AdS-GNN - a Conformally Equivariant Graph Neural Network
Authors: Maksim Zhdanov
Nabil Iqbal
Erik J Bekkers
Patrick Forré
Venue: ICLR 2026 Poster (2026)
Keywords: equivariance; conformal group; scale equivariance; ising model
Primary area: learning on graphs and other geometries & topologies
TL;DR: We suggest a framework for building conformal group equivariant models that are consistent under angle preserving transformations which include translations, rotations, reflections, scaling and special conformal transformation.

Abstract: Conformal symmetries, i.e.\ coordinate transformations that preserve angles, play a key role in many fields, including physics, mathematics, computer vision and (geometric) machine learning. Here we build a neural network that is equivariant under general conformal transformations. To achieve this, we lift data from flat Euclidean space to Anti de Sitter (AdS) space. This allows us to exploit a known correspondence between conformal transformations of flat space and isometric transformations on the Anti de Sitter space. We then build upon the fact that such isometric transformations have been extensively studied on general geometries in the geometric deep learning literature. In particular, we employ message-passing layers conditioned on the proper distance, yielding a computationally efficient framework. We validate our model on tasks from computer vision and statistical physics, demonstrating strong performance, improved generalization capacities, and the ability to extract conformal data such as scaling dimensions from the trained network.

Published as a conference paper at ICLR 2026
ADS-GNN- A CONFORMALLY EQUIVARIANT GRAPH NEURAL NETWORK
Maksim Zhdanov*
Nabil Igbal"
AMLab, University of Amsterdam m. ww.zaheeuuua....
Department of Mathematical Sciences, Durham University AMLab, University of Amsterdam
nabil. 1. iiba 1 @eweramaam.mmm acc.hham............mmm.........
Erik Bekkers
Patrick Forré
AMLab, University of Amsterdam j.bekeers@uva.cc
AMLab, Al4Science Lab, University of Amsterdam p. d. forre@uvaa.n
ABSTRACT
Conformal symmetries, i.e. coordinate transformations that preserve angles, play key role in many fields, including physics, mathematics, computer vision and (geometric) machine learning. Here we build a neural network that equivariant under general conformal transformations. To achieve this, lift data from flat Euclidean space to Anti de Sitter (AdS) space. This allows us to exploit known correspondence between conformal transformations of flat space and isometric transformations on the AdS space. We then build upon the fact that such isometric transformations have been extensively studied on general geometries in the geo- metric deep learning literature. We employ message- paasaggg layers conditioned on the proper distance, yielding computationally efficient framework. We validate our model on tasks from computer vision and statistical physics, demonstrating strong performance, improved generalization capacities, and the ability to extract conformal data such as scaling dimensions from the trained network.
INTRODUCTION
The notion of symmetry is key tool both in our understanding of nature and for the construction of machine learning systems that perceive nature. The construction of equivariant neural network archi- tectures that encode specific symmetries has powerful advantages both conceptual and computational. In particular, much work has been dedicated to building networks that equivariant under familiar symmetries such as rotations and translations.
In this work, will study the symmetry group of conformal transformations i.c. the set of trans- formations on I2d that preserve angles. This in- cludes translations, rotations, reflections, scalings and so called special conformal transformations. Scale transformations i.e. rigid rescalings of the whole system - are an extremely important subgroup of the conformal group. Conformal and scale invariance play a central role in many diverse fields. To give some examples: biolog- ical visual systems seem to exhibit insensitiy- ity to scale (Logothetis 2020). Physical systems undergoing a second- order phase transition have fluctuations all scales; they are generally conformally invariant the critical point (Cardy) (996). and can use- fully be described by conformal field theories
label: 0
X;, Xs. hu
AdS Embedding & (h., h1, ((X..X)) AdS-GNN Message
boundary
Conf. transf-n
al. 1995: Han al.,
Figure 1: AdS-GNN lifts points from Euclidean space to Anti de Sitter space and computes mes- sage passing conditioned the proper distance.
"Equal contribution.

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Published as a conference paper at ICLR 2026
(Di Francesco al. 1997). Indeed in physics most systems exhibiting scale invariance also exhibit conformal invariance "for free" (Polchinski 1988 Nakayamal 2013). Diverse applications also exist
in computational geometry and computer vision (see e.g. (Sharon & Mumford 2006 Lei al.
)023)
In this work, construct neural network that is equivariant under conformal transformations, motivated by applications to computer vision and to physical systems. Our approach acts on point clouds and lifts the data into an auxiliary higher dimensional space called Anti de Sitter (AdS) space. As outlined below, this approach is inspired by ideas in conformal field theory in theoretical physics. We validate our construction on some tasks drawn from computer vision and statistical physics.
PREVIOUS WORK
Construction of equivariant neural networks under general symmetry groups began with (Cohen &
Welling (0166) and has since evolved into a well-studied field (see, e.g., (Bronstein et al. 2021 Weiler ct al. [023) for reviews). Advancements include treatments of isometrie transformations on general Riemannian manifolds (Weiler et al. (023) and extensions to comi-Riemannial manifolds (Zhdanoy
ct al. 2023 2024). The significant benefits of incorporating equivariance, particularly for point cloud data and even at large computational scales, are increasingly demonstrated (Vadgama et al., 2025 Brehmer et al., 2024). highlighting advantages in performance and efficiency. Lie group theory has been employed to develop equivariant networks for broad classes of transformations, such as general affine transformations (Mironenco Forré 2024), or to leverage specific algebraic structures like adjoint actions on Lie algebras (Lin et al., 2024). However, despite progress in handling Euclidean symmetries, scale equivariance, or even these more general Lic groups, these methods do not address the full conformal group, which uniquely includes non-affine special conformal transformations.
The pursuit of scale equivariance, a critical component of conformal symmetry, is the closest related area to our work full conformal equivariance, with various approaches developed from specialized convolutional architectures (Bekkers) 2020 Sosnovik et al. 2019) to techniques like Fourier layers for robust scale handling (Rahman & Yeh, 2023). Our of Ads space connects conceptually to scale space theories (Witkin| 1987] Worrall & Welling, 2019), where the extra dimension corresponds
to scale. However, a crucial distinction that the geometry of AdS space enforces equivariance under the larger group of all conformal transformations by construction, not just scale or isometric transformations. This approach also finds resonance in physics with recent explorations of using neural networks to model aspects of conformal field theory (Halverson et al. 2024).
3 CONFORMAL SYMMETRY AND ANTI DE SITTER SPACE
3.1 CONFORMAL TRANSFORMATIONS
detailed and self-contained account of conformal transformations is given in Appendix [D] here we present a brief review. Formally, global conformal transformation of the Euclidean space IR is an injective smooth map p Rd { + IR,, x + sp(x), defined on id except on a possible point Te such that local angles are preserved, i.e. for all x E Rdd \ {a.p} and v1, 02 E Rd (0) we require
(()))1''' (z))2) (x)mll (x)02l
(v1,02)
cos(<(s' (x)e), o (x) - cos(<(e1, 02)) | [0a]]
where (,,-) and ||.I| denote the standard Euclidean scalar product and norm, resp., and v'(x)) the Jacobian matrix of & at x. Note, that, in contrast to isometric transformations, we do not require that the distances/norms are preserved.
The group of all global conformal transformations of Rd is denoted by Conf, (RR)). We also define the (restricted) conformal group Con((TRd) as the connected component of the identity of Conf, (R.d).
By allowing &o to not be defined on certain point T. E Rd we effectively allow yo to map x. to the "points infinity" in the conformal compactification gd of Rd. In fact, every global conformal transformation of Rd uniquely extends to an angle-presereee
diffcomorphism dd for d 2 2, see (Schottenloher 2008 Thm. 2.612.11, or D.4.2 and D.4.3) This is why we introduce the definition with J. in this way.
2 An equivalent, but more general and abstract, definition provided in Definition D. and Lemma D.1.1
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It is shown that for d 2 2 the global conformal group Conf Confg(RR)) is isomorphic the projective orthogonal group:
where the role of the quotient is described in Appendix Din Proposition D.4.1
is instructive to consider the action of the group on points in terms of separated parameters. In particular, general element G of the group can be written in terms of parameters (A, t, b, M) e R x Rd x Rd x ((d) and acts on a point x E R through the composition of maps:
x xt
x'
Mx
x² Ax
b
(2)
2 1
||=01 2
4
translations
rotations
scalings
special conformal transformations
resulting in a transformed point = =x. These (d + [) + 2)/2 parameters then assemble into an element of O(d + 1,,, see e.g. (Di Francesco al. 1997) for review.
Above we have explained how points x E IR transform under the global conformal group. In typical application we will often be dealing with conformal fields ((z) defined on this space. The transformation of these fields is governed by the representation theory of the global conformal group" which is non-trivial and we do not review here, except to state that privileged role is played by basis of fields called conformal primaries, which we denote by 0(r). Associated with each primary field number Do called the conformal dimension which characterizes the field in question Conformal primaries are distinguished by the fact that they transform simply as possible under the group, and in particular under scaling of the coordinate they transform by multiplicative factor:
Example. A simple example is the electric field in ordinary electrodynamics; the usual inverse square $
law near a point charge E(x) ~ 1a/2 is consistent with equation if Ag = 2. Another example is image data viewed scalar field p(z) denoting the pixel value; under a conformal transformation only the argument of the field changes and not its value, and thus we would take Apixel O.
In physical applications the values of Ag for a given conformally invariant system - e.g. a particular kind of phase transition are often of great interes they are pure numbers that are universal: they generally do not depend on the microscopic details of the system, only on the symmetries and the spatial dimensionality.
3.2 THE ANTI DE SITTER SPACE AdSo+1
To organize the data, we will instead lift it from IR to an auxiliary space with one higher dimension, ic. Anti de Sitter space AdSd+1 This space can conveniently be understood in terms of submanifold of Rd+1.1, equipped with its natural metric
1 diag( 11, 1... 1)
Consider the submanifold given by the constraint || -1, where Y E gd+1,1 As shown in Figure this submanifold has two connected components, each of which is defined be a copy of
see Theorem D.4.2 Appendix D Proposition D.4.1 and (Schottenloher 2008 Thm. 2.6-2.1 11 for details.
*Elementary reviews be found in (Simmous Duftin 2017 Di Francesco al. 1997).
'It may be helpful to consider fields transtorming under the rotation group SO(3): there are scalar fields,
spinor fields, vector fields, etc. which are characterized by discrete parameter called the spin s which takes discrete values
etc. A very loosely be thought of as the analogue of "spin" for scale transformations. The fact that is now continuous variable related the fact that the conformal group is non-compact.
& L.e. Sscalar O, Svector
'eee e.g. the 1982 Nobel Prize lecture by Wilson 1983 for historical overview, (Cardy 1996 Kardar
2007 Zinn-Justin 202T for textbook treatments.
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Ads +1 For concreteness, we will present all formulas working with the "top" branch, i... the one which has y > O, and we define:
Ads = (Y E Rd+11 | || = -1, -1, 0}.
(5)
There is a natural action of PO + 1.1) = O(d + 1, 1,1)/(+ [+1] on this submanifold, given by 0(d+1,1) rota- tions on Y on each connected component. This forms the isometry group Isom Ads4+1) of Ads see (McKay 2023) Theorem 7.5 or D.2.6 As mentioned before, note that PO + 1) is also precisely the global conformal group Conf, (R.) of TR. This not an accident: the fact that local operations in the interior Ads result in conformally invariant operations on R. is very well known in the context of the AdS/CFT correspondence in quantum gravity (Maldacena 1999 Gubser et al., 1998; Witten 1998). which states that under some circumstances quan- tum gravity theory on Ads is equivalent to a quantum field theory with conformal symmetry on IR.. Also, see Remark
Y
Y2
Y
Figure 2:
Example of hyperboloid | = -1 embedded in R2211 consti- tuting two copies of Ads?.
Here we will not use any of the dynamics of quantum gravity, but we will exploit some of the well-studied kinematics of that correspondence as a convenient tool to build convolutional kernels on Ads4+1 Indeed, general framework for constructing convolutional layers that are equivariant under isometry groups of any pseudo-Riemanne Riemannian manifold was given in (Weiler et al. 2023 Zhdanov al. 2024). and our work can be viewed as building on special case of that.
To be more precise, we first place explicit coordinates X = (X) (x Xd+1) on Ads It will be convenient to separate these coordinates as (X) (X'...., Xdii)= (a',... x² z) = (x, 2) R x 1R where E Rd and z E 2 is an extra dimension. We can now solve the hyperboloid constraint equation in terms of these coordinates as
2 + || +
22 +
|
YO
y
yd+1
(6)
2z
2z
The Riemannian metric on Adsd+1 is the induced metric on this hyperboloid
ds²
(X)dXXXX 22
Σ () + de²
Finally, we record how the isometry group acts on AdS as X = GX. Using the same parameters as in equation? 2 we have:
(x', (x)
(4',2') (Mx,2)
(I',2') Az),
(8)
x'
z' + 2/2 (1=])²
+ b,
11'1 ||
||-||1 + +
|| + 22
A key point is that the manifold has a d-dimensional boundary at = 0. This boundary is mapped itself under the isometries. Furthermore, the isometry group acts on the boundary points (z 2 0) precisely as in (2). Thus one should imagine that conformal data on IR "lives on the boundary of Ads In what follows key role will be played by the PO(d 1, --invarian proper distance between D(X, between two points in AdSd+1 which is related to the absolute inner product Y") and is given explicitly by:
cosh D(X, == KY, 22 + 22 + Dd (x
(10)
2zz'
7 For a more abstract and general definition AdS for general quadratic spaces we refer to Definition D.2.3 and the corresponding (partial) paramoterization presented here to Appendix D.6 Forr the details, Proposition 1
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4 ADS-GNN
We will now describe how to use these ideas to formulate a conformally equivariant neural network by extending the data from the boundary into the bulk of Ads
Algorithm 1 AdS Embedding
1=] C Tid hin E N, 20 E Require: X = [x;
1: for each point E (1,...,N) do
0.06
2: 24 f- 20
3: neighbors, < KNN(,;; X, k(tt
4: (iz, 21 & M
0.00
5: end for
6: return {(1;, 2 C AdS
0.4
0.0
0.4
0.4
4.1 EMBEDDING POINTS IN AdS
Consider point cloud N points in Rd, {0,). E (1,..., We would now like to lift this data into the bulk of Adsd+1 in manner that preserves the symmetries.
A first attempt from the correspondence of symmetries shown in (8) is to simply embed each of the
points directly into the boundary z = 0, i.e. with X = (xf = D). However the metric (7) has singularity at = 0 e.g. from TT) note that each point at = 0 is at infinite proper AdS distance from any points at = > 0 0-aan and thus such an attempt will require us to pick regulating value z. Using fixed constant will explicitly break the symmetries.
We instead determine a value 2; for each point using an approach outlined in Algorithm I embed each point into AdS using
We first
X = (zf = 2p)
(11)
with small regulator zg- For each point, then compute the AdS center of mass X (a,, 2,) of
its klin. nearest neighbours. The AdS center of mass is generalization to hyperbolic space of the familiar notion of the center of mass from flat Fuclidean space. approach due to (Galperin 1993) which we review in Appendix A.2
be computed easily using
The geometry of AdS implies that the center of mass will generally be deeper inside than the original points. Importantly the = value of the centroid now has finite limit as zu - 0, in which case depends on the (appropriately averaged) relative separation of the points. We then perform a final embedding of the point using this z value, i.e.
X = (1
(12)
Intuitively, the z coordinate corresponds the length scale of the degrees of freedom of system 9 Our choice above amounts to saying that the appropriate length scale for point is related to its distance from its neighbours. This exactly preserves scale invariance, but it gently breaks special conformal transformations. This is expected physical grounds, as generally any choice of regulator necessarily breaks conformal invariance (Cardy 1996). In experiments, we check generalization under special conformal transformations empirically and verify that the breaking is mild.
Proposition 4.1.1. The lifting procedure (i.e. the map from x Bdd X" E Adsd+1 described in equation (2) is equivariant under the subgroup of the conformal group generated by translations, rotations and dilatations.
This is proved in Appendix Finally, have discussed lifting the points v;. The input data may sample
also have some features h Iuput associated each point z;. They should be interpreted of an underlying conformal field O(z) with dimension A, where h iipput = O(z,). This boundary feature should be contrasted with the feature associated with a bulk point in Adsd-1, this is scalar
9This is familiar from the physics of the AdS/CFT correspondence, where it is well-understood that the infrared physics lives deeper the hulk (Susskind & Witten 1998).
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and does not transform with an associated factor of X in equation 3
Said differently, the full
dependence of bulk features under scaling arises from its dependence on an extra coordinate 2. This difference in representation is taken into account by lifting the feature follows
lifted h;
23 h;
input
(13)
For many cases (e.g. image data) the input feature will have A = 0 and this step may be skipped. This relation has an analogue in terms of bulk-to-boote -ob-odddrrrry propagators from the AdS/CFT correspondence. where such factors of A relate physics the bulk (i.e. h, hjfittd) to that of the boundary (i.e. h; imput 1o
Given this set of points {Xi} in AdS, now operate it using graph neural network.
4.2 INVARIANT MESSAGE PASSING
We begin by studying scalar features. To orient ourselves, recall first earlier model, that of E(n) Equívariant Graph Neural Networks (EGNNs) (Satorras et al. 2021 Liu et al.] 2024). These graph neural networks that equivariant flat-space rotations, translations, reflections and permutations. The input to the model is a graph G (V.E) E) whose vertices are embedded into Euclidean space 12d. We denote the position of node U, as P, IR and its latent D-ddimensional feature vector of node U/ as h; The 1-th layer of EGNN is then defined as
m,j dh (b by II: -Pyl12),
EGNN message
(14)
h [+1 = tha (h. m.), m, Σ (EN()
myj
aggregate + update
where here (i)) represents the set of neighbours of node u. the tha are message and update MLPs, which we see depend only the pairwise distance in Id between nodes.
We adopt the model above to operate on AdS where a graph G is embedded. As above, each node i; has a position X; E AdS and a latent feature vector h,. If edges & are not provided in the graph, we induce connectivity with kuon nearest neighbours using the AdS proper distance (10).
In the message function (14). we also use the AdS proper distance instead of the Euclidean one, i.e.
mij = (h h/,, X,),,
AdS-GNN message
(15)
which yields an efficient conformal group equivariant GNN without substantial computational overhead compared to its Euclidean counterpart. Note that by making use of the AdS proper distance, we introduce a notion of locality both in ordinary space and in scale (as represented by the z-coordinate). Although the embedding of the point cloud mildly breaks special conformal transformations, the graph neural network itself exactly invariant under all of (Rd)
The message function in Eq. 15 only depends on distance and therefore is restricted to invariant features. To enable more expressive equivariance, we want to exploit the fact that Ads is (Rn lequivariaayyy embedded sub-manifold of Rd-1111 as discussed Section A. This allows us employ O(d + 1, --eeuiivariant neural networks from (Ruhe et al. 2023), which operate on multivectors, ie. elements of the Clifford Algebra CI(Rd+1,1) This yields the following message structure:
Mi = (H H(,,,,,
AdS-CEGNN message
(16)
where M,,, H, E CI(Rd+1.1) X; Adsd+1 ç Rdd1,.1
We summarize the equivariance properties of both networks:
Proposition 4.2.1. After the lifting procedure described in equation T2). AdS-GNN is equivariant the global conformal group: Conf, & ((R=) PO(d 1,,,, and, conformal group: (RR) PO" (d 1, 1), the identity component of Confg (IR see D.5.1
to the restricted
The framework above will result in features h1 that manifestly invariant under the conformal group. If performing an invariant task (e.g. classification) we can now aggregate the information in a permutation-invant manner by summing over nodes usual the final layer. On the other hand,
Soeee e.g. Section 2.5 of Witten 1998) for more details.

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Table 1: Classification error on SuperPixel MNIST. We aug- ment the test set with random rotations and scaling. X indi- cates random-guen - guess performance. Baseline results are taken from (Bekkers et 2024).
Figure 3: Test emor on augmented data, SuperPixel
MNIST.
Test Ervor (%)
100
0000
Error rate, % rotated
Model
EGNN
non-augmented
rotated+ssalld
50
AdS GNN
MONET
8.89
X
X
SplineCNN
4.78
X
X
GCGP
4.2
X
X
0.01
0.10
1.00
10.00
GAT
3.81
X
X
Scaling factor
PNCNN
1.24 1.12
X
X
Tost Ervor (%)
100-
PONITA
0.11
1.17
X
EGNN
4.17
4.17
X
50-
AdS-GNN (Ours)
4.09 + 0.27
4.09
4.09
0-
Special conf. transf-n,
for regression task we should specify the transformation properties of the output: e.g. if the output from the network is a conformal field (x,,) living on the boundary it must transform under scale transformations with specified A as in equation 3 This can be accomplished by taking the output ((r)) be
O(a,) = - M
(17)
where f is the conformally invariant output from the node associated with æ; and 2; the coordinate of the embedding found equation 12 This relation is the inverse tranformation of equation 13 and guarantees that the output satisfies equation
Computational cost The computational cost of our approach decomposes into two components: the AdS embedding procedure (Algorithm 1) and applying the graph neural network. In the former, employ k-nearest neighbors which has O(N log N) complexity. An MPNN scales linearly with the number of nodes N in the graph. We note that in the when graph is available, potentially can use during the lifting, thus alleviating the overhead of k-nearest neighbors.
5 EXPERIMENTAL RESULTS
We test the framework above on tasks that are loosely divided into two types: computer vision tasks and applications from physics. For most of the tasks we use the scalar AdS-GNN.
5.1 COMPUTER VISION TASKS
SuperPixel MNIST We benchmark AdS-GNN on the super-pixel MNIST dataset (Monti et al. 2017), which consists of 2D point clouds of MNIST digits segmented into 75 superpixels. Results are given in Table For in-distribution data, AdS G-NN performs on par with its antieqquivariant counterparts. It does however fall slightly behind PONITA, which has orientational information; we feel this happens as AdS-GNN is unable to handle orientation and relies on invariant descriptors. We also study the response EGNN and find that AdS-GNN has much stronger generalization capacities. As expected, AdS-GNI is precisely scale-invariant. For special conformal transformations, there is a small breaking of symmetry arising from the uplift. We empirically measure the effect augmented performance in Figure for a special conformal transformation as in parametrized by (0, bz) and verify that it is very small.
model various augmentations (see Fig. 3) for this we compare to
Shape segmentation We further benchmark against a shape recognition dataset, where we sample points from a selection of randomly sized and rotated shapes (either a square, a circle, or a triangle). The task is to assign shape class to each point in the cloud. We benchmark against EGNN and message- message-passing neural network MPNN which conditions messages against æ; Iý (and thus has only translational equivariance, not rotational). Here, shown in the left panel of Figure 4 we find that AdS-GNN outperforms EGNN even on in-distribution performance. We believe that this happens the training data contains structure at different sizes to which AdS-GNN adapts more efficiently
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Shapes
Ising correlations
Ising correlations
0.0
MPNN
EGMN
Test Accuracy
0.8
Relative L2
0.4
AdS GNN (Ours)
Relatiee L2
0.3
0.7
Example
0.2
0..6
0.5
0.2
0.1
10m
0.4
1024
8192 32768
51
S12
2048
8192
Number of Training Points
Number of Training Points
System Size
Figure 4: Performance on shape segmentation and the Ising task. Left, test accuracy of shape segmentation function of the number of training points. Middle, relative L2 as a function of the number of training points, with system size fixed at N = 16. Right, loss as a function of system size with 8192 training points. Inset shows N 2.
than EGNN, particular when there is small number of training points. Both of them fall behind MPNN, which we again attribute to their lack of orientational information.
A further segmentation task VOC is discussed in the Appendix, where we find that AdS- UIS- is essentially the same as EGNN.
5.2 PHYSICS TASKS
2d Ising We next consider a task from statistical physics, that of predicting N-point correlation functions in the 2d Ising model. As we review in more detail in Appendix C this is simple model of magnetism with a tunable temperature. As one increases it the model undergoes a phase transition, and at this point fluctuations of the magnetization exhibit conformal symmetry and are precisely understood in terms of a continuum model called the 2d Ising conformal field theory. This model is exactly solvable and has two non-trivial types of conformal field, the spin (Ir) and the energy e ezz), which have conformal dimensions A. = and A. 1 respectively
To benchmark AdS-GNN we consider the task of predicting the N-th moment of the energy and spin operators (E(z1) (xx)) ... <(xN)) as a function of their input points. These functions are known explicitly in 2d (see equation 44 and are rather complicated: they have erratic pattern of spikes when two points come close, on top of a more gentle modulation arising from the background of the other points.
We create a training dataset where we sample the coordinates of input points uniformly in [-2, for various values of N and conformal field theory results for the N-th moment as the training data. We simultaneously predict the spin and energy moments, using the sum of relative 1.2 losses. As the output is the N-point function of conformal fields with non-trivial A we use N copies of equation 17 to find our prediction Preda ((x.)) in each channel a to be
log(Pred,, ((x:))) = ((z.)) A. Σ log(2;)
(18)
where a E (a,c) and the A₂'s 's are trainable parameters which can be interpreted as the learned conformal dimension of the a and fields respectively.
We benchmark against EGNN and a baseline message-passing neural network (MPNN) whose messages are conditioned against x; - Tyj and which thus only has translational equivariance. The results are shown in Figure for the choice bus = 1. We see that the performance of AdS-GNN is superior in all regimes. We also note various useful features. Note that for 2 points (inset on right panel of Figure 4 AdSGNN performs more than an order of magnitude better than the others. This is because the form of the two-point function is completely fixed by conformal invariance to be proportional to [r] =22 -22, and thus AdS-GNN need only learn two numbers, whereas the others must learn to reconstruct the functional form of the power law.
There are other slightly non-local fields that we will discuss here.
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In-distribution
Out-of-iistribution Domain
Out-of-distribution System Size
10-
Ground Truth
lnggEnggg Cocrelation)
14-
GNN-
bog (nguegg Cocriation)
12-
bug (egEeny Corvtation)
12-
13-
EGNN
12-
MPNN
11-
-1
10-
30
-10
coordinate first node
coordinate first
coordinate of first node
Figure 5: Visualization of the output from various models on the Ising task; here all points are fixed except for the x coordinate of the first, which varied. Left, models tested on in-distribution data.
Center, testing on values of z which are far out of the training range, which is shown with dashed lines. Right, testing a model trained on 8 on a test set with N 16.
Interpretability; from equation 18] the learned values of the conformal dimensions A. and A may be read off at the end of training, as shown in Table 3 They are very close to the ground truth, showing the useful ability of this model to extract conformal dimensions from data.
3D Ising model
0.40
MPNN
0.35
EGNN
Final Test Loss
0.30
AdS. GNN (Ours)
0.25
Generalization: AdS-GNN generalizes better than EGNN in two different directions. As shown in Figure 5 it gen- eralizes well values of that outside the training data, as one might have expected from ocale-equivariance, AdS-GNN also generalizes well across N; as shown in an example in Figure 5| AdS-GNI trained with one value of N works well when asked to predict correlation functions with different N, suggesting that it has robustly under- stood the underlying physics. It always generalizes better than EGNN, particularly when the graph connectivity is more dense. This is discussed quantitatively in Appendix B.2
0.20
0.15
0.10
0.05
64
512
2048
6400
Number of Training Points
Figure 6: Relative L2 loss for the Ising 3D task. AdS-GNN also correctly re- covers the conformal dimension of spin A, 0.518.
3d Ising We perform similar analysis for the 3d Ising model. This model is not exactly solvable, and there are no simple formulas for the N-point functions to use for training data. Nevertheless it is possible to use modern conformal bootstrap methods (reviewed in Appendix C.2) to compute a good approximation in the case N = for the spin operator. We use this to build training dataset and then perform the same regression as above. Our results are shown in Figure 6 Just as the case of the 2d Ising model, AdS-GNN clearly has superior performance.
0.14
CEGNN (Ruhe et al.)
AdS-CEGNN (Ours)
0.12
Relative
0.10
0.06
0:04
32
1024
10240
N-body simulation In the last experiment, we employ neural network model for learning the dynamics of system of 10 charged particles. The particles interact through electric fields in three spatial dimension; the inverse-square law of the electric field means that this a conformally invariant problem in 3d. The acceleration is expected to be conformal vector field with A. = 2 (as it is determined by electric field, whose conformal dimension discussed below equation B).
Number of Training Points
Figure 7: Relative L2 loss for the N- body task. The L2 error is computed across long trajectories.
We forecast acceleration value a, R 10x3 for each body given only the position zy R 10x3 at the time step t. At training, the model is learned from random slices of short (T' = 4) training trajectories. During evaluation, we generate set of very long trajectories (2,000 time steps) and measure the

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relative L2 error between the ground-truth truth trajectories and those unrolled by the models. The task involves handling vector information, hence we use AdS- CEGNN. We benchmark against CEGNN (Ruhe et al. )023), which is SOTA at N-body simulation.
AdS-CEGNN outperforms CEGNN, as evident from the results in Fig 7 Furthermore, it indeed correctly recovers the conformal dimension associated with acceleration A.. = 2. Therefore, AdS- CEGNN effectively provides useful feedback when the value of the conformal dimension is known a priori: if at the end of the training A is correct, the model has processed the data correctly. This is in general not the case for deep learning models whose quality is based exclusively on performance.
6 CONCLUSION
In this paper, we introduced AdS-GNN and Ads AIS-CEGNN neural networks that are equivariant with respect to conformal transformations. We have demonstrated strong performance on various tasks, including computer vision and N-hody simulation. We found particularly interesting an application to the Ising model, where the model exhibited impressive generalization capacities and interpretabiliiy, in that the conformal dimensions A. important universal quantities could be extracted from the trained model. One might ask what other such universal information exists: general conformally invariant field theory in physics is defined in terms of these dimensions A. and set of 3-point coefficients cabe which turn out to determine higher-order moments of the conformal fields is interesting to ask whether this information could be be usefully extracted from the trained network. We anticipate further applications to critical phenomena and long-range interactions in physical systems.
Future Work Besides physics-related tasks, we see potential applications of AdS-GNN and AdS- CEGNN to computed vision and roboties tasks due to the inherent consistency of our models to local scale variations. That is, proposed models maintain consistent predictions when objects appear at varying scales within scenes, and do not require extensive multi-scale training data to achieve robustness.
ACKNOWLEDGMENTS
MZ was supported by Microsoft Research A14Science. This work was supported by grant from the Simons Foundation (PD-Pivot Fellow-00004147, NI and EB). is supported in part by the STFC under grant number ST/T000708/1. This publication is part of the project SIGN with file number VI. Vidi.233.22 of the research programme Vidi which is (partly) financed by the Dutch Research Council (NWO) under the grant httss//ab..or/1111616666/0////11
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A Ads DETAILS
We provide a few further details about aspects of Ads
A.1 THE GROUP ACTION ON AdS
Recall that Ads can be understood the hyperboloid defined in equation 5 The action of [A] E O(d + 1, 1/} then works concretely as follows
[A].Y sign sign((AY))) (AY)
(19)
where AY is just vector matrix multiplication and where the scalar multiplication with sign((AY)) corrects the sign of the 0-th entry (the component of (- signatuur) Together we get:
d+1
(([A].)))} E (((A)))))> 1, ([AJ.))> > 0,
(20)
and thus: [A].Y E Ads This gives us a well-defined group action of O(d 1,1)/(#1) on Ads4+1 and, in particular, a well-defined group action of (Rd)) on Adsd+1 Note, however, that this action does act linearly (since - acts trivially). This shows that we have now two different group actions: i) the linear action, given just by matrix multiplication, and ii) the sign-corrected action, given by Equation 19. It is subtle, but important point to notice that for every E O connected component of the identity 7, we have sign sign((AY))) elements A E we can canonically identify the (restricted) conformal group with the above:
(d++ + 1, 1), the whenever y > 0. So for those
these two actions agree. Note that -I é (d 1, see D.I.5 Since
(R. (O 1) (d+1,11)
(21)
we get well-defined linear action of (R on Adsd+1 meaning that the inclusion map:
j Adsa-1 C gd+1,1
(22)
is equivariant w.r.t. (R"'( considered as the subgroup Conf( = O (d+ O(d+1,1) c(d++11))
This implies that any O(d 1, 1.1e)-quiiveeee equivariant map (like CGENNs) &i Rd+1,1 - 7. induces
(R=)) map:
v o Ad5 + Rd+11 H.
(23)
This insight will be used to construct conformally equivariant graph neural networks (with multi- vector features).
A.2 THE CENTER OF MASS OF SET OF POINTS ON ADS
We will require expression for the "center of mass" C((X,)) of set of points AdS. This problem was solved in (Galperin) 1993); the basic idea is to view the hyperboloid as a submanifold of Rd+1,1 as above, use additivity properties there find a vector, and then find the intersection of the ray in the direction of that vector with the hyperboloid.
In practice, this is quite simple to implement. Denote the center of mass by Y and the set of N points for which we want the centroid by (xr) 2t). We would like to find the analogous coordinates for the centroid (5.9 2).
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We have that
y
1+
Σ z 2
1+ 1
1++
xfae))
(24)
2
82
NN
F
Y
(25)
2
NN
24
yd+1
2
1
1 g2
(1
Σ
F x²)
1 NN
Σ 2
xfaa)) )
1 1- 2/
1
11
Σ
(26)
2
The first equality is the definition of the embedding, the second is the definition of the centroid from Galperin. Here N a normalization constant which is picked to guarantee that
(Y
-
(27)
So to find the centroid, the easiest thing do is to compute the sums the right hand side of the second equality, which thus determines the vector Y up to an overall scale N; then we enforce the norm constraint above which lets us find V and thus fixes the vector Y completely. We then express the answer in useful coordinates by solving for (z, x²) through
y
T
(28)
yu yd+1
y0 yd+1
To get some intuition for the procedure, we study it in the case of two points X1 = (xf, e) and X2 (22,e) e) starting at the same value of the coordinate. We find
X2), (5(08 x2),
1 2 21
22 4e
(29)
i.e. we simply take the average of the spatial coordinates and move inwards in = by an amount which depends on the separation between the two points in the spatial direction. In this case the center of mass is actually the midpoint of the geodesic that connects the two points.
A.3 EQUIVARIANCE OF LIFTING PROCEDURE
Here we prove Proposition 4.1.1 i.e. that the lifting procedure is equivariant under translations, rotations and dilatations but not special conformal transformations. We denote general element of the conformal group by G. G acts points xę Rd as in equation 2
and points in X!" E Add
as in equation g
The lifting begins by embedding a set of points {x.} E Rdd into Ads using a small regulator value 2g in equation 11
(xf := (-f z0)
(30)
We then compute the AdS center of mass C((X,o}) of the set of points using the algorithm above. The question is whether this approach is equivariant in the bulk when the conformal transformation acts only on the boundary coordinates I²" f.e. is it true that
C( ((x)))(Gaa)) - GC((X,,(<;))
(31)
(Note that if the conformal transformation also acted on zn then this would certainly be equivariant; the question at hand is how much the choice of fixed regulator zo spoils the equivariance).
is immediately clear that the procedure is equivariant under translations and rotations, as in those cases we from equation g that the coordinate is left invariant under the transformation. Through direct computation using equation
2 and equation g we can further explicitly verify that the equality in equation 31 holds for dilatations but does not hold for special conformal transformations.
This may seem somewhat surprising, and we try to understand it more abstractly by studying the geometry of the embedded points X2 in the A coordinates defined in equation
Y. == 2 ||
X
x
yd+1
2
|
1
(32)
2zo
2zo
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As we take 20 0 while holding 2; fixed, can extract the dependence 2g by writing these (.ulll Explicitly we then have
points as y
1 Y
|
yo E.uill
Y Emull
yd+1 I.null
(33)
Here the points Y lie on the null cone defined by [| = O. where the norm is taken with respect to the Rd+1,. metric defined in equation 4 The points this cone longer depend explicitly 20. and furthermore the cone has natural Old 1, action on it given by regular matrix multiplication
on y as in equation 19
Naively, one would now expect that one could just ignore the overall normalizing factor and sum up the points in the cone when computing the sum in equation 24 As y >> 0 for all points this would lead to point in the interior of the null cone (i.e. in Ads itself) and the procedure now appears to make no reference to Ey and is manifestly equivariant under all of O(d+ 1,1.
Interestingly, this is not the case. It turns out that under x + Gr the points on the cone do not transform with the usual action of Odd + 1, 1); instead they acquire an overall scalar rescaling factor which depends on the parameters of G and the point in question, i.e.
GY G (af) o (G, x Y (Gz,)
(34)
The scalar factor preserves the cone || = 0, but it means that the sum in equation 24 no longer transforms equivariantly.
The existence of this factor can already be seen in the simplest case d = 1, where the embedding simply becomes
+ x2
Y
(35)
2%
2zo
We can now compute the prefactor a for various transformations by computing both sides of equa- tion 34 Under a dilatation x + Ar the overall prefactor (A, x) is simply a constant A-1 which can indeed be ignored when computing the center of mass; thus the lifting procedure is equivariant under dilatations. However under a special conformal transformation x + be+I the prefactor is instead (b, .) (1 6z)² and thus depends on the point in question. This spoils the equivariance under special conformal transformations.
Our discussion here is somewhat formal but the conclusion is expected on physical grounds, as in general it is difficult preserve full conformal invariance in the presence of a regulator such as 20.
A.4 VECTOR BO-BUNAYY MAPPING
Given an O(d + 1. 1) vector feature bulk expressed in the Y coordinates in the bulk we may need to map it back to the boundary to interpret it as a neural network output. This first done by using the Jacobian associated with equation to map it to an orthonormal basis in the (x", 2)) coordinates:
0Y
bulk A OX U
bulk
(36)
where we have used the Binstein summation convention, where X²= (x=2;)) =), have denoted and where el is the inverse vielbein which satisfies:
e7e 8pc
(37)
where yar is the AdS metric. In the coordinates used in equation this vielbein is simply el 28 Finally the analogue of equation 17
is
output
bulk
(38)
#
Working out the Jacobian explicitly we find that the spatial components of the vector are
output a
(ubulk + xz (ubulk bulk
un =bull))
(39)
where we stress that the components on the right-hand side are of the vector ubulk expressed in the y'A coordinates, is natural for the Clifford algebra construction.
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B FURTHER EXPERIMENTAL DETAILS
B. PASCALVOC-SP
In a further experiment, we also compare AdS-GNN to EGNN on the LRGB data (wwivedi et al. 2022), see 2 This is pixel segmentation task; thus the output data at each node is conformally invariant and we take A 0 in equation 17, i.c. we read out the output from h, directly, The difference in performance between EGNN AdS-GNN is statistically insignificant, which indicates that in this task conformal equivariance does not constrain the model significantly and still allows for high expressivity.
Table 2: Classification error PascalVOC-SP.
AdS-GNN Test F1 27.80 0.74 28.07 0.57
Model
EGNN
B.2 2D ISING GENERALIZABILITY
Here we discuss further the generalization properties of the 2d Ising task discussed in the main text. In particular, we provide information how model trained on given number of nodes Nirain performs when evaluated on a differrnt number of nodes Neest The connectivity on the graph on AdS controlled by a parameter kcous the number of nearest neighbours which are connected to each node.
System size
A, (energy)
A, (spin)
1.0000 0.0000
0.1250 = 0.000
4
0.9998 + 0.0000
0.1250 = 0.000
0.9924 0.0010
0.1248 = 0.000
12
0.9894 + 0.0032
0.1247 + 0.000
16
0.9893 = 0.0007
0.1247 = 0.000
Table 3: Learned value of A's from AdS-GNN. Ground truth values are A, = 1 and A, = 0.125. Statistical uncertainties are standard deviations from runs; for A, they are 0(10-5) and are not quoted.
For most of our results in the Ising section, we pick hçon performance. In that case note from Figures [a]and [b that AdS-GNN generalizes dramatically better than EGNN. EGNN's s performance is particularly bad when taking a model trained on a larger system and evaluating it smaller one.
so that the graph is fully connected. This results in the best in-distribution
However if we reduce Kcon e.g. to 2 then this greatly increases the generalization ability both models. In particular, the gap between EGNN and AdS AdS-GNN (as seen in Figures &c and [d is much narrower, though still present. We see however that this choice hurts in-distribution performance; AdS-GNN is able generalize reasonably well in the fully connected case.
B.3 IMPLEMENTATION DETAILS
In every experiment, we use the AdamW optimizer (Loshchilov & Hutter 2019) with a learning rate 10-3, 3, Every model is trained
single Nvidia RTX6000 GPU. All models implemented in JAX. All experiments are run 5 times with different seeds. Models are trained until convergence with early stopping. The number of layers is fixed to with hidden dimension 32.
SuperITxel-MNNIT The task is to predict digit given point cloud representation. We compare
against MONET (Monti et al., 2017). SplineCNN (Fey et al. 2018). GCCP (Walker & Glocker 2019) GAT /Velickovic al.) 2018). PNCNN (Finzi et al., [022]] and PONITA (Bekkers 2024).
Every model is trained with batch size 128, baseline results are taken from (Bekkers et al. 2024
Kumn is set to 16, kis to Training time is approximately 10 minutes.
PoScal-VOC
The task is to predict a semantic segmentation label for each superpixel node (total
of 21 classes). Each graph is embedded in 2D Euclidean space, each node is associated with 12 scalar features. We used the batch size 96. hinn = 16, kin 5. Training time is approximately 30
minutes.
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adegnn
1,0422
8.3132
01106
5
8:003
8:7143
0,300
8.5129
0.122
0.2399
0.1021
8:3133
OL177
Tixtard
Trstand
(a) AdS-GNN with full connectivity
(b) EGNN with full connectivity
adeggn
0U317
E,200
0,2817
8.775
8.3154
0,2230
CLIDO
System Ma
System
2319
6.129
01P12
012443
04277
01244
8.3113
0L1225
0.30
01100
Thxtaed
Tratand
(c) AdS-GNN with kicon 2
(d) EGNN with Rcop
Figure 8: Generalization across system size; each square shows the relative L2 loss system trained system of size Ntrain (æ-axis) when tested system of size Nust (y-axis) for different level of graph connectivity.
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Shapes Given point cloud, predict for each point each shape it belongs (circle, square, triangle. intersection). We train by minimizing cross-entrops loss. ksen is set to 16, kun = 16. The number of testing and validation points is set to 512, while the number of training points varies from 64 to 8192. Training time is approximately minutes.
Ising Given a set of points, we predict two scalar values, one for energy correlation, one for spin correlation. We relative 1.2 error our training objective. keon equal the number of points (i.e. fully connected system), Aii = 1. The number of testing and validation points is set to 512, while the number of training points varies from 64 32768. Training time is approximately 3 minutes.
C ISING MODEL REVIEW
Here, for completeness, we provide an elementary review of the Ising model - which can be imagined the simplest example of model of magnetism and the physics at its critical point. This material is standard; see e.g. (Kardar 2007 Di Francesco et al. 1997) for textbook treatments. We begin by discussing the model in 2d.
C.1 2D ISING MODEL.
The model defined terms of binary variables called spins o, +1 sitting on the sites of square lattice with L sites on a side. The model is defined in terms of an energy function:
B[o]
Σ 0 (5)
(40)
where the notation (ij) means that one sums over nearest-neighhour links connecting two adjacent sites i, j.j. We can see that the energy is minimized when spins on two adjacent sites have the same value, i.e. "spins want to align". The energy is also invariant under Zg symmetry which acts by flipping all of the spins, a; + -0.
This energy defines statistical physics model in which the probability of obtaining given spin configuration (a) is given by
Pa(a] = exp(-
(41)
where I is the inverse temperature and Z the usual normalizing constant. This model has two phases, which we now describe.
Consider first taking 3 very large; in that case any increase in energy will be heavily penalized, and the most likely configurations will be those that minimize the energy, i.e. where spins have the same value, either a; +1 o; -1 for all i. This is called phase with spontaneous symmetry breaking, as a choice of cither of these configurations breaks the Ze symmetry. It is also called the ordered ferromagnetic phase.
Now consider taking 3 very small; in that case the system is very disordered, and all spins fluctuate strongly and randomly, and there is sense in which symmetry is spontaneously broken. This is called the symmetry unbroken or disordered phase or the paramagnet.
In the L oo limit there is sharp distinction between the two phases. A quantitative way to understand it is to imagine taking the system size to infinity while computing e.g.
(a)
lim 100000
1 1
Bampp
E
Σo:
(42)
iee the expectation value of the spatial average of all the spins. This is called the order parameter. This is nonzero in the ordered phase (where all the spins are aligned, resulting in a net contribution to the expectation value) and zero in the disordered phase (where all the spins fluctuate strongly, resulting in a cancellation across sites). In the infinite- L limit there is a analltillttt in the function (a at a critical value of B B. at the phase transition point. For the 2d Ising model the location of this point is known be B. N log(1
22) 0.441.
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A great deal is known about this critical point. Here fluctuations of the spins take place over all scales, and do not decay exponentially with distance as one might normally expect. It is possible to capture the long-disting long-distance statistics of these fluctuations in continuum limit where we formally take the lattice spacing to zero. The resulting structure is called the 2d Ising conformal field theory (CFT), and is example of quantum field theory that exhibits conformal symmetry. In partieular, one can define two operators in this continuum theory: the spin operator ((x) (which is the continuum limit of the spin operator o, defined on discrete lattice sites above) and the energy operator e(x) (which can be thought of as product of spins at adjacent sites). The 2d Ising CFT is completely solved and thus one can compute any arbitrary moments of any of these operators in closed form.
To get some intuition, the correlation function of two spins behaves as:
(z())
2
(43)
where A, 1. The power-law functional form is completely fixed by scale-invariance, and the only input from the theory here is the value of the conformal dimension Ag. A similar relation holds for the energy operator with A 1.
8.
In this work we build neural network to predict the N-point correlation functions of the spin and energy operators as a function of the positions of the operator insertions. To do this we use as ground-truth training data the following closed-form formulas from the theory of the 2d Ising CFT (Di Francesco et al. 1997):
2
(<(21)((22) ... (..(E))))..(N Pf
(44)
24
2j
IeijeN
(z(2))o(2)) ... ((N()))
Σ
II
23 23
(45)
22N
where Pf denotes the matrix Pfaffian. Despite their compact presentation, these are rather complicated functions; e.g. combinatorially the Pfaffian can be viewed as sum over all possible perfect matchings of N points. The number of these matchings grows factorially in N, and each of them contributes a (different) product of 2-body interactions. There are more efficient ways than this to actually compute the Pfaffian, but our neural networks will unable to realize that structure.
C.2
3D ISING MODEL
The lattice construction equation 40
may easily be generalized 3d. Unlike the 2d Ising model, the 3d Ising model has not been explicitly solved, and there are no simple exact formulas such as equation (45 for moments of the spin operator the critical point. Nevertheless, great deal is known about it, where the most precise information comes from the conformal bootstrap (El-Showk et al., 2012), i.e. a method which exploits the self-consistency of conformal field theory solve the system. For reviews see (Simmons- Duffin 2017 Poland et al. 2019).
Using these ideas, the 4-point function of the spins is given by
g(u, v)
(1(=)))((22)0(()))((a)))) |0o 22 24. (xg x 24.
(46)
where the scaling dimension of the spin operator is A, = 0.5181489(10) (10). u, are conformally invariant functions of the four insertion points called cross-ratios, and are
2ig0is 213
V
Tij = I, Zj
(47)
The non-trivial information in this point function is stored in the dependence on the cross-ration
g(u, v). We compute this following (Rychkov al. (007)). which we review briefly below.
This function g(u, v) can be decomposed in the following fashion:
g(u, v) = Σ c (oo
(48)
O
where the sum over O runs over the conformal primaries of the theory, Ao and 4p are the conformal dimension and spin of the operators respectively, and t (u,v) is theory-independent function
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called conformal block. which These functions are the same for all theories, and thus the information characterizing this function is encoded in the set {Ao, lo. Cooal, where Co is set of theory- specific numbers called operator product expansion coefficients. For the 3d Ising model this set of numbers is not known analytically, but they have been computed for the operators with lowest dimension to extremely high accuracy.
Following the method in (Rychkov et al. 2017). and using the lowest five operators and their associated {Ac, lo, Coso) from (Komargodski & Simmons-Dufinn 2017). we compute gu, v) using the Mathematica code of (Costa et al. 2016) to find the conformal blocks. We then create dataset of 4-point functions precisely as for the 2d Ising model above. This is not an exact solution to the 3d Ising model, but the error from truncating the operator sum has been conservatively estimated to be one part in 10-2 in (Rychkov et al., 2017). (We also note that for benchmarking our results presumably any choice of g(u, v) would suffice, but we think it is more physically reasonable to use the function that relevant known physical problem).
Such methods would be much harder to implement for higher-point functions, and thus our test dataset for the 3d Ising model is restricted to the 4-point function.
D THE CONFORMAL GROUP OF THE PSEUDO-EUCLIDEAN SPACE
In this section we will provide a self-contained introduction to conformal geometry the (pseudo-) Euclidean space in arbitrary signature and its conformal group. We provide definitions, examples and main theorems.
We follow the references of (SSchottenloher (2008) and McKay 2023)
D.1 CONFORMAL TRANSFORMATIOONS OF THE PSEUDO-EUCLIDEAN SPACE
Roughly speaking, a conformal map is map that preserves local angles. However, this definition is rather tedious write down and work with. A technically more convenient definition introduced below in Definition D.1.2 To motivate this definition, we first show in Lemma D.I.I that in the (linear) Euclidean case these two definitions are equivalent:
Lemma D.1.1. Let A Rdxd be real invertible (d x d)-matrix. Then the following statements equivalent:
1. angle preserving, i.e. for all 01, 122 E IRd
{0} we have:
(Ar1, Auz)
(01, 02)
(49)
| AAu2ll
||
||22|
2. A is conformal matrix, ie. A cA with scalar >> and orthogonal matrix
A E (dd)
Proof. "(": Let A cA with e e> and A E (d). Then we get for vn, U2 E Rd:
(Au), Au2) = vj AT Au
(50)
= 2 .. v.
(51)
= T
(52)
= 2. (11, v2))
(53)
This also implies:
Av || = e- ||al|
| = c. -/2l
(54)
Together this implies the claim:
(Av1, Aua)
(v1, U2)
(55)
Av
||Auzll
|
[
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"="; Assume that A is invertible and preserves angles. Then C := A is symmetric and positive definite. By the spectral theorem we can diagonalize C, i.e. there exists an orthonormal basis el...., Cd Rd (e)rez) dzg for all i,j [d], and positive scalars X1 X1...... da 0 such that for
all i E [d]:
CeN X3 C2.
(56)
We now claim that: d1 .... Ay. By way of contradiction assume that there exists E [d] such that A1 de. Then put: U1 61 and 222 e1 Ec. This gives:
|| = 1,
|| 12 = ||eı|l + = 2,
(57)
||A 1|2 ====A A11
A Aur = \1 e1 + de ee,
(58)
(Av1, Au2) = X11
| 1| = A1 + A2.
(59)
(wi U2) = 1.
(60)
This implies:
(Av), Auz)
(0l, U2)
(61)
VA v + Squaring and solving for d shows:
|| Acil |
|0l |02
1 v2
Ae A1,
(62)
which contradicts the assumption. So, we indeed have that:
c² = \1
Aa 0.
(63)
This shows that: C 1²1, Putting A == 1A we get:
A A - C-
1
(64)
which shows that A E ((d) and thus A = cA with c > 0 and A E O(d).
Definition D.1.2 (Conformal maps/conformal transformations). Let (M, T and (N, be two Greudo-Riemanen manifolds. A conformal map f M N is defined to be a smooth map such that there exists a smooth map w M + Rso such that for all x E M and v1, U/ E TyM we have:
( (11) (v)) wxx)² n (V1, 82),
(65)
where dfz T. + Te is the differential of f at M. The above map w is called the conformal factor of f.
In case the conformal factor w of J equals the constant one, w((a) = 1, then we call an isometric map or isometric transformation.
Definition D.1.3 (Conformal diffeomorphisms and isometries). Let (M, be a pseudo- Riemannian manifold. A conformal diffeomorphism f M + M is conformal smooth map that has a conformal smooth inverse f M + M:
fo
(66)
If. in addition, its conformal factor factor of f is the contant 1, then we call f an isometry of (M, MT) The group of conformal diffeomorphisms of (M, 1²²) denoted as:
(M,
{ M M M conformal map with conformal inverse).
(67)
The isometry group of (M M) is denoted as:
Isom M == {f :MMM> M isometric map with isometric inverse).
(68)
Notation D.1.4. In the following we will denote by:
1. (TRP,, be the standard pseudo-Fuctidean space of signature (p, q):
P- U2) = Tt A P-922-
A P-9 diag( (1, +1,..,, +1, -
(69)
xP
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2. Opp, q) the ppseudo-Jorthogoon group of signature (p. q):
O(p, q) == (A e GL((P.9) A A = APA),
(70)
3. SO(p, q) the special (pseudo-Jorthooonn) group of signature (p,g):
SO q) == {A O(p. 9) det A = 13.
(71)
4. (p q) SO 9) the identity component (=connected component of the identity) of (p, q). Note that we have the inclusions of groups:
(p. S SO(p. 9) S O(p.g).
(72)
Lemma D.1.5 (The connected components of O(p. q)). Let p, 2 0. Then we have the following:
1. pf = = then O(p, q) = {0} has only one connected component.
2. !fp + q 2 and pq then O(p, q) has 2 connected components:
S0(p,)) g) (A E O(p. g) det A 1) = (p,9), {A E ((p.g) = -1}. (73)
3. p/ 2 then OOp, q) has 4 connected components:
A
B
0 O++(p.)) = A =
E O(p,9)
det 42 +1, det D 2 +1
(p).)), (74)
C
D
[
B
0+ (p.0) = A =
E OOp,.)
del 12 +1, del -1
(75)
C
D
A
B
0 (p.g) = A =
E O(p.4) 4)
det A -1, det D 2 +1
(76)
C
D
A
B
0
(p.0)
A
E (Op,.)) 4)
det -1, det D S -1
(77)
C
D
We have the following open-and-closed subgroups Of 9):
A
B
(+ A
E O(p.g)
det A +1
(78)
C
D
= (p. (2 0
(p.9).
(79)
O (p.9) (A= [A B
E O(p.g)
det 2 +1 21
(80)
C
D
= O h+ ( (p.q),
(81)
SO(p.9) 9) {A 0(p. det =
(82)
0000000 (p,9),
(83)
(p.p.) 0* (2.4)
(84)
= 0 -0(2
(85)
= 0 (500))))))
(86)
In particular, if 9 2 1 and p or q is odd number then -I $ (p.
Lemma D.1.6 (Polar representation). Let p.g 2 1. Then every A E O(p,4) uniquely be written as:
T 0
I + BBT BT
(87)
0
V
VI, VL+BBBB B
with unique U E O(p), V e O(g). E R2x,, Conversely, every such product of matrices lies in O(p. q).
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Example D.1.7 (Affine conformal diffeomorphisms). Ler RP-9, 0 A E O(p. q). Consider the affine map:
f 1:RP4 -> RP.9,
cAx b.
(88)
Then is a conformal diffeomorphism of RP-9 with constant factor w(x) c.
Proof. The differential at x: is given by the matrix:
d/z= = cA.
(89)
So for v1, 02 TgIPP 12-9 we get:
(P(d(a(r). = (cAv1) AAP((AAV2)
(90)
= =. A A
(91)
c² c. A
(92)
c (U1 U2).
(93)
So. if then define the conformal factor RM - R>o to be the constant map w(x) have shown that f is a conformal smooth map. Its inverse f is given by:
(x) c - c -b
(94)
which is thus of the same form f and thus also conformal smooth map. This shows the claim.
O
Theorem D.1.8 (Affine conformal diffcomorphisms, see Amir-Moéz (1967). Consider the affine map on IP4:;
f 12P.9 R24
f(x) := Ax b
(95)
with square matrix A and translation vector b RP.9 Then f is conformal map (w.r.. 178-9) iff there exists a E Rz0 and a A E O(p, q) such that A = cA. If this is the case, I is a conformal diffeomorphism and both e >> 0 and A E O(p.q) are uniquely determined by A as: = // A with d : + and, A = A.A
Proof. One direction is proven in Example D.1.7 For the other direction assume that f is a conformal map. Then we have dfz= = A and the conformal relation (in matrix form):
A A A w(x)² P
(96)
Taking determinants on both sides gives:
(det A) (2)x) > 0,
(97)
showing that A is invertible and that (xx) = s/det A =: e is not dependent on æ and thus equal to the constant > 0. Dividing the first equation on both sides by c² and rearranging gives:
LA
P.O [_A)
AP..
(98)
This shows that A := A E O(p, 4) and thus the claim: A = cA. The rest follows from Exam- pleD..17
Definition D.1.9 (The linear and affine conformal group). We define the linear and affine conformal group, resp., of signature (p. 9) as follows:
CO(p.g) q) := Ryo x O(p.9).
CE(p.q) := = q) x BP4
(99)
Note that the entries correspond to scaling factor > reflection-rotatio matrix A E O(p, q) and translation vector b TRP- in the conformal affine maps from Example D.I.7 and Theorem D.I.8
We also define their identity components:
CO (p 9)
Rz0 0 (p..g)
CF. CO%(p, RP..
(100)
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Example D.1.10 (The inversion the pseudo-sphere). The following (partial) map:
< R + IR
(1)(()))
2
(101)
1/p-0 (x,,)) x)
is called the inversion (at the pseudo-sphere of signature (p: q). Note that the inversion here is only defined for x E == {a E RM 1p Φ) + 0}. is an involution (self-invers) on as (with 17 ))
n(s(z),s(x)) = x[1
1
+ 0,
(102)
x1, (2r,
<(x)
s(s(x)) ((<((),((x))
2.
(103)
We now claim that is conformal diffeomorphism. For this we compute its Jacobian matrix (differential), which, in matrix form, is given as:
ds π< 2)2 (((,,)) 2xx T AP.)..
1
(104)
To check the conformal relation we thus compute:
(dsx) (d.9(ds=)= = x)+ (11l2,2)1 2xr T AP. T AP (n(x, x) - 2xx T AP-))
1
(105)
1
((n(xx))1 - -2P P x)(x,,)) - 2xx T AP.))
(106)
7)x,, 2)²²
(1(2,x))
4n () A A 4
A
(107)
1)( x)2 (n(a,,)))2 4n(x,x)AP9I1 T
AP. + An(xx, x)
AP:9)
(108)
1/x,, x) A
(109)
This shows that the inversion Ç - U is a conformal diffeomorphism with conformal factor w(x) [...]]] We continue the analysis in Example D.4.5
Example D.1.11 (Special conformal transformations). For b E RP-9 we define the (partial) map:
17(2, x)-
Tb : 12-99 + IRP.
(b(x) = 1 - 2 r)(,,b) + 1/ b) _)(,xx) (s(x)= b)
(110)
where < = <P. RR-.- + RP-9 denotes the inversion from Example D.1.10 of signature (p.g). Maps the form as are called special conformal transformations of T.. and are conformal maps on their domain of definition:
Un == {x R ((x,b) #0}
(111)
with conformal factor wx)
p( where we abbreviated the above denominator as:
ulr, b) = - 2. n1,,, b) + 17(b,b) b) (xxxx))
(112)
Furthermore, a U-D + Up is the inverse of ob Up + U-b
Proof. We first check that = (x(x)) lies in U-b for x E U.. For this we compute:
1a 17 x)b, x 711,, b) n({,zz) - 27(2, x (m b) + n(x, (b,,))
(113)
= -n(x,z)v(z, b).
(114)
11 -b) b-)(z,b) + 17 (b.
(115)
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Dividing both sides by p(x, b)2 and v(z, b), resp., shows:
m( 7xx,2) x)
mly b) u(r, (-n(xx)) +1(,,))(((,)) b)).
(116)
u(x, b)
With this get:
lly -) -
- 2 nyy, bb) + m)(b,b) b) n(y.v)
(117)
b) (u(x, b) + u(x,b, b) - 2 1,(1,, x) n)(b,b) + ((b,b) -((,)))))))))))))))
(118)
(u(x,))) b) + ((,,b) (b,b)) ,)(x,)))
(119)
((xbb)
(120)
(xx
# O.
(121)
This shows: y E U-6 for Ub. Plugging the relation vly. v(2,6) into Equation
116 shows
that:
o-v(y) = y + ny, y)
(122)
VI. -b)
1
V 7 v)
vly, -b) b
(123)
vly -b)
I ((=n(z,z)
v(x,b) b)
+ ( x)
(124)
b)
x.
(125)
This shows that a-6 : U-b + U3 is the inverse of σb Ub + U_b.
To show that σ Ub + R2-9 is a conformal map we need to compute its differential. Computing the differential directly gives:
(dor). v(z,b) (1 - 2 b. x T AP9) - (x - 71x,(( x) 6) (-2- 6 p(x, b)2
AP + 2 2.,((bb))
x A
(126)
which is difficult to work with. Instead, we use the fact that Ty is smooth extension of the map as(x) s (s(x) -b), b). With the chain rule we then get:
(doo)z (ds)(c(e) dsx
(127)
With this and Example D.1.10 we get:
(dor)) _ A A (dor)) = (dsx) ((()) PP (ds) dss
(128)
x
1
(dss)) T AP- dsr
(129)
(s(x)) - b, 5(x) - b)²
1
n),x))² (s(()) b, 5,x(z) - -)) PM
(130)
v(x, 6)2 Ano.
(131)
For the last step note that:
17 (<(x) - -b,((x) b) v(2,b) 170,, x)
(132)
So the conformal factor of σo at E Un is wb(x)= p(
We continue the analysis in Exam-
ple D.4.9
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Remark D.1.12 (Extending conformal maps to conformal compactification) The examples D.1.7 [...8] 1.11000.11111111111 ( have given us several conformal transformations of B.... However, for us to be able to define the inversion <: U IBP-9 and the special conformal transformations ob Ub - RR we had to restrict their domain of definition to an open subset of Re... The question is now if we can actually extend those maps like and o, to conformal map on the whole of TRe. by possibly enlarging their codomain, e.g. by putting:
if)(2,)) +
<: R299 IP U
<xx)
n(Z,a)
(133)
0O:
ifT(2,z)=0?
However, for this would need to turn the extended codomain TRP- U {oo} into proper pseudo- Riemannian manifold MP Furthermore, we then would also like to properly define ç on the added new points MP-9 X IR24 and turn it into a conformal diffeomorphism of MP.., Note that in the non-Euclidean case, i.e. if p.q > 0, the space {xr E [RP-9 n)æ,, x) = 0} consists of more than one point and thus needs more consideration.
The above is the main question (of the existence) of conformal compactification of RP.9, Below we follow an ad hoc approach to this question.
Remark D.1.13. Even after the question of conformal compactification. as described in Re- mark markDD1.12 is solved, we are still faced with the ambiguity of how define the conformal group of RR..9 There are several non- eenivaiuvalens options:
1. as the group (RF.))) of all conformal diffeomorphisms of R2-9, which would include all affine conformal transformations, but exclude the inversion and the special conformal transformations:
2. the group (MP-9) of all conformal diffeomorphisms of a conformal compactifi- cation MP of RP which would include all affine and special conformal transformations and the inversion. However, one can show that for p + q 2 some of the properties of the conformal compactification will break down and that ConfDiff (MP-9) can become pathologically big in the case (p, 9) (1,1), see Schottenloher (2008);
3. as the connected component of the identity of one of the above groups, in order to stick to the main conformal diffeomorphisms that have descriptions with help of a Lie algebra and Lie exponential map, etc.:
4. as the subgroup of (partially defined) conformal diffeomorphisms of B2-9 that is generated by all affine and special conformal transformations (excluding the inversion) of RP-;;
5. as the subgroup of conformal diffeomorphisms of MPP- that is generated by all affine and special conformal transformations (excluding the inversion) of Ry.;;
For this draft we will settle with one of the last definitions, which are mostly equivalent, and call it the (restricted) conformal group of TR. and denote by (R For more details see further below.
To define the conformal compactification of RP-9 we first need to introduce the bigger pseudo- Euclidean space Ig++1 and the projective space pete+1
D.2 THE PROJECTIVE SPACE, DE SITTER SPACE AND ANTI-DE SITTER SPACE
Definition D.2.1 (The projective space). Let V be a (real) finite dimensional vector space. We then define the projective space associated with V as the space of equivalence classes:
P(V)= = (V \ {0))/
(134)
where we define two non-zero vectors 1 I2 V to be equivalent if they lie on the same straight line through the origin:
22
3c E R (0). 21 C-
(135)
We then define the standard projective space of dimensions d 2 0 as: Pd = +)).
(136)
Note that d denotes the dimension I'd which thus requires the Euclidean space Id+1 to be of one dimension higher.
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Remark D.2.2. W.r.t. our pseudo-Euclidenn space IR::9 we will in the following mostly consider the projective space of dimension d +1, where == p + q:
pd+1 : IP(
(137)
Elements of [pd+1 will be either denoted by [2] with
Ret1.9+1 with coordinates
t.".z...... ,,,,,,,, =d+1) or directly as = z=: :.:..:: zd =d Note that we endow 1R++1.++
with the standard metric rptl1tt of the preudo-Euclideee (p + 1.q + 1) and that we consider the coordinates ,.....,"" to belong to the +-ig-signature and
space of dimension d + 2 and signature
2" the coordinates 2P+1 ...... 2 to belong to the (-1)-signature.
Definition D.2.3 (de Sitter and anti-de Sitter space). Let (V,1) be non-degenerate finite dimensional real quadratic vector space. Then we define the de Sitter space and the anti-de Sitter space and the (projective) zero quadric associated (V, .1) as:
aS(V, n) {[=] E P(V) 2(2,2) > 0}.
(138)
AdS(V, 7) = {[=] E P(V) [122 < 0) M((,,) n) {[=]] E P(V) (1222) = 0).
(139)
(140)
Note that these are all well-defined subsets of (VV). The ppeudo-Riemamiia. metrics on the first two spaces are induced by the following subspaces V:
Y+ = E m(y =)
Y- = tu E V (n(u,y) -11,
(141)
and the 1 locally diffeomorphic surjective maps:
x Y dS(V,n) S P(V),
ty r [y],
(142)
T= Y_ + Ads(V,n) CPV)),
ty + [y],
(143)
where both Y_ and Y- are endowed with the pull-back metrics n1th and n- from n of V to those subspaces. More explicitely, the tangent spaces and local metrics are given as follows:
Tie dS(V, { (12, =0,, TH Ad)(V,n) = (u E 0) 00}
Mpl (01, 022) m711, U2),
aS(V.n)
(144)
Ads(Von) (V1, 122 (7(1,
M[=]
(145)
We define the (standard) de Sitter space and the (standard) anti-de Sitter space and (standard) zero quadric of signatures (p, 9). resp., as follows:
dSP.9 = dS dS( 1.)),
(146)
Ads = AdS( spp.ot2).
(147)
MP = (RP+ spttltt))
(148)
The pseudo-Rienee Riemannian metric on MP-9 will be defined later in Definition D.3.5
Definition D.2.4 (Projective orthogonal group). Let (V,1) be non-degenerate finite dimensional real quadratic vector space. Then the projective orthogonal group of (V,1) is defined be:
PO(V.)) m)) = D(V.m)/(#id/
(149)
The (standard) projective orthogonal group of signature (p. q) is:
PO(p.4) 4) == O(p, ()}
(150)
We denote the identity component of those groups as: and
(V.(V,
PO' 9).
(151)
Lemma D.2.5. Let (V.1) be a non- non- non-degene finite dimensional real quadratic vector space. Then the projective orthogonal group PO(V, 1)) acts on the spaces P(V, 11), dS(V. m), AdS(V, n) and on M((V n) via matrix multipicaation:
[A][=] = [A2].
(152)
in well-defined way.
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Theorem D.2.6 (See McKay
(2023) Thm. 7.5). Let (V.n) be non-degenerate finite dimensional
real quadratic vector space. Then the standard action of PO(V, n) on dS(V, n) and AdS(V, 77) acts isometricallyy and get the following identification with their isometry groups:
Isom(dS(V,, n)) P))V,n),
Isom (AdS(V, n)) = PO(V.n).
(153)
Remark D.2.7. For our pseudo-Euce Euclidean space IR we will use the projective zero quadric MP as its conformal compactification, see later in Definition D.3.3 For this we need to consider the bigger pseudo-Euce Euclidean vector space (RR+11.++ 1 +.+)), and, the projective space pd+1 F(R++19tl) Here we get the disjoint decomposition of pd+ into anti-de Sitter space, conformal compactification and de Sitter space:
pd+1
Ads UMP. Ö dSP-2+1
(154)
Note the different signatures on each of the spaces. With Theorem D.2.6 thus get the identifications:
Isom (som(dSP- PO(p + + 1) IsommAds
(155)
In the following, we further want to show that the action of PO(p + 1, 1) on MP-- induces conformal diffeomorphisms of MP.-a This identification via PO(p + 1) will thus induce correspondence between (certain) conformal diffeommopphsmms of the conformal compactification MPP-9 of ig--9 and the isometries of the corresponding Ads 1.9 -space (or, also the dS -space).
D.3 THE CONFORMAL COMPACTIЯCCATION OF THE PSEUDO-BUCLIDEAN SPACE
Notation/LLemaa D.3.1 (The isometric embedding). The following map:
7p (an.ar)
L: RP4. Retllt1
(xx)
(156)
1+ (20.))
an isometric embedding and satisfies for all TR..
(xx) + 0,
and
pp p (1(x),((x)) =0.
(157)
Proof. For the latter consider the computation for x E RP:.:
4n 1.0 1 (((x),((x))
(158)
= (1 n (a,, )² 2 + 1p (2r, 2x) - (1 +
(159)
= (I, + up 8 x) + 41p (z, x) 1 -1, - (x,
(160)
0.
(161)
This shows: (1))) = ((x) * is clear. To see that i is an isometric embedding compute the differential (Jacobian matrix):
-x - P..
dl
T
(162)
x P.a
With this we get:
r PM1
(diz) Antllatl den =
A A 1
I
(163)
x T A.9
A.9
|-A
AP4
-AP-2]
(164)
P-g
P- A +
APT
A A-9.gm
A P.0
(165)
P.9
(166)
This shows that is isometric embedding.
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Definttions/Notttions D.3.2. For p.q 2 0, d p q, introduce the following notations:
1. We introduce the affine zero quadric as the following subspace Yo of R 1gt11 via:
Yo == {y Ret \ [0) | 17 11ptt.tt1000 (y. v) = 0}
(167)
and endow the tangent spaces for y E Yo:
T,Yo = {0 E 1 upttoot (y. = 03 S Cgptllatt with the pull-back metric io of spptlatt
(168)
Yo (v1, v2)= = np pptltt (2) By
for U2 E T.Yo.
(169)
2. We also introduce the double sphere:
p
d+1
PP.9
x S
E 1+tl1
Σ
1v01 =
Σ
10-12 = 1 5 Yo.
(170)
j=p+1
and endow the tangent spaces for y E SP
p
d+1
T TySp.g
1. Rp+ll+1
Σ
y w²
Σ j=p-1
y² v = 0
C Rottltt
(171)
with the pull-back metric 1p80 of 1/p+1.001
Tl (v1, v2) p 02)
for 00 E T,SPa.
(172)
Note that we have the inclusions:
((RP-9) S Yo S Bettlgt
SP4 CY S CR+llgt1
(173)
Notation/Lemma D.3.3. Consider the maps:
p: Yo + R.
ply) VEA l
(174)
b: Yo + SP.
(ly) = p(y)
(175)
Then v is a well-defined map and conformal with factor (y) = -(z))
Proof. First note that for y E Yo we have:
p 1
0 p (v.
12
(176)
2=
jmp+1
and thus:
1
p(y)
(177)
Ei=o 1/112
v - 1
This shows that for E Yn we have:
(1y) = p(y) y SP x 89 = SP.
(178)
The differential of de at y E Yo:
du Tyyo + So C Rw
(179)
is given with help of the product rule as the matrix:
dby = y. (v) + plv) I.
(180)
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With this get for all y Yo and vi, 02 Ty
vi (diby) T A (d1y
(181)
= =v (v ('(y) + ply) 1) Antlost (v. p'())))) + ply) 1)02
(182)
=(y)) ui A Aptlot U2 ui p'()² y A yope-yyypuu (2)02
(183)
+ p(v) 0/ p'v)' y A + A p(u) o (2)
(184)
0. TT
=0, E
= - 0i Aptlatt
(185)
This shows the conformal factor of ply). Notation/Lemma D.3.4. Consider the following map:
T :P _4 Yo SP,,
(186)
(
2
T(x) := (((x)))
(187)
1 a 1² + Di=n 1o-12
1+1 "1)00-9(000 (w.
Then 7 is a well-defined conformal map with conformal factor:
p(((x))
(188)
V - (2, Ep |zا|2
Definition D.3.5 (The standard conformal compactification of the pseudo- Euclidean space). For p.q 2 0, d := p + q. the (standard) conformal compactification of (1899)))p-9))) is defined to be the following projective zero quadric subspace of the (d 1)-dimensional (real) projective space:
MP-a = {[$] = [z" 2² : 2d 2d+ E pd+1 npttaat (2, = 0} C pd+1
(189)
Its metric p induced with help of the double sphere:
:= = S" 99 y RP+1.0+1
Σ 1
Σ j=pt1
1
C R++1
(190)
and via the 2: 1 locally diffeomorphic surjective map: π. SP4 + MP4.
My [p].
(191)
Note that [2) E MP-- then we can put:
y = (/2)) = p(z) 22 Ro+1.011
(192)
leading to the following propertie
E SP-,
[ = [z] E MP-.
(193)
More explicitly, the tangent space of MPP- at [2] is given by:
d+1
T MPP
v E 1p+
v = 0
(194)
==
j=p+1
p
d+1
11 E 1++
Σ
y 1 =
(195)
j=pt1
Sp99 C T,potlott 1.
(196)
and the metric of MPP9 at fzr U1 U2 E MPP by:
MP.. (ui, 02) i gp (vi, v2) tl1tt (v), v2) 0j A A 02
(197)
By also multiplying & with sg'(>') for index 1, d, d 11 c.g. d 1. in
addition, arrange that y 2 0 for this one fixed index j. For example. if q = then SP = 8" x {+1). and we
always arrange the representative y to have: nd +1. In this case: Ma.0 ~ s (+1).
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Definition D.3.6 (The conformal embedding of the compactification). The conformal embedding of IR-. into MP-9 is defined to be:
space into its conformal
[c(z)] = 1 - 1p 2))
1 (I,
[c] : IR + MP
(198)
2
1 - (x.,)) 2x + 17 (1,
(199)
Note that this is well-defined map, as by Notation/Lemma D.3.7 we have:
(xx)= # O,
1p (1(x), ((x) = O.
(200)
Note that we can also factorize the conformal embedding as follows:
: RP In Sna = MP
(201)
leading to the identity:
[f(x)] = [v(z)],
(202)
and showing that the conformal embedding [t] is a conformal map with conformal factor:
w()(2) w+(x) p(((x))
(203)
V +111 - (1)- Ei
D.4 CONFORMAL TRANSFORMATIONS OF THE CONFORMAL COMPACTIHICATION OF THE SBEUDOEEUCLIDER SPACE
Proposition D.4.1 (See Schottenloher 2008 Thm. 2.6). Let p.q 2 0. Then every [A] E PO(p - 1.9 + 1) acts as conformal diffeomorphism on the conformal compactification MPP4 of R2-9 via matrix multiplication and with the conformal factor:
E 12
([]((=)) V EE (AZ)
(204)
The inverse of [A] is given by |1 Furthermore, exactly the two matrices +A E O(p + 1, + 1)
induce the same conformal diffeomorphism on MP.., thus inducing an embedding/inclusion of groups:
PO(p + 1, 1) S (on
(205)
Proof. For every A E 1(p+1pp1.0) 1) it is clear that Alva is an isometric map from Yo to Yo. Now define the map:
a = -b(y)
(206)
As the composition:
th : SP Y
A Yo
SP
(207)
of the isometry A (with conformal factor 1) and the conformal map e (with conformal factor p(y))
also A' is a conformal map with the conformal factor p(Ay).
If now [z] E MP-9 then := p(z) SP.9, This then shows that:
MIP-v - MP.-9
([]] = [Ae] [((())).
(208)
a conformal map with conformal factor:
([=] = p(Ay) p(A(p(=))) = ((Az) D 12412 p(z) V Σ (Az)*
(209)
Since that action is induced through matrix multiplication it is clear that [A= induces the inverse map to [A].
Now assume that A1 and Ag induce the same map MP-9, Then for every y E SP-i we have: [w,, (v) = ( (v)). So (xx(v) = A (y). Since this holds for all y E SP.9, we get: A₁ = = This shows the claim.
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Theorem D.4.2 (See Schottenloher (2008) Thm. 2.6. Thm. 2.9, Thm. 2.11). If either + 9 3 or (p. 9) = =2, (2,0) then the inclusion from Proposition D.4.7 is already an isomorphism:
PO(p + COMPDiT(MPP.))
(210)
This means, in those (p, -asess, that every conformal diffeomorphism MP + MP-- is given by the matrix multiplication with an (up sign) unique matrix +A g, 1), Le.: ([[=]) [Az] for all [2] E WIPd
Theorem D.4.3 (See Schottenloher (2008) Thm. 2.6, Thm. 2.9, Thm. 2.11). In addition to the isomorphism in Equation (210) in Theorem D.4.2 we get the following stronger statements:
1. Let p 2 3, then for every conformal map @ U - R defined on connected open subset U _ TRP. there exists an (up to sign) unique matrix +A E O(p + 1,9 1) such that the following diagram commutes:
U
4
R...
(211)
M[P-
MP.99
[A]
Le. for all x E U we have:
[1o)(()]] = [Ac(z)].
(212)
In particular, yp is injective.
2. Lel (p.q) = (2, 0), then for every injective conformal map U + TR2.00 defined either on
U = 2.0 on any punctured plane U RR200 {5), there exists an (up to sign) unique
matrix =A E O(3, 1) such that the corresponding diagram from 211 commutes]
Remark D.4.4. For (p. For details see Schottenloher (2008) section 2.5.
1) the group ConfDiff ConfDif((MMP-1) is much bigger than PO(p 1.9
Example D.4.5 (The inversion at the useudo-spheree, We continue the discussion about the inversion (at the -peude-sphere) I2-- from Example D.1.10
p. s U + RP.9
(x)
I
(213)
(2)
defined on U = fo E RP-9 (,, x) # 0}. Consider the (d+ + 2) x (d + 2) -matrix:
A
0
0
(214)
0
1
We now claim that the map:
: [Ag] MP + M
[ = [A,²],
(215)
Note that, for the case (p. q) (2, 0). the injectivity of yo needs to be assumed and we also only allow for open subsets U of 2.00 where most point is removed from 2.0
32

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is the conformal extension of ffrrm U to whole MP3, Indeed. first note that A, E O(p 1, 1)
and det A₂ = -1 and det( (-A,) = (-1)d-1. Then compute:
- ()rs()), (()))
d((x))
(x)2)
(216)
1 + ((s(z)),(())
yy 2
(217)
2
(
1 + ((2,z)
(1(,,, x))
1
2x
(218)
72 x)
MM(,, x)
+1
7 x)
1
0
0
2r
(219)
(22
1 + ( x)
Aq((x),
(220)
which shows the claim:
[A,((x) [M<(x))) EMP
(221)
Note that for RP. with x2), O get:
((x) [1 2x 1],
(((()))=[-)) 2x: 1],
(222)
(0) = 00:011 1].
5(((0))= = --::
(223)
Example D.4.6 (Linear conformal transformations, see Schottenloher (2008) Thm. 2.9). We continue from Example D.I.7 and Theorem D.I.8 the linear conformal map: + Ar with the matrix
A cΛ, where A = cA E CO(p, ②) with > O and A O(p.)) q). Then we can define the (d + 2) (d+ 2)-matrix:
2c
2c
P' ==
0
A
0
(224)
1-n
1
2c
2c
Then les E O(p + 1, 1) as:
1
[
0
0
1+c2
1-e²
PT AT
2c
2c
0
AP-9
0
2c
0
A
2c
0
(225)
1-c²
Lie2
0
1- 2
1+e22
2c
2c
2n
2c
1+c2
1-c2
2c
2c
2c
2c
A
0
A P-9
A
(226)
1
2c
2c
2c
2c
1
1-c2
0
2c
ATAPA APMA
0
(227)
1-c2
2
1+c
2
2c
2c
0
A.4
0
(228)
0
1
(229)
Also note that if E (p q) then Dea 000pp++p1ppp as + provides path to the identity component
We now claim that:
[Ta]= P + MP..
[<]] [T.
(230)
The reverse is also Tex
A (p.g))
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is the conformal extension of cA cA from RM MP.
1+c²
1-ca
(a,a)
2c
2c
2
[2)(
0
A
0
(231)
1-c2
1+c2
(2.,e)
20
2c
1+c
1- 1p 1-np" (..w)
1-c²
+
20
20
Ax
(232)
c-c²
- (2,
1+c²)
+
20
2c
c²
1p (x, x)
(x, x)) + c² (,p c (p ())
1
4cAr
4e
2 (p,)(1))) x) + + mp
(x, x) + +++ 0² 1,, + 8 (2, x)
(233)
2 2 c² 1p (1,
4cAx
(234)
4c
2+ 2 0² 178-01 x x))
1- (cAz,
cÂ. 1+ (cAA,,AA))))
(235)
1 (cAr).
(236)
This shows:
[Tac(<]] [c(cAr)] MP..,
(237)
and thus the claim.
Example D.4.7 (Translations, see Schottenloher (2008) Thm. 2.9). We now continue from Exam-
ple D.1.7 and Theorem D.I.8 for case of translation: + b for b RRP-, Then we can define the [d + 2) x (d + 2) 2)-matrix:
Tı,: := I S4b
(238)
(b,b)
-b A 0
I+
(239)
(b,,
bt AP
(b,
(b,
-6 AP
(6,6)
(240)
2 (p b)
ATAPO
1 (p, b)
We first show that 1,, E O(p 1, + 1). For this consider:
AT T 1.b APt 1 + S2 A 1+1 + Apt
S4 + 22J A Us.
(241)
We compute terms separately:
0
2 (b,,6
-
AMM
- 1 b)]
Aptlag =
P-9
0
b
b
(242)
0
(B,)) b)
1
b AP.
(b,b)
2
1 1 (b. b)
ATAPP
2 1 b)
O
AP.ab
(243)
a b)
-TTAPT AP4
- (b, b)
2.J Apt1g++1
A Aptlot2 200)
(244)
( b)
bT ATA
- (6, b)
-Apinb
0
(245)
(b.b) b)
DTAPO
(b. b)
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With this compute:
0
2 Artlott1 + AptLotlss (b,
0
0
0
(246)
0
1
(247)
We also get:
(.. b)
TT APT
22 b)
2 (b, b)
-b -b A
2 b)
A
A
O
b
(b, b)
6T API
(6, 6)
(b,
6 A
96, b)
(248)
bT Ap9b O 6T AP.9
A
0
(249)
0
6
A
[
0
17
(b..)).)) b)
(250)
1
0
1
Together this shows: I APt Le 1,6 Artlet1 + 22g Aptl,g11
Artlatl
22y A.tilattton Ω
(251)
0
17
0
1
A - (p. - b)
0
0
0
+ 17
(b,, (b,b)-
0
0
(252)
0
1
1 0
1
A
(253)
This thus shows the claim: r, E Opp + 1,.
Furthermore, for b - 0 we see that I - I.
This thus even shows that: r E (p + 1,9
We now claim that r3] is the conformal extension of the translation map x + x + b from IRP-9 to MP-9. For this compute:
PT,a(x) = (I+ (6))((r) = =((x)= + -sp((r),
(254)
(255)
with:
2 (b, b)
T AM
22 (b, (b)
[ (2,
Q5((2)
0
b
2x
(256)
(b, b)
bT AP.
(b, b)
1 + 1
(p.9(b,b) b) 26
A
2h nma b) + 2b Z
(257)
(p b) - (b, 2h
=
(258)
(b))(,,)) + 20pp.0 ((b,z)
With this we get:
1 (r b, x + b)
(x + b)
2(x + b)
(259)
2
+ p- b, + b)
1 1 - 1p (,,60 b) - (b) x)
2r + 2b
(260)
2
1 + p 17) + 17 1p-9((,bb b) + (b, xx)
[ (I, x))
Pb. b) - (1, x²)
1
1
2x
+
2b
(261)
2
1 +
2
(b, b) + 2n ((,
= (xx) + 2bc(x)
(262)
TT))(x))
(263)
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This then implies:
[Trx [1(x + 6)] MP
(264)
This shows that [Fr,o] conformally extends the translation map x + b from RP-4 to WP.-.
Example D.4.8 (Affine conformal transformations, see |Schottenloher (2008) Thm. 2.9). We continue from Example D.1.7 and Theorem [DI.8 for the affine conformal map: m Ax + with the matrix cλ and b 1844, where A cλ E CO(p. with e and A E Op, q). Then we can define the (d + 2) x (d + (d++2)-mzz) -mmttii:::
Ce r rr
(265)
1 (b,6)
-TT A I
L+e2
2c
25
b
0
(266)
m b)
6 A
1+ 1+ m
1-c
1tc²
2c
2c
Then have: T O(p+ 1, 1).
This is clear. as both matrices I' TA E O(p 1, 1)
by Example D.4.6 and Example D.4.7 map + CAr + from IR--9 to MIPM
thus also clear that the matrix [T (T..,) extends the affine
Example D.4.9 (Special conformal transformations, see Schhotttenll (0088) Thm. 2.9). We continue from Example D. about the special conformal transformations. Recall that we defined the special conformal transformations for b 12-99 on an open subset Ub S IRP.9 as:
17 x)
Tb : Ub + R.P.
(b(x) = 1 1 2 n1 b) + (b, b) (1(II)) (s(()) -b)
(267)
where 5 is the inversion at the pseudo-sphere from Example D.1.I0 and Example D.4.5 With the
latter representation and Example D.4.7 7 we immediately get that the (d + 2) x (d + 2)-matrix:
1,
(268)
0
0
(b,.)
T AP I
(6, b) -
0 I
0
1
0
-b
(269)
0
0
1
(b, b)
-b And
1+ (b, b)
1
1
(b,
-bTAPA
(b,
I
(270)
-b AP.
1+ (b
lies in O(p + 1,9 1) and that [Es] extends the special conformal transformation 0; from RR.-9 to MP.., Furthermore, for 0 we see that En + I. This then even shows that: Σg E 0 (p+1..+1))
D.5 THE DEFINITION OF THE CONFORMAL GROUP OF THE PSEUDO-EUCLIDEAN SPACE
Definition D.5.1 (The global and restricted conformal group of the pseudo-Euclidean space). Based on Theorem D.4.2 we define for all p, q 2 0 the global conformal group RP-4 be:
Conf g (R.-9) == PO(p + 1,
(271)
and the (restricted) conformal group (IP..))) of I2P-9 to be the connected component of the identity of PO(p 1, 1):
(RP = Conf,,(P'g = Conf =
(272)
Note that by Proposition D.4.1 we always have the inclusions:
Conf, C Couf, C CoMIDi
(273)
Theorem D.5.2. (R contains all affine conformal transformations
Ar + of TR. and A O' (p. q) and b E B.- in form of the map [T (e MP-. + MP-- from Example D.4.8 and, also all the special conformal transformations o in form of the map [Ea] MP-4 - MP from Example D.4.9
where A cA CO (p. q) with
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D.6 A (PARTIAL) PARAMETERIZATION OF THE ANTI-DE SITTER SPACE
Proposition D.6.1. Let eetp, 2 0 and == p+ + 9. Consider the following space:
A
R2 x BR
(274)
which endow with the metric of t which is given at Aptla and with v1, U2 E T,4pt1.
pe+la as follows:
Aptl. (U1, = 1 U2
Ne
1x:02
(275)
Then consider the map:
② APtLa + Retlgt x+ 2+ Φ(x) =: V,
(276)
(277)
(x()) = 1 1pttt.0 1.9 (x,
1
x
2, 11" pi (x (xlidd xlid),
(278)
2x
2x
2
2lid
(x) = 2
(279)
dd+1(a)
(r) 1 ap 1.0 (x,
1
x
n (x s
(280)
2x
2x.0
2
2n
Then is a well-defined isometric embedding and its range coincides with the subspace:
AdS 1+0
= {v E Rp+lo+1
1pptlat1 (y. yd + y > 0).
(281)
p+ln
An inverse o-1, AdS Aptl. of given by:
20 = (0-) 8d + y > 0,
(282)
yird
Lid = (6 (6-1)ird(1) = 8d + y
(283)
Proof. We first show that & is well-defined. For this we compute for y = 6(x):
1/10|2 p (g..++(..))) v) (1 (z, x))² + 9 adid) (1 + p2 xx)
(284)
= --nppti. x) + 4np ,.1 = 12
(285)
(286)
implying: p (v.v) = -1. Furthermore:
1 + m (2
y + p -
z0 0.
(287)
2r
2x
Together, this shows: ((x) E Ads 1.1 to σ. i.e.:
C 1p+ is easy to see that Φ is the left-inverse
c 6(xx = x E 1t4
(288)
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To show that Φ-1 is also right-invers let y E AdS
Ads
p+1.0
and :== (v). Then compute:
(x)
1
lidd = (v² + ydtt) 30 + y =yldd
y lid
(289)
x0
x
1
('(x)=
2x 1p (x (xllin
1:d ,l!l
(290)
2x
2
1 + yet
2 (v0 + ydtl) 2 (h)) + (othh (y yltd)
1
(291)
2
1
(v" 1
80 + yott (1 + pald ( (y) yهبر)
(292)
2
1
(v" + (dt1)
40 + ty++ (1141112 12)
(293)
2
1
(0 + tott)
- -(0 - ))
(294)
2
y".
(295)
Similarly:
1
(x)
1
+
x
+
2x 77 (x (11d
(296)
2x
2
14h
1
2 2yy + ydt1) + (n) + yd+l) ( (y) (d
(297)
(10 + gdtl) + y + yo+! (1 + و 1nd y(دد))
(298)
(v" + y 2)) = 8 + yo+i (l1dt1pe 12993)
1
(299)
1 ((v 1) + (1 - -0))
(300)
y d+1
(301)
This shows that:
pook 2oo (w) = E Ads 1
(302)
We now compute the differential of cb:
1 + 2 ao
a (1o2id) (A AP
2
2
2/2
dØ
# + (x T
(303)
T
( g 2/
2k + 1
An
and the metric at x E Apt14;
I
0
(dex) A dob. (dob.) TT
0
dø
(304)
0
We compute the entries of the last matrix separately. First, top left entry:
1
17 (xxlod 1:d x 1:dy
2 +
(x Lid 1od
1
(1
170 1td x Ilid)
2
4
12
10:0
4
1
|2
(305)
17 (2r lnd, 1o lod
+ 1 ltd, x lod)
1
(306)
12
1
l
(307)
38

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Published as a conference paper at ICLR 2026
Note that in the middle we used the formula: (a + b)2 - (a (b2 = 4ab. Next, top right entry:
(1 Lidd
1
(x!²) P.
(xtid) API
(308)
1x0
1=012
12013
(1
178-0
x 1:d)
1
(a lad T TAP
(309)
12002
12=01
x
(x T
AP.9
(a 1dd)
T
AP.4
(310)
2 [2:012
x
1:013
0.
(311)
By symmetry also the bottom left equals O. Finally, bottom right entry:
A lnd
(x
A-9
I
Ip
A lid (x!1dd
ld) T
P.
(312)
x00
x.0
x0
x0
x
Together we get:
(dox) A
doz
12
1
0
=
Aptlog
(313)
This shows the claim.
Theorem D.6.2. The map:
Artl.g Ads 1.g
Adg
2 + x+z) =: y ly],
(314)
is an isometric embedding. The complement its range is given by the closed subspace:
2: {[=] Ads 22 + 2 d+1 =0}.
(315)
The left-inverse for [z] E Ads (with z0 + 2d+1 # 0) is given by:
sgn (2) + xdt1)
2 E Ads 1+1,0
= v //ptl.gg (z, [2
(316)
and to get x E Ap+1a we put:
x" = (0 (0-1))(v)- = v /bppt1gt (z.
> 0,
(317)
120+1 + 10
2 1:dd
x l:d = (sb (()1))=(()))
(318)
2d-1 20
Remark D.6.3. Note that in Theorem D.6.2 in the Euclidean case, i.e. if O, the subspace Z is empty; as 20 + gd+1 = O would imply:
np p+1.9 (2,2)
(319)
12 + 1p ltd 12 1.d] - -120+112
(320)
(2" + ddt1 2 (2 - 2 21 + 1p-- (2 1nd)
(321)
(2 1:d 2ldd)
(322)
which is not possible for = O. It follows that for q = 0 we get isometries:
A A+110 Ads 1.0
(323)
Remark D.6.4. For p.g 2 0, = + g, consider the map:
R20 x BP4
pd+1
x + [1 - p x) I: 1 1p (x,
(324)
Then o is well-defined. For d = 0 the map coincides with the conformal embedding of IRP4 into the conformal compactification MP-1 pd-1 of RP-9, and, for d > 0, & coincides with the
isometric embedding & of 4P+14 into Ad C pd+11
39

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Published as a conference paper at ICLR 2026
D.7 THE EUCLIDEAN CASE 2 AND 9 0)
In the following we now consider the Euclidean case, i.e. 0, for p 2 2, d = Remark D.7.1. Note that we have:
g0 = {=1},
gd = 3d x {
(325)
forfor q 0, This thus simplifies the description of the conformal compactification, the corresponding de Sitter and anti-de Sitter space.
Corollary D.7.2 (The conformal compactification of the Euclidean space). We have the isometry for d 2 0 and q 0, p d:
M
S {+1}
= Sd
(326)
20d
0nd
[ (d+11+11
+
(327)
2d+1
The conformal embedding is then given as:
T+(x): (1 (1-1pt (2,
2x
T4 :R MI = gd C Rd+1
(328)
1+ rfd 1 1d (x,
which coincides with the invers stereographic projection (from the "south pole") ") The group PO(d + 1,1) then acts S follows, [A] PO(d + 1, y yodd Sd:
0:d
A
y
1
[A].
(329)
d+1
A
where A y denotes the matrix vector product of (d 2) x (d ( ma-mati A and the vector y with
1 appended the last component. Note that the indices of the components of y Sd C Rd+1 range from 0, 1....,dd. Remark D.7.3. Note that we always have:
Isom (Sd) = 0(d+1)) 1),
(330)
via the usual action and for 2 2:
ConfDiff(S = PO 1,1),
(331)
via the action above. The latter is part of Theorem D.4.2
Remark D.7.4. Recall that by Proposition D.6. have isometries:
Theorem D.6.2
and Remark D.6.3
3.6.3r for q = 0 we
d+100
AdS
d+1.0
Adgd+1.0
(332)
1.0
where we can slighly re-write
AdS
as:
1+10
AdS
{y
gd+11
17 (..1.
-1, ydt >0}.
(333)
With this we get the isometry:
Ad
AdS
1+1.0
Rdd111
[2]
sgu( 2d+11
2 =: y.
(334)
11d+1,1 1, (2, 2)
Note that then:
y [z] Ads pd+11
(335)
Note that for E 1d+1.1 with 1) 1 (y. w) < we have: 11 > 12. So, for those y, we have the
cquivalence: dd+ < 8d d+1
40

---

Published as a conference paper at ICLR 2026
d+1.0
Furthermore, for [A] E PO(d+ 1) we define the action AdS
[A].y = sgn((Ay)) (Ay).
(336)
where Ay denotes the usual matrix product. Note that this is well-defined and that:
[A]-y E Ads 1,0
[[]]b]] = [Ay] = [A][a] E Ads
(337)
This shows that the above isometry AdS
d+1,0
Adg is also PO(d + 1, -equivvaaiian
d+1,0
Definition/Lemma D.7.5 (Geodesic distance of Ads The geodesic distance AdS Ad$4+1,0 and A 1,00, resp-, is given by:
A 11.00
(11, == arccosh (177 d+1,1 (111 82221
(338)
(21,
Ads
([21), [22]) == arccosh
(339)
/1d+1,11 (=1, 21) (1pttil و2ح
012 + 1 + spd (zit 2ههيهح
dadt1.0 (x1, := arccosh
(340)
2x x2
41


Reviews:
[{"review_id": "xpeVZNPVXs", "reviewer": "Reviewer_HgBi", "summary": "This paper proposes \"AdS-GNN,\" a new graph neural network that is equivariant to conformal transformations (translations, rotations, scale transformations, and special conformal transformations). To achieve this equivariance, the authors, inspired by insights from the AdS/CFT correspondence in physics, introduce a method to lift input data from flat Euclidean space $\\mathbb{R}^d$ to Anti-de Sitter (AdS) space $AdS_{d+1}$, which has one additional dimension. On AdS space, the conformal transformations of the original space manifest as isometric transformations (distance-preserving transformations). Therefore, the paper efficiently achieves conformal equivariance by constructing a message-passing GNN that utilizes the proper distance of AdS space as an invariant. The proposed method was evaluated on tasks from computer vision (e.g., SuperPixel MNIST) and statistical physics (e.g., 2D/3D Ising models). Particularly in the physics tasks, it was confirmed to show high generalization performance even under extrapolation (OOD) scenarios, such as scale transformations or changes in system size (number of points). Furthermore, the interpretability of the model was also demonstrated, as it was able to extract a physical universal quantity—the scaling dimension of the Ising model—from the trained network with high precision.", "questions": "**Regarding generality:** While I understand the method's primary contribution lies in physics tasks, what advantages do the authors believe this conformal equivariance approach offers over existing equivariant GNNs (e.g., those with rotational or translational equivariance) in domains outside of physics, such as general computer vision or robotics? I would also like to ask for the authors' insights into why the method failed to achieve SOTA performance on SuperPixel MNIST. Are there specific reasons to consider, such as the approximation error from the lifting procedure, the nature of the point cloud data, or a potential lack of expressive power in the architecture?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "**Novelty of the Idea**: The application of the profound idea of AdS/CFT correspondence from physics to the context of geometric deep learning, thereby constructing a GNN architecture with conformal equivariance, is highly original and commendable.\n\n**High Affinity with Physics Tasks**: Due to its design background, the proposed method is an excellent fit for tasks in statistical physics where conformal symmetry plays a dominant role (e.g., the Ising model near its critical point).\n\n**Excellent Generalization and Interpretability**: In experiments on physics tasks, AdS-GNN demonstrated performance superior to existing equivariant GNNs (like EGNN). Its robustness to extrapolation tasks, such as changing the system size (number of points), is particularly noteworthy. Furthermore, the fact that the model can automatically learn and extract a physically meaningful universal quantity—the scaling dimension—from data and recover its true value with high precision is a testament to the model's high interpretability and a significant contribution.", "weaknesses": "**Lack of Generality and Performance**: The main contribution of this method appears to be limited to physics tasks. As shown by the experimental results (Table 1), in standard image (point cloud) classification tasks like SuperPixel MNIST, the performance does not reach that of existing SOTA methods (e.g., PONITA), and the method's superiority in general-purpose benchmarks has not been demonstrated.\n\n**Limited Applicability**: This method requires prior knowledge that the target data or task possesses \"conformal symmetry.\" Its application to many general machine learning tasks where such strong symmetry does not exist or is unknown is difficult, and the method's utility is inherently restricted.\n\n**Insufficient Appeal to the ICLR Community**: As a result of the above two points, the paper's contribution feels strongly directed primarily at the physics community. There is a lack of discussion or evidence regarding what new possibilities this conformal equivariance approach could bring to the broader ICLR audience (including computer vision, reinforcement learning, natural language processing, etc.).", "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": "JHYs9GVdJ7", "reviewer": "Reviewer_jAnW", "summary": "The paper presents a conformally equivariant graph neural network through showing how to lift data from Euclidean space to Anti de Sitter (AdS) space. Distance metrics in AdS space can then be used in conjunction with invariant message passing to create neural networks that are equivariant to the conformal group. Experimental results on computer vision tasks (SuperPixel MNIST and shape segmentation) and physics tasks (2d/3d Ising model and N-body simulation) are presented. In particular, for the physics tasks, AdS-GNN seems to require less data and generalizes better. Additionally, it recovers the correct conformal dimension, illustrating interpretability.", "questions": "What is the computational cost of these models compared to other invariant models (e.g. EGNN)? Does the lifting procedure incur significant computational costs?\n\nAnother scientific domain of interest could be fluid dynamics. At sufficiently high Reynolds number, the statistics of turbulent motions in the so-called “inertial range” become universal and exhibit scale invariance properties [2]. I would be interested to see the performance of these models on turbulence modeling tasks.\n\nIn what settings would conformal equivariance be useful for images? It seems like it would be more useful in dynamical systems/more-physics motivated problems as shown in the physics task experiments. Is there further motivation for why one would want robustness to conformal transformations in computer vision tasks?\n\nWould it be possible to have non-scalar features in message passing? Were there any experiments done extending the model to include non-scalar features?\n\nHow is the regulator $z_0$ chosen, and how does this impact the resulting lifting procedure? Are there other ways that one could lift data into AdS/why was this way in particular chosen? If the lifting procedure breaks certain conformal transformations, which transformations does it break? Figure 3 shows one broken transformation, but what do these “special” transformations correspond to physically, and why would this not be a problem that these are broken? Is it accurate to call the model conformally equivariant if this is the case (maybe approximately conformally equivariant)? Correct me if I'm wrong, but currently it seems conformally invariant rather than equivariant?\n\nWere any comparisons done to pre-existing scale invariant models? This would be interesting to include with comparison to the rotationally/translationally invariant models.\n\n[1] Batatia et al. MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields (2022).\n\n[2] Pope, Stephen. Turbulent Flows. (Cambridge University Press, 2000).", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper presents an interesting new framework for conformal group equivariance, using ideas from theoretical physics (and building open pre-existing work for constructing group equivariant convolutional layers). It is well-organized with experiments in multiple sub-domains, and provides a self contained introduction to the conformal group. I particularly like the figure at the top of pg. 3 showing possible transformations under the conformal group. Experimental results show that AdS-GNN maintains scale invariance (e.g. SuperPixel MNIST). The physics tasks are good illustrations of the utility of AdS-GNN, showing that AdS-GNN outperforms other models (EGNN and MPNN) for predicting N-point correlation functions for the 2D Ising model and requires less training data. I found it particularly interesting as well that the learned values of the conformal dimensions match the ground truth, and it would be interesting to explore this point further in more complex physics datasets.", "weaknesses": "I am not sure of the usefulness of the image experiments. I wouldn’t expect image datasets to require conformal equivariance/to me the orientation of the image would probably matter the most. However, the AdS-GNN model is invariant, so this orientation is not taken into account. It does not seem to be persuasive that one would use AdS-GNN in a computer vision setting.\n\nI found the part about embedding points in AdS somewhat confusing, and the choice of regular $z_0$ is unclear. I am concerned that this would impact the dataset/break certain symmetries (see questions).  I think further clarification is needed in this section, perhaps an additional figure showing another embedded shape (or embeddings of circles, triangles as in the shape analysis data) would be helpful.\n\nAdS is unable to handle orientation and relies on invariant descriptors. This seems to be a significant limitation, for both image datasets and physics datasets, in light of other work on equivariant neural networks with message passing of higher-order tensorial features (e.g. [1]). \n\nThe shape segmentation task is quite toy, the authors could consider exploring a more realistic dataset such as ModelNet.\n\nOverall, most of the experiments seem to test scale invariance, but there are pre-existing scale invariant models (from my understanding). It may be good to benchmark against these pre-existing scale invariant models and see what conformal invariance actually gives us.", "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": "guGyzdz3Io", "reviewer": "Reviewer_PdJG", "summary": "This paper introduces a new architecture, AdS-GNN, which is equivariant under conformal symmetries (preserving angles). \nConformal symmetries include scaling and a type of inversion, called the special conformal transformation, in addition to rotation and translations. \nThe key idea is to first map the data onto an Anti-de Sitter space, a uniform negatively curved space. The reason for this mapping is that at its boundary, AdS has conformal symmetries. The mapping to AdS far from its boundary breaks invariance under the special conformal transformations, but the rest remain intact. This mapping is motivated by the so-called AdS/CFT correspondence. \nThe approximate conformal inviariance on AdS corresponds to a certain norm being invariant and that norm is used to define equivariant and invariant features for the GNN. \n\nThey apply AdS-GNN to a few problems and show strong results in physics examples, especially those enjoying some scale invariance.", "questions": "1. How does the run time, including AdS embedding, scale with size of the systems? \n2. Do you only use scalars and vectors as features in AdS-GNN? This should work fine for Ising which has i=only quadratic interactions. But what if you had a spin-glass with higher order interactions? Do you think higher tensors would be needed to learn correlations there? Is it known which of those systems, or in what regime they would be scale-invariant? \n3. Any good criteria for deciding when to use AdS-GNN, other than obvious scale-invariance?  \n4. Some of the scale-invariant systems here seem fully conformal invariant. How much does the breaking of special conformal cause a problem? How do you measure the effect of this breaking?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. AdS-GNN outperforms other baseline on the physics tasks, such as Ising and N-body electrostatic simulation\n2. It performs well even in small data regimes.\n3. The idea of first mapping to AdS before applying the model is interesting and inspired by known fact from AdS/CFT. \n4. This is a nice model for equivariant model scale-invariant systems. \n5. The mapping to AdS is fairly easy, though it involves some preprocessing. Other than that, the model seems easy to implement.", "weaknesses": "1. Given the fact that special conformal is broken by the AdS bulk embedding, what is left is scaling + rotation + translation. So, really the model is adding scale equivariance to EGNN. This should be discussed or emphasized. \n2. The choice of experiments isn't explained much. As above, I think AdS-GNN works best for scale-invariant problems. In physics, these are the systems which we expect to have conformal invariance, as discussed in the intro. This may be why on Superpixel MNIST they don't outperform EGNN. \n3. A general discussion of limitations and realm of applicability is needed. \n4. All systems and datasets used in experiments seem very small. It is fine as a proof of concept, but scalability needs to be discussed.", "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)