.n, RR R))-ivarrian measure on Pos(n) is 5m
d[ppos()) (5) = - as ((ntt(s))((+ll)) 1)/2
dSij,
AS E Pos(n) (124)
Isisisn
The following result can be found in more general setting, often expressed using the 'right' polar coordinates of the decomposition A = R.S1/2 Let H = Vn.m {R E Mnm(R) (R) R R = mm for n 2 m and = Mm (I))* {A Mnn (R) rank(A) = m] the set m. Vn.m is the Stiefel manifold of orthonormal m-frames in RR
m matrices of rank
which O(n) acts transitively by left multiplication such that Vn.m = O(n)/O(n - m). with special cases Vn,n = O(n) and Vo.n-1 = SO(n). The complement Mnm (R)* in Mnm. (IR has Lebesgue measure zerq and Man (R)* GL(n, RR. In this case it will correspond to (Herz 1955 Lemma 1.4) (Murhead 2009 Theorem 2.1.14).
Theorem 4.3. Denote G GL(n, R), H O(n), and let na be the Haar measure on G and
yn the Haar measure on H normalized by Vol(H) = 1. For A E G, under the decomposition = 51/2 R, S E Pos(n), R E H, the measure on G splits as dpa(A) (A) = BB Baddppo(n)) (i) (S)du (R),
normalizing constant. Restricting to G GL (n, R) and SO(n) and ignoring constants, we have:
where 8n vol(O(n))
2n
f 1 f(A))do(A)) Pos(()) J
/
5(S\/2R)dpa (R)dnn(())ddaaaoo R)dpppss(s)) (S), Af E Cc(G)
(24)
Bourbaki & Berberian
2004
VII, $3. No. 3, Example 1).
Warner 1983 Theorem
3.68).
Bourbaki & Berberian
2004
VII, $3, No. 3, Example 8). (ros(n.)) is bi-GL(n, Ri-invariant, as well as
invariant under P + p
Gross & Kunze
1976
Example 5.5) or Faton 1983
Proposition 7.1).
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Proof. A proof is given in (Gross & Kunze 1976 Prop. 5.6) for the decomposition of the form
A = RS1/2. In the form R it is proven for example in (Faraut & Travaglini 1987 Section 4).
In the context of multivariate statisties see Theorem 5.2.2 and Remark 5.2.3 of Farrell (2012).
recent reference is (Chirikjian 2012, Section 16.7.2).
Note that the constant Bn is independent of E C.(G). From (Chirikjian 2012 (16.36)):
121
Vol(O(n)) = Vol(SO(n)) (n/2))
(125)
Where (.) denotes the multivariate Gamma function. From (Chirikjian 2012 (16.55) & (16.56)),
if dA is the Lebesgue measure on R=",, under the decomposition S1/2 R we have:
dA d(s- R)- [.]lded
(126)
Then considering that AAA" the Haar measure duCL(N.R) = |det can be expressed:
((GI)( = | " ((S = 3r (aaeAA44))==dddee((()))))
(127)
We use this decomposition treating G (GL (n, I, R SL,,, RR) as our sample space. From Section 5.2 of Farrell (2012), for the case G = Mum (R)* = Vn. x Pos(m) it can be shown that if A E Mnm (R) is a random matrix with (nn)-eft invariant distribution, then for (o()) RS the corresponding random variables R E Vn,m and S Pos(m) will be independent, and R will have a uniform distribution on Vn.m- Furthermore, there exists a relationship between the density function A = RS E Mom (R))* with respect to the (n)) ((v)--ivaria measure and that of E Pos(n) with respect to 124 If G = GL*(n, IR) or = (n, IR), the Haar measure Pa is bi-O(n)- (invariit (respectively (i-SO(m)-invarian) We can then work with either decomposition 52 / R R/
Theorem 4.4. afa random matrix E GL(n, I) has a left-O(n)) invariant density function relative to I d.A then = and R = (AAT)-1/22 are independent random matrices and R has uniform probability distribution O(n). The uniform distribution on O(n) will be the normalized Haar measure Po(m)- Conversely, if E Pos(n) has density function f Pos(n) - R20 relative to plos(n)) and R O(n) is uniformly distributed with respect to the Haar measure Mo(n), then A = S has a density function 8 f(Af(AAA)) (d)))))))))))) relative to d.A.
5
Proof. This theorem collects Lemma 5.2.4 & 5.2.8 of Farrell (2012) applied to the case where are working with random matrices in GL (n, and using the left polar decomposition. See also (Eaton 1983 Proposition 7.4).
Restricting only the connected component GL Gn, R), the task is now specify proba- bility distribution on Pos(n) relative to the measure dppos(n)- For the case of SL(n, R), us- ing the isomorphism 117). we can define a SL, Ri-invariant measure on SPos(n). More pre- cisely, P = (det(P)2/n) P) for P E Pos(n), P = (et(P)-1/mP E SPos(n), which we write P = 11/n P, t >> O, such that ddppatt)) (P) dt d[isppamm)) and for f (Pos(n) (Po(())
53
J s(P)dppontn)) (P) S (Pos(()
dt f(1/1 /(P/")) dpsppssn) (P)
(128)
B.6.1 SAMPLING ON THE SPD MANIFOLD
Following Said et al. (2117). Riemannian Gaussian Distribution denoted G(P, ar) depends on parameters P E (os(n) and volume element ())pos(()) by:
0 to define a probability density function with respect to the
exp[ d(P.P) 20
p(P Z(o)
P Pos(n)
(129)
52 in the sccond case, one considers the right action PA, for A GL(n, R) and P E Pos(n).
Teras 2016 (1.21) & (1.39)).
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Here, d (n) x Pos(n) + R20 is the Riemannian distance corresponding to the metric (113). The metric plays a key role, as the measure (124) the Riemannian volume element associated to it. The distance can be expressed by:
d(X,Y) = tr[logm( X 1/2XX 1/)
AX, Y E Pos(n)
(130)
and (() is normalization factor given in Said et al.
2017 by:
d² dP
JPss(M) expl
[ddppatt)) (P)
(131)
20²
For 2 an analytic expression ((r) exists, otherwise can be approximated by Monte Carlo integration. et al. 2017 Prop. 5 & 6) describe an algorithm for sampling from this distribution.
Alternatively, an approximate solution when sampling close the identity is given by the Log- normal distribution defined in (Schwartzman, 2016 Sec. 4.4). From (Schwartzman, 2016 Def. 4.4.2), X E Pos(n) has Log-normal distribution X (N Σ) with mean M Pos(n) and covariance Σ if logm( (M 1 M-1/2) ~ ((0, E). This definition assumes that M is the empirical Riemannian center of mass, corresponding the random variable X.
B.6.2 ALTERNATIVE DECOMPOSITION BASED ON THE QR FACTORIZATION
There are several choices available for decomposing GL aL**0 (n, R) and SL I)) such that invariant integration can be made easier while working with the smaller factors. The primary tools of interest are the Iwasawa and the Cartan decomposition, and one possibility given by the Gram decompo- sition (QR factorization). Let T(n, R) {X E GL(n, R)] X,j if >> j} be the group of real upper triangular matrices and T(n, [)) S T(n, I)) its subgroup whose diagonal entries are positive. Every matrix A GL(n, R) has unique decomposition A RT or A TR for T E T(n,R)+ and R E (n).
Under this decomposition, Theorem 4.1 (2) is applicable. The orthogonal factor becomes R E SO(n) restricted to A E GL* (n, R). For A E SL(n, R) the decomposition is given by replacing T(n, IR)+ with its subgroup ST(n, R)+ S T(n, R)+ of matrices with unit determinant.
B.7 MORE DETAILS ON THE LIE ALGEBRA PARAMETRIZATION
Any A G can be expressed uniquely as = e R for x E m and R E H. Since SO(n) in both cases, the fact that expm so(n) + SO(n) is surjective allows to write it A = xx Y E so(n). The factors and f e² are obtained using 4 22 Then by taking the principal branch of the matrix logarithm on H SO(n), Y logm(A) R). map 5-1 G g described in Section 4 is therefore constructed as 5 = (idm x logm) Φ- More precisely, for any A = exe" G, using& & we obtain the horizontal/vertical tangent vectors XY so(n) x and since = so()) @ m we have unique 7 = X + E g-
If is the dimension of G, the tangent space g is d-dimensional vector space isomorphic Rd, with basis elements denoted by (E,...., Ez). Once a basis is chosen we can concretely represent linear combination of the 'generators' such that v > ny F, for
any element of (or h. m) any v E The vee and hat functions (denoted V and A) are used to map tangent vectors to their coordinates in this basis and back:
A: RRd + g.
A:V= (0), U2, Ua...,,,a,") va)² + v
E
(132)
V: V V (V-V
(133)
The basis (Et)aej is chosen be orthonormal with respect to the inner product (09) which is used to construct the invariant metric. Going forward is understood that functions parametrized on the Lie algebra, such as the kemnel he g + I, take as input the vector of scalar coefficients of the tangent vector expressed in the chosen basis (the result of the map).
Galllerr Quaintance 2020 Theorem 2.6).
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To summarize, the map 5- G + g is implemented for any A E G
1. Mapping A to its product space representation in m x (O(n) using + (A) = (X, R).
2. Using the matrix logarithm on R = ex (which is available in closed form for the cases of interest S0(2) and SO(3)) to obtain (X, logm R)) = (X, Y).
3. Expressing the tangent vector Z X Y using the chosen basis Z E Rd.
C ARCHITECTURE & TRAINING DETAILS
All experiments will use the same ResNet-like architecture He et al. (2016). and it will consist of a lifting cross-correlation layer, a single residual block and a final cross-correlation layer. Finally, to achieve invariance global pooling is applied over the spatial and group dimensions. The (lifting) cross-correlation layers are always followed by normalization and non-linear activation layers. the case of the affine robustness task, we use GeLU nonlinearities and 'LayerNorm' normalization The residual block contains 2 group cross-correlation layers and we apply max-poot max-pooling over the spatial dimension of the feature maps after each block to increase the robustness of the model. For all experiments, the kernels Kn g + R are parametrized using 'SIREN networks', introduced in Sitzmann et al. (2020). SIREN networks can be considered as one example of Implicit Neural Representation (INR) model. These models have seen widespread use in various areas of computer vision and graphics, e.g. Mildenhall et al. (202I). INRs can be formalized as learned continuous function approximators based on MLPs. They can be described simply as MLP layers of the form:
Ym = (Wmym-1+ bm)
(134)
where a a non-linearity. In case of SIRENS we have (xx) sin(wbrr) where wh RR is a multiplier controlling the frequency of the sinusoid. We emphasize again that the proposed method- ology is not dependent on the specific parametrization other activation functions such as the (complex) Gabor wavelet [Saaaadaam et al. (2023) offer com- parable results. We set We = 10 for all experiments. We use 42 output channels in both the lifting and cross correloorne layers. Each SIREN network consists of 2 layers of size 60.
ka, and have experimentally found that
A key hyperparameter to consider is the number of group elements that will be sampled in the Monte Carlo approximation of each of the cross-correlation layers. Empirically, we have found that 10 12 samples are enough to achieve a better performance compared to the previously described models. The models are trained for 100 epochs, with batch size of 128, and the Adam optimizer of Kingma & Ba (2014) with a standard learning rate of 0.0001. Sampling from Pos(2, R) is done using the log-Normal distribution of Schwartzman (2016) centered at the identity while for SO(2) we work with discretization of equi-distant points in 00, 2T).
D EQUIVARIANCE ERROR ANALYSIS
Equivariance error Since our models are only equivariant expectation, we validate this prop- erty numerically by measuring their equivariance error following the same approach as Sosnovik ct al. (2020), where we look to quantify:
A = [[C][((f) &
(135)
where 9 G and & represents network. We evaluate the equivariance error before training the network, i.e. + is convolutional network with randomly initialized weights. We take to be simple convolutional network composed of lifting map (5) a cross correlation (7) and projection correlation mapping our data back to a scalar field (which transforms under the trivial representation). The same normalization and nonlinearities described previously are employed. In Figure I we plot the equivariance error of क. for different choices of ki, and compare our model to a standard CNN with the same input-output dimensionality for its layers. For the evaluation we use 100 samples from the Haar measure of S2, RR,, using the tools described in this thesis, and obtain an average estimate over 10 random seeds.
Similar interpolation methods have been explored in the context of numerical linear algebra and computa- tional mathematics. See See/Munthe- Kaas et al. (2001) (014) and especially Gawlik & T.ook] (018), as this method can in part be seen as an instance of their more general framework.
See Hendrycks& & Gimpel (0166) and Ba al. (016).
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10
Fox Tortanree 4522
SL/2.R) WIRE
SUZZRR SIREN
SL/2.8) MLP
Sanvber Corglar
Figure 1: Equivariance enror as a function of the number of MC samples.
We compare three possible choices of kernel parametrizations, namely a standard MLP with Swish non-linearities employed by Finzi al. (2020), well the Siren Sitzmann et al. 2020) and WIRE Saragadam et al. (2023) INRs. Note that this choice will also have an effect on the equivari- ance emrrr,
working with discrete pixel grid when representing images, and any symmetry breaking operations will propagate the loss cquivaraince through the network. Figure quantifiess the degree which the performance of the model described in the previous section is affected by the number of MC samples used when approximating the convolutionneros--orellatt general, we observe significant performance degradation when employing S 6 samples and in previous work on integral approximations of continuous convolutions Knigge et al. (2022) observe no additional benefits beyond 12 14 samples. However, an exact specification of the approximation bounds corresponding to the groups employed is missing in our presentation.
integral. In
SL(2, R) affNIST
SL(2, R) homNIST
Figure 2: Test error on affNIST/hoNNIST as function of MC samples.
35


Reviews:
[{"review_id": "5x3lbD0vvs", "reviewer": "Reviewer_gQEm", "summary": "This paper proposes a new approach to efficiently generalize group equivariant convolutions for the  $\\text{GL}^{+}(n, \\mathbb{R})$ and $\\text{SL}(n, \\mathbb{R})$ groups. While the Haar measure for such non-compact groups is ill-defined, the approach proposes to work with the decomposition of these groups into factors that have well-behaved Haar measures. Factorizing the convolution integral enables the computation of group equivariant convolutions on these groups. By using results from random matrix theory to efficiently sample elements of these groups, the authors propose an efficient way to compute these integrals via Monte Carlo (MC) integration. This results in an order of magnitude improvement in the number of samples needed for a converged integral. The method is tested on two image classification tasks under affine and homographic transformations. In both benchmarks, the method is shown to outperform previous approaches.", "questions": "- See my point regarding numerical stability.\n\n- This might be a standard result already, but I would like to have a formal view of the convergence of your approach. Do you think it would be possible to obtain an explicit bound on the equivariance error as a function of the number of Monte Carlo samples, the sizes of the factors in the group decomposition, and the size of the representation? To be more precise, I am looking for something akin to this bound: $|| \\int_{H} \\int_{K} f(\\rho(h) \\rho(k) \\cdot x) d\\mu_{H}(\\rho(h)) d\\mu_{K}(\\rho(k)) - \\frac{V}{N} \\sum_{i} f(\\rho(h_{i}) \\rho(k_{i}) \\cdot x), \\rho(h_{i}) \\sim \\mu_{H}, \\rho(k_{i}) \\sim \\mu_{K}|| \\leq f(N, dim(\\rho), dim(H), dim(K) )$\nUsing that, one could ask what kind of factorization gives the best bound.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 2, "contribution": 3, "strengths": "- The paper is clearly written, and the contributions are made clear.\n- The mathematical concepts are well-introduced, precise, and effectively utilized.\n- The approach is novel and original, making elegant use of existing results in classical group theory and random matrix theory within the context of geometric deep learning.\n- The two groups covered in the paper ($\\text{GL}^{+}(n, \\mathbb{R})$, $\\text{SL}(n, \\mathbb{R})$), are two important groups for image-related tasks.\n- The sample efficiency gains might prove to be a key element for the future adoption of group equivariant convolutions in image-related tasks.\n- The benchmark results are convincing, comparing favorably to strong baselines in the field.", "weaknesses": "- The analysis of the numerical stability of the method is largely missing. In particular, methods that rely on approximating integrals over groups will always be approximately equivariant. While increasing the number of samples could reduce this error below floating-point precision, such an analysis is not present in the paper. A curve illustrating the equivariance error as a function of the number of Monte Carlo samples would be pertinent in this context. Although sizeable equivariance errors might be manageable in the context of image classification tasks, this is not the case for all types of application tasks where group convolutions are relevant. Specifically, in physics simulations, equivariance errors are related to the non-conservation of physical quantities. The proposed approach should be contrasted with approaches that use linear operators to evaluate the group integral, such as [1] and [2], which are exactly equivariant (even for non-compact groups) but require large tensorial operations (for [1]) or ad hoc representation theory for each group (for [2]). I encourage the authors to briefly discuss these points in the text and add an analysis of numerical stability in the form of a plot to the appendix.\n\n- In the paper, only the symmetrization to the canonical group action is explicitly mentioned. Extending the matrix elements to other finite-dimensional group representations should be straightforward, provided that the correct basis is determined. I am curious why this was not presented in full generality.\n\n[1] \"A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups,\" by Finzi et al.\n\n[2] \"A General Framework for Equivariant Neural Networks on Reductive Lie Groups,\" by Batatia et al.", "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": "u456WnY3bM", "reviewer": "Reviewer_yeU2", "summary": "The paper considers the problem of designing equivariant neural networks when the group may not be compact or abelian, and when the exponential map for the Lie group may not be surjective. In an earlier work which makes use of the Lie algebra to construct the kernels, when the exponential map is not surjective, the log map is only applicable around the identity in the lie algebra, and is thus restrictive. Furthermore, this method has exponential memory complexity in the group dimension. The focus of the paper is two such groups: $GL^+(n,\\mathbb{R})$ and $SL(n,\\mathbb{R})$. The key idea is to decompose these groups into subgroups and submanifolds where sampling the group elements is easier and well defined. Experiments on small standard datasets shows improvements over earlier methods.", "questions": "No additional questions.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. The paper is mathematically well-grounded. \n\n2. The contributions in the paper clearly improve on existing works in more than one way: applicable to groups like $GL^+(n,\\mathbb{R})$ and $SL(n,\\mathbb{R})$\n\n3. The ideas of group decomposition as a way to build equivariance to such groups is an interesting and general idea, and will hopefully be useful in other cases. \n\n4. Experimental results show improvements over state-of-the-art methods.", "weaknesses": "1. The number of samples for Monte Carlo integration is not clear to me. When the authors say 10 samples are used for the affine group, do they mean 10 samples for both rotations and scales combined? How many rotations and scales?\n\n2. Although clear theoretically, experiments don't seem to have tested the settings that were setting this apart from the others, particularly, by MacDonald et al. 2022. Perhaps at least one experiment with n=3 with larger scale changes as part of the test set could be interesting. Even a different version of affNIST with larger scale changes could be interesting. \n\n3. Having numerical values of equivariance/invariance error, and as a function of the number of Monte Carlo samples should also be useful here.\n\n3. I don't fully understand why the proposed group decomposition technique is able to estimate the correlation integral well with only 10 samples while the work by MacDonald et al. need 100.", "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": "GKKqeneTsz", "reviewer": "Reviewer_kmML", "summary": "This paper introduces a way to build group convolutional (Gconv) layers for groups in which the exponential map from the Lie algebra to the group is not surjective (e.g. $GL(n,mathbb{R})$). The problem is that Gconv is an integral over the group, which is often approximated by discrete sampling done using the exp map. When the exp map is not surjective, we can't rely on it to get good coverage in the sampling.\nTo mitigate this, this paper suggests to find a decomposition of the group in the form $G=PK$ such that the integration (Haar) measure over the group also factorizes $d\\mu_G = d\\mu_P d\\mu_K \\delta(\\cdot)$. In certain cases, such as the Affine group, this decomposition can yield familiar subgroups for which we have either a surjective exp (e.g. $SO(n)$) or efficient sampling strategies (e.g. diffeomorphic to $\\mathbb{R}^k$). \nAfter deriving the theory behind such decomposition, they introduce a Monte Carlo sampling scheme for it. Then they conduct a couple of toy experiments to show potential benefit of their design.", "questions": "1. How are methods relying on irreps affected by the expm not being surjective? Does it matter in those cases? \n2. For the Haar measures to factorize to $\\mu_K \\otimes \\mu_P$, do we need $G=PK$ or semidirect or direct product? A number of different decompositions are used in 4.1 and 4.2 and the requirements are not fully clear. \n3. I see that $G=PK$ can be done when $P$ or $K$ is a normal subgroup. In the $SL(n)$ to $\\mathrm{SPos} \\times SO(n)$, is $SO(n)$, is one of the factors also a normal subgroup? \n4. Even when expm is not surjective, the product of multiple $g_i = \\mathrm{expm}(X_i)$ can produce any arbitrary group element, as I understand it. Doesn't this mean that multi-layer Gconv using expm should in principle be sufficient to get good sample coverage on the group? In other words, is your decomposition really necessary?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. Mathematical rigor: goes through the details of the structure of Gconv to pinpoint the issue arising from non-surjective expm. \n2. Has extensive derivations, appendices and concrete calculations of the integration measures, rarely stated in other works. \n3. The Monte Carlo sampling scheme seems to work well with few samples and may alleviate memory issues of others. \n4. The experiments, though limited, are useful.", "weaknesses": "1. Mathematical rigor: very dense, difficult to read, too much re-derivation of previous results. A good part of sec. 3, Background, in __continuous group equivariance__ and some of __Lie algebra parametrization__ can be cut and moved to the appendix. Sec. 4 can also be shortened, like derivations in __layer definition__ in 4.2. The density of the math makes the paper hard to read. \n2. The paper calls $SO(n)$ abelian, which is incorrect for $n>2$ (page 4, sec 4).\n3. Other literature: Many Gconv approaches do not sample the group this way (ses question): Clebsch-Gordon Nets (Kondor et al NeurIPS 2018), Lorentz group (Bogatskiy, ICML 2020), EMLP (Finzi 2021) or Lie algebra conv (Dehmamy, NeurIPS 2021).", "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": "AmI3Q01npc", "reviewer": "Reviewer_azQE", "summary": "This paper presents a method for constructing approximately Lie group equivariant neural nets. Like previous work, their method relies on Monte Carlo approximation of infinite integrals; however, they extend this methodology to Lie groups without surjective exponential maps. Their critical insight is to decompose certain Lie groups of practical interest into semidirect products, which simplifies the form of the Haar measure. They evaluate their method in modifications of MNIST and outperform baselines.", "questions": "1. My understanding was that non-compact groups do not have finite Haar measures (i.e. the measure of the entire group is infinite). I am therefore confused how e.g. equation 13, and really all the integrals, are well-defined without additional assumptions on the bounds of integration or the decay of the function being integrated.\n2. Could the authors elaborate on the primary advantages of this integration method over that of MacDonald et al 2022, which also handles the case of Lie groups with non-surjective exponential maps? The related work mentions memory requirements, but it is not clear to me how this method circumvents these issues (or what the precise issues are).\n3. Assuming the main advantage over MacDonald et al 2022 is in memory requirements, why then do the authors think that MacDonald et al had slightly worse performance in the experiments? \n4. The papers by Finzi et al and Bogatskiy et al, noted under “weaknesses” above, are not integration methods. Are they suited for the input and output representations that this task considers? If so, can the authors compare to these methods as well?\n5. At the top of page 8, the authors note that they sample the translation factor from $[-1,1]^n \\subset \\mathbb{R}^n$. How is this justified, when the integral is defined over all of $\\mathbb{R}^n$? Doesn’t this induce some error, which is compounded in subsequent layers?\nIs this construction universal? I.e. ignoring the Monte Carlo integral approximation error, can any continuous equivariant function be represented?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 2, "contribution": 3, "strengths": "The framework is quite general, in that it applies to Lie groups with non-surjective exponential maps. The experimental performance is strong relative to baselines, including other work that handles groups with non-surjective exponential maps. The presentation of the underlying math is quite detailed, and is a good resource on its own (especially the appendix). The decomposition of a Lie group into groups via the semidirect product is novel in a machine learning context, as far as I know.", "weaknesses": "1. From what I can tell, the input and output spaces/representations are restricted to either the group itself under the regular or trivial representation, or a homogeneous space of the group. This does not seem as general as e.g. the two papers I mention in point (3), which can work with any finite-dimensional representation.\n\n2. The advantage of this approach over MacDonald et al 2022 is not made sufficiently clear, since both approaches can also handle Lie groups with non-surjective exponential maps. \n3. The related work is missing at least two important papers on Lie group equivariance: “A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups” by Finzi et al 2021, and “Lorentz Group Equivariant Neural Network for Particle Physics” by Bogatskiy et al 2020. It would be important to compare to these papers (both in theory, in terms of the Lie groups and input/output representations to which they apply, and in the experiments), since they are exactly equivariant. \n4. To a similar point, this method is only approximately equivariant (due to the sampling in the integral computation), rather than exactly equivariant. This could be expressed more clearly in the introduction of the paper. It would also be helpful to have computational tests of equivariance, which assess to what degree the learned network is actually equivariant after integration error. \n5. The paper’s presentation could be significantly improved. In addition to typos, some of which are noted below, the writing leaves something to be desired overall. For example, the main body of the paper is very dense in mathematical background and details. It would be helpful to clearly and concisely express the main ideas of the paper earlier on and to give only the important intuitions, and to defer many such details to later in the paper or the appendix.\n6. Enforcing equivariance to Lie groups with non-surjective exponential maps could be much better motivated — applications are not discussed very often, and the datasets used are synthetic (e.g. affine transformations of MNIST). When does such equivariance arise in practice?\n\nHere are several typos I encountered while reading:\n* Page 1, abstract: backwards quotation mark on the word larger\n* Page 2, fourth line: backwards quotation mark on the word larger\n* Page 2, related work: “Recent proposal” —> “Recent proposals”\n* Page 2, bottom line: “maps can be defined AS k-channel”\n* Page 3, between equations 5 and 6: “ecompasses”\n* Page 4, in Lie algebra parametrization: “approxiamtion” and “encompasses recent proposalS”\n* Page 7 before Theorem 4.4: “to obtain GL(n,R)-sampleS”", "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": "7AZgaGMfpd", "reviewer": "Reviewer_gz8n", "summary": "This paper presents a framework for the development of equivariant (regular) neural networks, specifically tailored to matrix Lie groups that may not be compact or abelian. The authors explain a strategy for the development of equivariant layers by subdividing larger groups into smaller subgroups or subspaces. This partitioning facilitates efficient sampling of group elements, conducted in accordance with the Haar measure, by allowing independent sampling from these defined subsets. Grounded in the theory of Lie group structure and the geometry of their homogeneous spaces, the paper focuses its analytical scope primarily on the groups $G = GL^+(n, R)$ or $G = SL(n, R)$.", "questions": "See the previous section", "limitations": "", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. While (regular) group convolution for non-abelian and non-compact groups have been proposed using basis functions on the Lie algebra. The exponential map, however, is only diffeomorphic locally around 0. In addition, the paper also aims the address the issue of efficient sampling of group elements according to the Haar measure for regular group convolutions. In this aspect, the paper is novel.\n2. Preliminary experiments are conducted to verify the potential of the framework.", "weaknesses": "1. My main concern of the paper is its mathematical denseness. I would recommend the authors to focus on the main message, and defer unimportant details to the appendix. For example, the condition of part (2) of Theorem 4.1 is probably to general, and it is not helpful for understanding the paper.\n2. The notation of the paper is also sometimes confusing. The (larger) group $G$ is sometimes refer to the affine group $G = R^d\\rtimes H$ (mostly in Section 4). However, $G$ in Section 4 is mostly refer to the subgroup $H = GL^+(n, d)$.\n3. One of the issue the paper aims to address is the global non-surjectiveness of the exponential map. However, since most (group) convolutional filters are only defined locally (on a neighborhood of the identity), does the existence of local diffeomorphism already suffice?\n4. Page 5, line 4 after \"Factorizing the Haar measure\", for general subsets $P\\subset G$ and $K\\subset G$, what does it mean to have \"G-invariant\" Radom measure. Although the authors later say that $P$ will always be a $G$-homogeneous space (so the $G$-action on $P$ is essential an action on the quotient space), it would be much better to state this assumption in the beginning.\n5. On page 4, the authors say $H = SO(n)$ is abelian. Is this the case for $n\\ge 3$?\n6. The theory seems to apply only to $H = GL^+(n, R)$ or $H = SL(n, R)$. Does it apply to a general $H\\subset GL(n, R)$?\n7. Is the Lie bracket of the Lie algebra used anywhere for the construction?\n8. Can this construction be extended to representations beyond regular representations of $G$? For high-dimensional $G$, such as $E(n)$ with large $n$, sampling from $G$ will suffer from COD.", "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": "KEkIOKL5UY", "reviewer": "Reviewer_8ViQ", "summary": "The authors propose to use Lie groups and Lie algebra to develop networks that can be equivariant with respect to affine transformations. They provide the theoretical framework to work with Lie groups that surpasses some of the issues encountered due to their surjective nature. \\\nThe way they managed to enable invariant integration was achieved by decomposing the larger groups into subgroups and manifolds that can be treated individually. \\\n\nMain contributions are two-fold: \\\na) they managed to develop a non-surjective method of the exponential map \\\nb) they achieved invariant integration with respect to the Haar measure in a principled manner\\\n----\nMany thanks for you response", "questions": "1) To what extent have you based the novelty of your paper on achieving SOTA performance on Affnist? \\\n2) Is there a scope that your method can be used in tandem with CapsNets?\\\n3) Would a dataset such as SmallNorb be relevant in this context?\\", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "This is a robust and well-written paper that is mathematically sound (as far as I can tell given it is almost impossible to check everything in the absence of source code). The ablations studies are good, but am not sure whether the experiments on affnist are enough. Nevertheless affnist remains the gold standard - more or less - for evaluating such properties.", "weaknesses": "-- It does not matter too much to be frank, as SOTA is not everything, but please have a look at this paper which to my knowledge remains the SOTA on Affnist (figure 5 for instance)\nhttps://aaai.org/ojs/index.php/AAAI/article/download/5785/5641 \\\n\nWith 174K parameters (64,16,16,16,10) they have achieved 98.1% accuracy on Affnist when trained at 99.7% of MNIST accuracy.\nSo you might want to update that, considering the number of parameters entailed too. \\\n\n-- I do not think the paper needs to be that heavy in maths - too many derivations that could have given space to further experiments and ablations. \\\n-- I would expand the related work, and flesh out the contributions in the introduction.", "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)