Projective Equivariant Networks via Second-order Fundamental Differential Invariants
Authors: Yikang Li
Yeqing Qiu
Yuxuan Chen
Lingshen He
Lexiang Hu
Zhouchen Lin
Venue: NeurIPS 2025 spotlight ()
Keywords: equivariant networks
differential invariants
projective group
Primary area: deep_learning

Abstract: Equivariant networks enhance model efficiency and generalization by embedding symmetry priors into their architectures. However, most existing methods, primarily based on group convolutions and steerable convolutions, face significant limitations when dealing with complex transformation groups, particularly the projective group, which plays a crucial role in vision. In this work, we tackle the challenge by constructing projective equivariant networks based on differential invariants. Using the moving frame method with a carefully selected cross section tailored for multi-dimensional functions, we derive a complete and concise set of second-order fundamental differential invariants of the projective group. We provide a rigorous analysis of the properties and transformation relationships of their underlying components, yielding a further simplified and unified set of fundamental differential invariants, which facilitates both theoretical analysis and practical applications. Building on this foundation, we develop PDINet, the first framework for deep projective equivariant networks, achieving full projective equivariance without discretizing or sampling the group. Empirical results on the projectively transformed STL-10 and Imagenette datasets show that PDINet achieves improvements of 11.39\% and 5.66\% in accuracy over the respective standard baselines under out-of-distribution settings, demonstrating its strong generalization to complex geometric transformations.

Projective Equivariant Networks via Second-order Fundamental Differential Invariants
Yikang Li', Yeging Qiu Yuxuan Chen', Lingshen He', Lexiang Hu', Zhouchen Lin 1,5+ State Key Lab of General AI, School of Intelligence Science and Technology, Peking University 2ccooo Science and Engineering. The Chinese University of Hong Kong, Shenzhen "Shenzhen Research Institute of Big Data
'Khoury College of Computer Sciences, Northeastern University
Institute for Artificial Intelligence, Peking University 1iyk18@pku . edu. en, yeqingqiudlink cuhk. edu. en, the..yuuuanndhoorteaatteee edu 1 lingahenhe@pku. edu. en, hulxdatu .ku edu. en, zlin@pku edu. cn
Abstract
Equivariant networks enhance model efficiency and generalization by embedding symmetry priors into their architectures. However, most existing methods, pri- marily based on group convolutions and steerable convolutions, face significant limitations when dealing with complex transformation groups, particularly the projective group, which plays a crucial role vision. In this work, we tackle the challenge by constructing projective equivariant networks based differential invariants. Using the moving frame method with a carefully selected cross section tailored for multi-ddimensoonl functions, derive complete and concise of second-order fundamental differential invariants of the projective group. We pro- vide rigorous analysis of the properties and transformation relationships of their underlying components, yielding a further simplified and unified set of fundamental differential invariants, which facilitates both theoretical analysis and practical appli- cations. Building on this foundation, we develop PDINet, the first framework for deep projective equivariant networks, achieving full projective equivariance without discretizing or sampling the group. Empirical results on the projectively trans- formed STL-10 and Imagenette datasets show that PDINet achieves improvements of 1..9% and 5.66% in accuracy over the respective standard baselines under outtr--distribution settings, demonstrating its strong generalization complex geometric transformations.
Introduction
Incorporating symmetry as an inductive bias into neural networks has emerged as powerful approach enhance model efficiency and generalization. Convolutional neural networks (CNNs) KKrizhessky et al. 2012, Simonyan and Zisserman 2013 He et al. 2016, Chen et al. 2017]. which are among the most widely used architectures deep learning, owe much of their success to the inherent translational equivariance. Building on this idea, Cohhnn and Welling (2016a] proposed Group Equivariant CNNs (G-CNNs), which generalize equivariance to broader transformations like rotations and reflections. Equivariant networks achieve symmetry incorporation by constructing network layers whose outputs transform in a predictable pattern under group actions applied to the inputs.
The development of cquivariant networks began with G-CNNs, whose feature map can be seen as function on group. Although G-CNNs have proven effective in various tasks (Worrlll and Brostow 2018 Esteves et al. 2019 Lafarge et al. 2021 Shamsolmoali et al. 2021 they are less suited
*Corresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Figure 1: An illustration of a projective transformation: when a planar object projected onto another plane, a projective transformation occurs.
to continuous groups, as handling such groups typically requires group sampling or discretization. which introduces approximation errors and computational complexity. Then, steerable CNNs (Cohen and Welling 2016b Weiler and Cesa 2019] were proposed overcome these limitations by viewing features as fields that transform according to specified group representations. In this framework, G-CNNs can be interpreted as special case where the group representation is chosen be the regular representation. Steerable CNNs are capable of handling continuous groups such as SO(2) and (3(3) directly, thereby significantly broadening the scope of equivariant networks to include real-world symmetries beyond discrete groups [Weiler et al.) 2018 Wang et al. 2020 Wang and Walters 2022 However, for more complex non-compact Lie groups such as the projective group, deriving closed- form steerable basis filters becomes intractable, inherently limiting the applicability of steerable CNNs. To address this, this, Manald et al. [2022] enabled group convolutions over dintenddmmniionall Lie groups by computing the integral on the Lie algebra, thus introducing projective equivariant model, homConv. However, this method still relies on group sampling, which results in exponential memory growth with increasing network depth, thereby hindering scalability deeper architectures. Mironenco and Forré (224) improved sampling efficiency via group decompositions, but focused solely on affine subgroups like R²" x GL GL*(nn, (n., RR and I SL (n. R). without addressing more complex groups such as the projective group. Recently, ei al. [2024 2025 proposed mnvaLLayer and steerable EquivarLayer that construct affine equivariant networks based invariants, enabling closed-form and sampling-fres affine equivariance. However, these works remain specialized to the affine group and do not yet generalize to more complex non-compact groups like the projective group.
Actually, projective transformations play a fundamental role in computer vision [Mohr and Triggs 1996 [Brchhield 1998, Hartley and Zisserman 2003]. as they capture the relationships between objects and their images under perspective projections (see Figure T). Achieving equivariance on projective transformations is especially critical in practical applications such as mobile robot naviga- tion, 3D scene analysis, and camera pose estimation, where accurately handling perspective effects and viewpoint changes can significantly enhance model robustness and accuracy [Lee al. 2000 Hartley and Zisserman 2003 Mur-Artal et al.) 2015 Schönberger et al. [0066). Early on, /uk and Flusser 2004 proposed projective invariant feature extraction method based projective moment invariants. Nevertheless, the invariants are formulated as infinite series of moment products, leading to significant computational overhead and intractable eитоr analysis in practical implementations. To overcome this limitation,) Li et al. (2018) proposed an alternative framework that constructs projective invariants using finite combinations of weighted moments, where the weights are derived from relative projective differential invariants. Note that moment-based projective invariants are essentially global image descriptors, which makes them inappropriate for constructing equivariant operators that act locally on feature fields for capturing fine-grained spatial patterns. Instead, differential invariants inherently hold the property of acting locally at each spatial position, which makes them natural foundation for building equivariant operators (Sangalli et al. 2022 2023 Li et al. 2024 2025). While Olver [2023) has proposed a systematic framework for the computation projective differential invariants via the moving frame method, it requires at least third-order derivatives because

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the projective action is not free at second order for scalar functions. Besides, they only consider single-channel cases, and the extension multi-dimensional cases needs more complex expressions involving high-order derivatives, which limits their practicality in computation and applications on color images.
In this work, we construct projective equivariant networks based on differential invariants, achieving full projective equivariance without relying on group discretization or sampling. This overcomes the depth limitations of homConv [Maalonald et al. 2022] and enables effective scaling to deeper architectures. core challenge lies in deriving concise and practical projective differential invariants. To support color images and multi-channel intermediate features in modern neural networks, we focus on the differential invariants for multi-dimmensonall functions. In this case, the projective group acts freely on the second-ordet jet space, allowing us to derive a complete set of second-order fundamental differential invariants using the moving frame method [Olver, 2015), which can express any second- order invariant of the projective group. However, the choice of cross section used to define the moving frame significantly affects the form of the resulting invariants. While a direct extension of the cross section in [Olver 2023] to the multi-dimensional case is theoretically valid, it leads to prohibitively long expressions with hundreds of terms, rendering them impractical. Instead, we propose a new cross section tailored to the multi-dimensional structure, which involves up to second-order derivatives, yielding a much more concise set of fundamental differential invariants. Further analysis reveals that these fundamental invariants are composed of a set of simpler components. By exploring the algebraic properties and transformation relationships of these components, we further simplify invariants into a unified set of fundamental invariants, facilitating both practical use and theoretical analysis. Based upon these simplified invariants, we design learnable equivariant operators by combining them with parameterized multi-layer perceptrons (MLPs), and embed the operators into standard neural network backbones to build PDINet, the first framework for deep projective equivariant networks free from group sampling. Empirical evaluations under challenging out-of-distribution settings demonstrate the strong generalization ability of our model to complex geometric transformations.
We summarize our main contributions follows:
We employ the moving frame method to derive a complete set of second-orde fundamental differential invariants of the projective group for multi-diimensooal functions, enabling support for color images and multi multi-channel features.
We conduct an in-depth analysis of the algebraic structure and transformation properties of these invariants, resulting in a further simplified and unified set of fundamental invariants that facilitate both theoretical understanding and practical computation.
We develop PDINet based on second-order projective differential invariants. It is the first time that deep networks achieve full projective equivariance without relying on group discretization or sampling. thus allowing effective scaling to deeper architectures.
Numerical experiments projectively deformed STL-10 and Imagenette under out-of- distribution settings demonstrate the effectiveness of our model, with improvements of 11 11.9% and 5.66% over the standard bascline results, showcasing its strong generalization capability under complex geometric transformations
2 Method
2.1 Basic concepts and notations
To begin with, we introduce some basic concepts and notations necessary for our formulation. An image can be viewed as continuous function u(x,y) defined 2D plane. For example, an RGB image corresponds to a three-dimensional function. Likewise, intermediate features in neural networks also be interpreted as functions, and each layer can be seen as operator that maps one function to another.
A central concept in this work is equivariance. If the output of an operator undergoes a corresponding transformation when the input is transformed, it is referred to as equivariance. The formal definition is as follows:
2 IImagenette is publicly available dataset downloaded from https//gghhabcccmm/ssaa/mmgggntt Our code is available at http://gitubb.cm//iyy////Z
3

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Definition An operator v : Fi + F2 is said to be equivariant with respect to group G if
g (u) = u), Vg E
(1)
where Fi and F2 are the input and ouipul function spaces, respectively.
R" be the range of function, and U U U1 x... Us be the derivative space up order d. A group action g on the domain naturally induces an action on functions, defined as (g u)(x) = u(g-1 x), which models how geometric transformations deform images. This action further extends to derivatives through prolongation to the jet space X x Uld), For example, the first-order prolongation of the action on X x U()) can be expressed as: (x, u(x), Vu(x)) + (x, ( Vū(x)), where x and ú - g with ū(x) = u(x).
Let X denote the domain,
A differential invariant is quantity that remains unchanged under the prolonged group action. The definition is given below.
Definition Given group G acting on X, a d-th order differential invariant is a function X x Uid) + R such that
I(g (x, )) I(x, ) Eg E G, (x, (()) E x U
(2)
where g (x, ( denotes the prolonged group action on the jet space X U
The definition can be extended the an k-dimensional differential invariant. In addition, we define relative differential invariants, which may transform with a weight function under the group action:
case. Specifically, we call (II..... Ix)
R(g (x, u ()) = w(g.x] R(x, u (d)),
where w(g,x) x) is a scalar weight depending on the group element and the point x. Notably, differential invariants are closely tied to equivariance,
(multi-dimensional) differential invariant yields an equivariant operator [xu)(x) & I(x, u(d) satisfying I(g u) = 9 I(u).
In this work, we focus on constructing such differential invariants and using them to build projective equivariant operators for neural networks.
2.2 Method of moving frames
The method of moving frames is powerful technique for deriving differential invariants Oller 2003 2015). We begin with the definition of moving frame.
Definition 3 |Olver 2015) Let G be a Lie group acting on a manifold M. A moving frame is map 7 .M + G such that
n(g 2)
g geG, G M.
(4)
Given moving frame, the invariantization of function F M - R is defined
(=)) e F(1(2))22)
(5)
which converts an arbitrary function F into group-invariant function satisfying o = More generally, we can define an invariant as I(g z) I(=), E G, E M. In our context, the manifold of interest is the jet space M = X x (n) and we focus on differential invariants.
A necessary and sufficient condition for the existence of moving frame is that the group G acts freely and regularly on the manifold M. Under this condition, moving frame can be constructed via a cross section, described below:
Theorem 4 Olver 2015 Let G be r-dimensional Lie group acting freely and regularly on m-dimensional manifold M. Given local coordinates z = (Z1,..., , Zm) on M, let K. a cross section of the form K. = (z1 C1, z2 . cr} C M, where are constants. Then for E M, there exists a unique g E such that g.2 E K. Defining r/()) = g. namely 71()) z E IC, yields map n M G. which is moving frame.

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Here, the group action is said to be free if for any E G. g. implies e, where is the identity element the group. Usually, the group action can be made free by increasing the order of the jet space. The action is regular if the orbits form a regular foliation, which is typically satisfied in common groups.
With a moving frame obtained from Theorem 4 we can construct a complete set of fundamental invariants, meaning any invariant can be expressed as a combination of these fundamental invariants.
Theorem (Olver 2015 Let vj : M - be a moving frame from Theorem 4 g Then
and define wlg z)
w(n(=), (21, C2....... Crs W (T(=), (r
Wm (n())))
(6)
where (2) & (1))),()),2),, .... Im-r Wm (n)(2), 2) constitute complete system of func- tionally independent invariants, called fundamental invariants.
This theorem provides a method to construct fundamental invariants via the moving frame and indicates that the number of fundamental invariants is r. In the following sections, will leverage these results to derive projective differential invariants.
2.3 Projective transformation
Projective transformations are ubiquitous in the visual world as two different views of the same planar object can be related by a 2D projective transformation. standard projective group action is described by the projective special linear group PSI((3, (3, IR) acting on the 2D projective plane RP2, which can be interpreted as the set of equivalence classes of points (r, (cr, cy, cp) for any + O. Points in IR22 with p * 0 can be represented in inhomogeneous coordinates as (xc, y), corresponding to the homogeneous coordinate (xr, y: 1). Thus, the action projective transformation on 2D coordinates can be written as
I a. + By +Y
AE py + pur oy +
pz оy TT 1
(7)
where the transformation parameterized by the coefficient matrix
a 8
P
1
(8)
P
0
Since the transformation is defined up to nonzero scaling factor, we can normalize by requiring the determinant of P to be 1, i.c., A = det(P) = 1. Thus, there are independent degrees of freedom. The transformation reduces an affine transformation when p = O. while a pure projective transformation, characterized by p² o2 * O, exhibits nonlinear behavior. Thus, projective general and complex class of geometrie transformations.
transformations represent
For an r--imensional function u(x, y), the projective transformation of coordinates induces a natural action on the function, ü(ā, Ø) u(z, w). which can be further prolonged to its derivatives. We denote the derivatives of the i-th component function 1 as
where D, and D, are the differentiation operators with respect to a and y, respectively. Under projective transformation, these derivatives transform as
u D'
(10)
with the transformed differential operators given by
D3 = po + σy + T (((pp λσ) + μr Do) + ((pp dooy - Xr - up))
(11)
A
Dj pr + σy + ((lao 3pp) BT 70) Dr (lao 3p)y + - -p)
(12)
A
5

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2.4 Projective differential invariants of multi- functions
The projective group action is not free on the second-order jet space for scalar functions, requiring prolongation to the third-order jet space to achieve freeness. This leads to complex formulations [Olver 2023], which may limit the practicality of the resulting invariants due to their complexity and computational cost. Moreover, in practice, third-order derivatives are harder to estimate reliably from data than lower-order ones. In this work, we focus on multi-dimensional functions, which naturally align with applications such as color image processing. In this setting, the group action free on the second-order jet space, allowing the existence of second-order differential invariants. Using the method of moving frames, we can derive these invariants, where the choice of cross section significantly influences the simplicity of the resulting expressions. Although the cross section proposed by Olver [2023] can be extended to the multi-dimensional case, the resulting invariants tend to be lengthy, typically involving hundreds of terms, which makes them less practical. Instead, we propose an alternative cross section that leverages multiple dimensions while relying only on derivatives up to second order, yielding invariants with significantly more concise and tractable forms.
Specifically, we choose the following cross section:
y - K= fz=z=y=ol 0, uf = 1, u!!' = 0, u! = 0, u 11 uPy = 0}.
(13)
Hereafter, we use the standard shorthand notation for partial derivatives, e.g.. Ug, By Uzzz May: Myy- The above cross section defines 8 normalization equations, which, together with the constraint A = 1, determine all group parameters, thereby establishing the moving frame 1). The detailed derivation of the moving frame is provided in the Appendix.
With the moving frame 1) constructed, we can then apply the invariantization process according to Theorem to obtain a complete set of fundamental differential invariants. For the coordinates involved in the cross section, we have
(x) = 0, (ly) 0, (u(l)) = 1, (((())) = 0,
<( ((ll)) = 0, ((ul5)) 1, ((!ا))))) ((ul))) =.
The remaining coordinates of the second-order jet space yield the following differential invariants:
((w!))= Tni Ji2
(14)
T222
(15)
J82"
(())) = T212 + 2T122
(16)
2J7
T121 + 2TT12
((uB))
(17)
Ti
Jzi
((u)))
3gisn.
(18)
J12
Tii
((u)))
3sisn,
(19)
J12'
((li)) J12 + T212
3 35ism n
(20)
T3
2J12Tr + 22T1 + 33J22 Tue
35i5 n,
(21)
Ti
((ul)) J12T111 +2JJ1
3 35isn,
(22)
TE
where Tij and Tizk are two key quantities defined as:
Jij A u (
11| [i]
[8]
(23)
Task a uld u
u
[R + u [8] 11) D
yy
ml (ubflu + +u =llll
(24)
satisfying Tn = 0, Ju JJ, and Typ Taji-

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The invariants 14
together with the obvious zeroth- order invariants
So = (ul) Sinn)
form complete of second-order fundamental differential invariants of the projective group. Compared to projective invariants for scalar functions [Olver] 2023), our results involve up to second-order derivatives and are expressed more concise form.
2.5 Fundamental components of projective differential invariants
In the previous subsection, we have derived a complete set of second-order fundamental differential invariants. While relatively concise, their expressions are asymmetric and depend on the specific choice of the first two dimensions used the cross section. To obtain a simpler, more unified, and elegant formulation, This enables us to further simplify their structure while preserving completeness.
conduct deeper analysis of the fundamental components of these invariants.
Note that the numerators and denominators in (4) are all relative invariants. Thus, we focus on the properties of these relative invariants, as absolute invariants can be obtained by taking the ratio of two relative invariants with the same weight. Moreover, since the expressions are built from the basic quantities Jij and Tiyk, we will delve into their transformation properties and algebraic relationships.
We first present three classes of simplified relative differential invariants of the projective group.
Theorem 6 Let W (pz+aytt)* Then the following quantities are relative differential invariants of the projective group:
For i + j. Tig is relative differential invariant of weight W.
.
For 1 Forlsis Tai is relative differential invariant of weight W
For 1 S i.j S n, Taja + 2Tu is a relative differential invariant of weight W
These relative invariants are not functionally independent; rather, they can be transformed into one another. Given that there are 6n second-order fundamental differential invariants according to Section 2.4 expect complete and independent set of relative invariants to contain 6n elements. To this end, we investigate the transformation rules among the relative invariants and aim to identify minimal generating set sufficient to express all fundamental differential invariants. We start with the transformation properties of Tij-
Theorem For any indices 1 S i1, b2, i3 ds S n. the following equation holds:
Jais Tests + Jnsis Juis + Jeis Tezis 0.
(25)
This implies that for any four distinct indices i,, i2, 13, 14, the six pairwise combinations of Jij dependent such that once any five known, the remaining one be determined. Based on Theorem we can construct a subset of (Ti (i + j} that is sufficient to express all Tij-
Theorem Define the following sets of relative invariants:
S1 A JJ12, In-1,n)
(26)
S₂ a {J13. Jza..... In Jn-
(27)
Then S₁ U S2 is generating set for the collection (Ji, 1 i + j}. meaning that any Jip can be expressed as a finctional combination of these elements.
Jis+3 can be written in terms of Jt,+11, J++1,142 J422.439 Ts 2,4 and Ji+1,+4, all of which belong to S U Sz. By induction, any Tn can thus be recovered from the generating set. complete and rigorous proof is provided in the Appendix.
According Theorem 7
Before establishing the transformation relationships for Tyjk. we provide more compact representa- tion of Júj and Tizk to clarify their structural relationships:
Jij = g, Qg,
(28)
Tize = g. Q H,Qg:
(29)
7

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Relative Projective Invarients
Unuu
Hidden Layer
R
Projective Invariants
Figure 2:
single layer of PDINet. (1) Compute projective invariants from relative projective invariants. (2) Combine these invariants with an MLP to produce equivariant outputs.
where & (wbb b is the gradient of the i-th component function, H, is the Hessian matrix of the j-th component function, and Q fixed orthogonal matrix defined as:
uy
(-
0 1
Q
Notably, {Jry can be used to establish the transformation relationships between the gradients. Lemma For any distinct indices i, ki, the gradient gi can expressed terms of gi and gj as:
Jkj
Tui
gk
g:
J 8j
(30)
Jij
Using Lemma and the form of Tiyk: in (29). derive the transformation rule for Tijk given {Ty} }.
Theorem 10 Let i + k. and let i' K',jK W. be arbitrary indices. Then Tr can be expressed as:
Te = J.kJwK Taga
JvkJwi + JviJwK
Tigk + IviJwi
Task.
(31)
T3
T2
T2
This result implies that, for fixed the triplet (Teje: Tisks Tayke serves generating set for (Tejk 115 8 K I n). provided {Tir is known.
With the transformation relationships for Jij and Tisk established, we can now construct a minimal set of relative invariants that suffices to express all the relative invariants Theorem
Theorem 11 Define the following sets of relative invariants:
S3 & [T11, Tez2. .... Jnnn
(32)
S, A [T121 + 2T112, Tiar + 2T T2-11 T + 2T 2T
(33)
S3 _ {T212 + 2Tz21, Taz3 + . + 2Tn.
(34)
Then Si U S2 U S3 U S4 S5 can express all the relative invariants in Theorem
be shown (see the Appendix) that the fundamental differential invariants (4 22 can be expressed via the relative invariants in Theorem 6 Therefore, we arrive at the following conclusion:
Theorem The union S A So U.S US invariants express all second-order differential invariants of the projective group.
S.. forms a complete set of relative differential
This set contains exactly 6n elements, matching the expected minimal number needed to express all second-orde fundamental differential invariants. Compared to the invariants derived in Section 2.4 the current formulation is further simplified and independent of the specific choice of the first two dimensions in the cross section, exhibiting a more unified structure. Moreover, while the original invariants involve polynomials of degree up to five, the present set only contains most cubic expressions, resulting in lower computational complexity.

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2.6 Projective equivariant networks
As discussed before, with a set of relative invariants obtained, we can convert them into absolute invariants by dividing each by another relative invariant with the same weight. We apply this procedure to relative invariants S to construct a complete set of fundamental differential invariants.
as the denominator, which is a relative invariant of weight W. We keep the elements in So unchanged, divide the elements in S1 U S2 by Ro, and divide those in S3 U.S U U.; by R.j This yields set of differential invariants sufficient to express all second-order fundamental differential invariants. In fact, this set with 6n 5 invariants may contain one redundant element, but completeness is of greater concern. To avoid division by zero, add positive constant the denominator during division, enhancing numerical stability.
Specifically, we select R4 = n (J12 + F23 +.... + Tn1)
Let = ( IN) ) denote the set of differential invariants we obtained, which naturally induces an equivariant operator I. Theoretically, the invariants I .Iv are sufficient to express all second-order differential invariants. In practice, leverage the expressive power of neural networks
to combine , In using a two-layer MLP to produce the output [Li el al. 22024] 2025
This
leads to a learnable equivariant operator:
Uout
hg I(u.))
(35)
where he is MLP parameterized by 0. By integrating this operator into standard network architec- tures, we can build projective equivariant models. We refer to the resulting model as the Projective Differential Invariant Network (PDINet), as illustrated in Figure 2
3
Experiments
For empirical evaluation, we conduct image classification tasks under out-of-distribtiion settings, where models trained the original dataset and tested images deformed by projective transformations. We adopt ResNet-18 [He et al. (016) as the backbone and replace its convolutional layers with our equivariant operators defined in (35) to construct projective equivariant network, PDINet. As the main counterpart, we consider homConv, a projective equivariant model proposed by MacDonald al. [0022]. To ensure fair comparison, using the same backbone. However, homConv relies on group sampling, which leads to exponential memory growth with network depth, resulting in out-of-memory (OOM) issues. Therefore, we follow the original network configuration of Dannalld et al. [2022] and reduce the number of samples to avoid OOM. In addition, we also include ResNet- 18 trained with projective data augmentation (DA) as reference baseline.
attempted implement homCony
3.1 Proj-STL--10
STL-10 [Coates et al. 2011 is dataset containing 5000 training images and 8000 test images. Each image has a resolution of 96 x 96 with RGB channels. We apply random projective transformations to the test set to generate the PT dataset. Models are trained the original STL-10 dataset (or with projective data augmentation for the DA baseline) and evaluated on Proj-STI--10, forming challenging dfitt--diittibution setting that assesses the model's ability generalize beyond the training distribution.
Table 1: Test accuracy (%) on Proj-STLL-10.
Model
Accuracy
# Params
ResNet-18
39.7310.33
11.18M
ResNet-18 with DA
48.73
11.18M
homCony
20.88+0.32
376K
PDINet (ours)
51.12++..17
8.90M
Each experiment is repeated five times with different random seeds, and we report the average accuracy and standard deviation in [ PDINet substantially outperforms the standard ResNet- 18 on the transformed test set, achieving an 11.39% improvement in accuracy. While projective data augmentation considerably enhances ResNet- 18's robustness, PDINet still achieves the highest accuracy. In contrast, homCony performs poorly due to its restriction network depth.

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3.2 Proj-Imagenette
Imagenette is a ten-class subset of the ImageNet dataset [Deng et al. 2009 consisting of 9469 training images and 3925 test images. All images are adapted to uniform resolution of 256 256 for model input. We apply random projective transformations to the test set to generate the Proj- Irag-emaeeeette dataset, while keeping the training set unchanged. This setup simulates and evaluates the model's ability to generalize to geometric transformations.
scenario
Table 2: Test accuracy (%) on moo-elmagee
Model
Accuracy 65.64.0.39
# Params
ResNet-18
11.18M
ResNet- 18 with DA
70.77
11.18M
homConv
25. 25.72
376K
PDINet (ours)
71.30
8.90M
Each experiment is repeated five times with different random seeds, and we report the mean + standard deviation of test accuracy in T The results demonstrate that PDINet retains strong performance under distribution shift, outperforming the standard R by 5.66%. confirms the effectiveness of the projective equivariance of our model in enhancing out-of out-of-distribution generalization. Although ResNet- 18 with DA reduces the performance gap, it still lags behind PDINet. Meanwhile, the shallow homConv model struggles to handle higher-resolution inputs due to its limited depth.
Detailed experimental settings and implementation details can be found in the Appendix.
4 Conclusion
In this work, we propose PDINet, framework for projective equivariant networks, based on second-order differential invariants of the projective group. Our method overcomes the exponential memory growth encountered by homConv [Mac (MaaDonald et al., 2022], enabling effective scaling to deeper networks. Leveraging the moving frame method and carefully chosen cross section tailored to muuti-diimesioonl functions, we derive complete and concise set of second-order projective fundamental differential invariants. Further analysis reveals transformation relationships among projective invariants, allowing us obtain unified and simplified formulation that enhances both theoretical clarity and computational efficiency. Building upon these invariants, we design a learnable projective equivariant operator that can be seamlessly integrated into various network architectures. is the first time to achieve full projective equivariance in deep networks without group sampling or discretization. Experiments under dittof-distributionn settings demonstrate the strong generalization ability of our model. With the prevalence and significance of projective transformations in vision, PDINet holds promising potential for broader applications in computer vision.
One limitation of our approach is that the second-order invariants we derive vanish the one- dimensional case, preventing the direct application of PDINet to grayscale images. In addition, this work focuses on group actions on scalar fields and does not yet cover more general cases involving arbitrary group representations, which we consider valuable direction for future research.
Acknowledgments
Z. Lin was supported by National Key R&D Program of China (2022D0160000)) the NSF China (No. 62276004) and the State Key Laboratory of General Artificial Intelligence.
References
Stan Birchfield. An introduction to projective geometry (for computer vision). Unpublished note, Stanford University. 14, 1998.
Georg Bökman, Axel Flinth, and Fredrik Kahl. In search of projectively equivariant networks. Transactions Machine Learning Research, 2023.
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A Related work
A.1 Differential invariants
Differential invariants are quantities involving derivatives that remain unchanged under the action of given transformation group [Olver] 1993]. They have proven useful in various image analysis
applications [Mundy and Zisserman 1992 Olver et al. 1999 Hartley and Zisserman, 2003 Li et al.
2018). Leveraging the connection between differential invariants and symmetrie partial differential equations (PDEs) [Olver 1993). Liu et al. 2010 2013] constructed learnable PDEs as linear combinations of differential invariants, achieving shift and rotation equivariance. This connection has also been explored for symmetry informmed discovery of governing PDEs [Hu et al. (202dd]. The method of moving frames offers a systematic framework for deriving differential invariants given transformation group [Fels and Olver, 1999. OOver 2003 2015). Based on this approach, OOlerr 2023) provided a characterization of differential invariants of the projective group. However, since the projective group does not act freely on the second-order jet space of scalar functions, [2023) prolonged the group action to the third-order jet space, resulting in complex expressions that are computationally expensive and difficult to estimate from data due to the involvement of higher-order derivatives. Differently, our work focuses on multi-dimensional functions, where the projective group acts freely the second-order jet space. This allows to derive complete and concise set of second-order fundamental differential invariants, which we further simplify through structural analysis.
A.2 Equivariant networks
Early developments in equivariant networks primarily focused on designing architectures that are equivariant to discrete groups, such as cyclic or dihedral groups. A representative approach is the G-CNN [Cohen and Welling 2016aa framework, which models feature maps as functions defined on group and implements convolution-like operations via discrete permutations over the input domain. Building upon this viewpoint, subsequent works (Shen et al. 2020 Romero and Cordonnier He et al. 202T] developed a broader range of group-equivariant operators beyond standard group convolutions. To achieve equivariance with respect to continuous groups, steerable CNNs [Cohen and Welling, 2016b Weiler and Cesal [019]] were proposed, which treat feature maps as
2021
fields transforming according to specified group representations. This framework enables principled
handling of Euclidean groups in 2D and 3D settings [Weiler and Cesa 2019 Fuchs et al.
2020 Wang
jet al. 2020 Zhao et al. 2022] Shen et al. 2022 Liao and Smidt] 2022 Hu et al. 2024 2025a
by
explicitly encoding the group action the feature spaces.
Symmetries are ubiquitous and can be leveraged in model design, either from prior knowledge or discovered from data (Yang et al. 2023 2024 Hu et al., [025cle]. However, generalizing equiv- ariant architectures to high-dimensional Lie groups, such as the projective group, poses significant challenges. For such groups, deriving explicit steerable basis filters is often intractable, limiting the ap- plicability of stecrable CÑNs, while G-CNN based methods require group discretization or sampling, introducing scalability issues. Bökman al. 2023 explored equivariance in a projective sense and achieved equivariant models with respect to projective representations of certain simple groups, rather than the full projective group. Another related effort is Möbius Convolution (MC) [Mitchel et al. 2022]. which achieves equivariance to spherical Möbius transformations for geometry and spherical image processing tasks. While both MC and our method aim build equivariant models under certain projective group actions, MC operates on the Riemann sphere under the complex-valle vvauudd SL C), whereas our work considers planar projective transformations governed by SL, R), corresponding to the commonly studied projective transformations in vision. Additionally, EMIP [Finzi et al.) 2021 and EKAN (Hu al. 20251] incorporate arbitrary matrix group equivariance into MLPs and KANs [Liu et al. 2025). respectively, but they target linear transformations on vectors or tensors and are not directly suitable for image inputs. [Shen et al. [2024) combined otation-equivarian with data augmentation to obtain nearly affine invariant features, without achieving full equivariance. LieConv (Finzi et al., 2020) attempts to handle Lie group equivariance by sampling from the Haar measure, but struggles to extend to complex groups due to inaccessibility to the Haar measure. To mitigate this problem, MacDonald et al. [2022] performed group convolutions by computing the integral on the Lie algebra, resulting in a projective equivariant model, homConv. Nevertheless, it suffers from exponential memory growth as network depth increases, making it impractical for deep networks. Mironenco and Forré 2024] improved sampling efficiency via group decompositions, but
networks

---

their method remains constrained to affine subgroups. Besides these two mainstream approaches, G-CNNs and steerable CNNs. [i et al. [2024] 2025] resorted another route by constructing equiv- ariant networks based on differential invariants, achieving affine equivariance without requiring group sampling or discretization. However, their method confined to affine groups and does not generalize to the projective group. In particular, projective differential invariants for multi-channel inputs have not yet been developed, and the SupNorm normalization technique used to construct affine invariants cannot be directly extended to the projective group. In our work, we target projective equivariance by deriving a complete set of second-order fundamental differential invariants for multi-channel inputs using tailored moving frame construction. Through algebraic analysis, we obtain simplified projective invariants for practical use, which enable the construction of deep projective equivariant networks without relying sampling discretizing the group.
B Detailed proofs
In this section, we provide the detailed derivation of the moving frame and complete proof's of Theorems 8 and
For completeness, we briefly comment on the results that can be verified primarily through direct computation, while omitting the detailed algebraic steps:
Theorem 6 follows directly from the definitions of Jij and Tajk- together with the trans- formation rules for first- and second-order derivatives under the group action derived from
11) 12 followed by straightforward simplification.
Theorem 7 and Lemma can can be verified by substituting the definition of Tiy and expanding the expressions algebraically.
Theorem 10 can be proved by rewriting Teyk in the compact form g. Q Qgw shown in (29). then applying Lemma[9 to express the gradient vectors ge and ga in terms of g and gk. and simplifying the resulting expression.
B.1 Derivation of the moving frame
We begin by setting D 0, yielding
? = O.T -yy,
(36)
V = Ax 1.y.
(37)
Next, imposing u 1, ag. 0 gives
a u 11l (px + oy + T);
(38)
8 = u (px + oy + T).
(39)
Similarly, enforcing a|!1 0, u
[2]
1 leads
u (px + σy + T), 2
(40)
p = u (pr oy + T).
(41)
We then apply the conditions
[1]
[1]
O, which yield the equations for and T.
Bg
T12(
wy U
Uzy tl
[2] [1] [2] + My
T122
p
11
Tz12 + 2u (2) T1
(42)
T x(ul (T 2 T11) + y(
11] T212
2u (1llTT Ti12)
(43)
wiz T + 2u
[2] Ti12
Finally, setting A = determines the parameter p
[1] T212 2uz [2] T112 the
(44)
2J 12
/3
With all the parameters now fully determined, we obtain the moving frame 77.

---

B.2 Proof of Theorem 8
1. To prove the theorem, it suffices Proof. Define Jr. - (Jraik 1 can be expressed in terms of elements from Si S2. show that each Js for 1 s k S
We proceed by mathematical induction on k.
Base case (k = 1, 2): This holds by definition, as J11 = S1 and J/2 = S2.
Inductive step: Assume that for all & 5L-1 1 with / 3, each set Jj be expressed in terms of elements from S1 U S2. We aim to show that Jj can also be written using elements from S1 U S2.
By the induction hypothesis, the following quantities can all be expressed by elements in Si U S2:
Tis+1: J1,4t-1, J1+1,441-1, I441541: J:41-1,44-
For any J.s-1 E Jr, note that when 3, the indices i, 1, 1, + I are distinct. By Theorem 7 it follows that Jast1 can be expressed as function of these five relative invariants. Hence, Ter44 can be expressed by elements in S₁ U S2.
By the principle of mathematical induction, all Jg for any S k S n can be generated from Si U S2, completing the proof.
B.3 Proof of Theorem
Proof. Recall that Jig has been established in Theorem 8 and Taa already included in S3. So we focus proving that Tije 2Tz; be expressed in terms of elements from S1 U U.S. U.S. U.S.s. Define
T A (Tei4R, 27,,,44 1 i k}. TE - [T, 2T.,, k + 1 1 51 n)
(45)
(46)
Our goal is to show that each element in 7 and TV for S k S 1 can be expressed in terms of elements from S₁ U S2 U S3 U S4 U Sg. We present the proof for T the case of T follows analogously due index symmetry.
We proceed by mathematical induction on k.
Base case (k = 1): This holds by definition, T S4.
Inductive step: Assume that for all k < 1-1 with 2 2, each element in T can be expressed terms of elements from Si US2 US U U.S.s We aim to show that every element Te,+1, + 2Tr, E T also be expressed using this union. From the transformation rule in Theorem 10, we have:
+ 2Tr..44
(47)
(GTT=411HHHHH11111CC21 C2T + + C5TTA.H-11)
(48)
1 Cg(2TT-1--sHiat
-C(TH--IHHHH-11+77 - + TeH1st-1.44))
+ +CTTHHLAHHHHH + 2C4Tm C5 Ti,14111.1)
-20TTH-L C2T + C3
(49)
where the coefficients defined as:
C = Taat
J2 J
C2
C3
J2L-1,
T2t-1.14
T22 J2411,841
Jitts
C.
Cs
(50)
Jr,tt-11
J1,44-11
By the induction hypothesis, 2T. + Tr,+t-1,5 E T can be expressed using elements from S US, In addition, it holds that Tai E S3, T--1-144.41111 2 1/111 St.

---

and 27/+1-1,4414 + 711,641-1664 E S5. The only remaining terms are:
-2C -11 C2 T2HHH1t-114 C3T,,t1-1.
(51)
1
C2 C2T
2
+ +C5(CTT=H-14H-11itt -1T----1- + C5THHtst-1144)))
(52)
(53)
which can be expressed in terms of S1 US U US U.; well Hence, the full expression for Tis+1,5 + 27 is combination of elements in S₁ US2 U US USS
Therefore, by the principle of mathematical induction, all elements in T for any S can be generated from the given sets.
B.4 Proof of Theorem 12
Proof. To prove Theorem 12 it suffices to show that all second-order fundamental differential invariants of the projective group, namely the quantities in (14)- along with the set So, can be expressed in terms of the elements in S. Since So S, may disregard So in the following discussion.
Moreover, Theorem 11 has established that all relative invariants given in Theorem 6 be generated by elements in the union S₁ U S. U S3 U S. U S5 C S. Therefore, it remains to prove that the expressions (14) 22 can be written in terms of the relative invariants from Theorem
Among these fundamental invariants,
19p can be directly expressed using the relative invariants from Theorem [ Thus, it remains to focus on the final three expressions (20) (22) Since the
denominators are already included in the set S, it suffices to consider only the numerators:
T22 + T212 3 ssis n, 2J12T1, + J12T21 + 3J2TT12 35isn.
(54)
(55)
Ta + TT,;
3 iiis
(56)
We now show that each of these numerators can indeed be expressed using the relative invariants in Theorem
6
For
54
have
J12T2 + J2:T212
(57)
-I22(T222 2T22;) Izi (T212 + 2T 2T12T22: -2Jz:
(58)
J22 + 2T22)) + J2i (T212 + 2T221) - 2To2((CoTn T221 221 + C7 C2T22) 222:
(59)
J12(222 2T22)) Jzi (T212 + 2Tz21 22T12 TT2222
(60)
where the coefficients are defined as:
Ja J12"
Jn1
C
C7
(61)
J21
For 55
we first rewrite Tixe as:
Tui =aa + CoTi
(62)
=Cs (Tin + +TTm.) + C2Tis + Tad) 2C&TT
CTTT
(63)
2
-Ca(Tia + 2TTm:) + Co + Tri)
- -2Cs(CoTin + CTT12) Co + 2C5C7T12 + T22)),
(64)
where the coefficients are defined as:
Tzi
Jn
Cs
Cg
(65)
Jn
In
IV

---

Similarly, we rewrite T21 as:
T21 C6T211 + C7T212
(66)
Substituting these into 155 we have:
2 + T12 + 3T2:2112
(67)
-211
CC<(Tia + 2T11)) + Co(2Tia + +Tas)
+ 21 (2CC((CTTTu+ C7T112) 1 ColC - 200rTT1 C7
TI2(C6T221 + C7T212) + +3J2:711
(68)
-211 Cs + 27n)) C2(TT + Tasi) - (4J12668 +xCCCC) Tmn
(69)
For 56
have:
J1n + +211:111
(70)
ITnn 27Tn:) +2,, 7112 2512TH
(71)
I12( + +2Tm1) + 2Jr, 2T2(C67T + C7T112
(72)
I12(( 2T11i) -2J:26
(73)
This completes the proof that all second-order fundamental differential invariants in 14 22 are expressible in terms of the relative invariants in Theorem and thus can be generated by S
C Implementation details of PDINet
While our theoretical foundation is developed in the continuous setting, practical applications involve discrete image data defined on grid points, where derivatives must be approximated numerically. To this end, estimate spatial derivatives using Gaussian derivatives (Li al. 2018 He et al. 2022 Li al. 2024]. For example, the partial derivative with respect to x is computed via convolution Ex set 0.99 and use kernel size 9. 9.
as Be ~3 f JG2,, where G2 is a Gaussian kernel with zero mean and standard deviation a. In our implementation,
As mentioned in Subsection 2.6 we construct projective invariants by dividing each relative invariant by designated relative invariant Ro or R2. which is computed from the input image. To prevent division by zero, we add positive constant e to the denominator. We set = 1. After obtaining the invariants, apply SupNorm normalization [Li et al.) 2024 2025), which preserves equivariance and is beneficial for training stability. Then we combine the invariants using two-layer MLP, which implemented as sequence of two
convolutions with ReLU activation function in between.
Given a standard convolutional network architecture, we construct projective equivariant network by replacing each convolutional layer with our equivariant operator. If a convolutional layer has stride greater than 1, we insert an average pooling layer with a kernel size equal to the stride before the second 1 x 1 convolutional layer in the equivariant operator. Additionally, we also apply the same pooling when computing Ro to ensure resolution consistency between the numerator and denominator. Figure 3 shows the training loss curve the STL-10 dataset, demonstrating stable optimization behavior of PDINet during training.
Experimental details
All experiments are conducted on a single NVIDIA RTX 3090 GPU. Each experiment is repeated five times with independently generated test sets using random projective transformations, and we report the mean accuracy and standard deviation.
Experiments on Proj-STL-10. Models are trained on the 5000 samples of the STL-10 training set and tested on the 8000 samples of the Proj-STL--00 test set, which is generated by applying projective transformations to each sample the STL-10 test set. Specificallyy, we decompose projective transformation into affine transformation followed by horizontal and vertical pure projections,

---

2.5
2.0
Training Loss
1.0
0.5
0.0
200
400
600
800
1000
Epoch
Figure 3: Training loss curve of PDINet on the STL-10 dataset.
[Ite defined (z. - ( Itcrz and (z, 5) T (I+eay' I+eay respectively. The affine transformation consists random rotation between -90 and 90' scaling in the range [0.9, 1.1], shear within =4', and translation within [-0.1, 0.1], while the projection parameters 0₁ and c2 are uniformly sampled from 1-0.0001, 0.0001]. All images are normalized by channel-wise mean subtraction and standard deviation division. Following (Sosnovik et al.) 2019], data augmentation during training includes 12-pixel zero- sero-padding followed by random cropping to 96 x 96, random horizontal flipping, and Cutout (DeVries and Taylor 2017] with single 32 x 32 hole. We train the models for 1000 epochs using SGD optimizer with Nesterov momentum of 0.9 and a batch size 0.1 and decayed by factor of 0.2 at epochs 300, 400, 600,
of 64. The initial learning rate is and 800. For the DA baseline, since the projective group is a complex non-compact group and the specific range or distribution of transformation parameters test scenarios typically unknown, we adopt a considerably wide range of geometric transformations. Specifically, we apply random rotation between 180 and 180%, scaling in the range [0.3, 1.7]. shear within =445, translation within [-0.1, 0.1], and projection parameters C12 C2 uniformly sampled from [-0.002, 0.002). For homConv, use the Adam optimizer following the original setup of [MacDonald al. 2022] as training with SGD fails to converge, while keeping other hyperparameters identical to those described above.
Experiments on moog-netgeeetee.. Models are trained on the 9469 samples of the Imagenette training and evaluated the 3925 samples of the Proj-Imagenette test The Proj-Imagenettee dataset generated by applying projective transformations to each test image in the original Imagenette dataset, following the same procedure Proj-STT--10. All images normalized by subtracting the per- channel mean and dividing by the per-channel standard deviation. During training, data augmentation includes random resized cropping 224 224 and random horizontal flipping. We train the models for 100 epochs using AdamW optimizer with batch size of 64. The initial learning rate is set to 0.002 and decayed via a cosine annealing scheduler. The same strategy as in experiments on Proj-STL--00 used for the DA baseline, while homConv is also trained with Adam as in [MacDonald et al. 2022
E Additional experiments
E.1
Equivariance error
Table Equivariance error across different image resolutions.
Image size
16 x 16
32 x 32
64 64
128 x 128
256 x 256
Equivariance error (%)
0.34
0.14
0.08
0.04
0.02
Theoretically, the projective equivariance of our operators is rigorously guaranteed by the fundamental properties of differential invariants. implementation, however, derivatives are estimated on discrete grids, which inevitably introduces minor equivariance errors. To quantitatively evaluate this effect,
VI

---

we follow the protocol in MacDonald et al. 2022] and define the equivariance enror as
lgg
(u) (g u)ll lgg
Error
(u)l
where v denotes the equívariant layer and y is random projective transformation. Since the projective equivariance PDINet is intrinsic and does not rely on training, we measure this error using randomly initialized equivariant layer. We compute the error on the Imagenette test set, resizing images to multiple resolutions. As shown in T the equivariance error remains consistently small across all resolutions and decreases monotonically with increasing image size, which is expected due to more accurate derivative approximation at higher resolutions.
To further visualize equivariance, Figure 4 compares the features 9 (u) and (ggg u), showing that they are nearly identical and thus confirming the commutativity between the equivariant layer and projective transformations.
Projective
Transformation
Equiv ariant Layer
Projective Transformation
Equivariant Layer
Figure Visualization projective equivariance: the features g- --b (u( and (g u) remain consistent, illustrating the commutativity between the equivariant layer of PDINet and projective transformations.
E.2 Computational complexity
Our projective equivariant layer exhibits linear growth in both time and space complexity with respect to the input size. Specifically, the computation includes estimating derivatives, computing differential invariants, and combining differential invariants, each with linear complexity. To quantify the overhead, we compare PDINet and ResNet-18 in terms of memory usage and FLOPs using torchstat, and report the results in Table 4 As shown, both models exhibit línear scaling with input resolution, highlighting the scalability of our model to higher-resolution inputs. While PDINet consumes slightly more memory than ResNet-18, it requires fewer FLOPs.
To provide an explicit runtime comparison, we further evaluate both models on the same hardware by measuring the total time required to process 1000 RGB images (224 x 224) with batch size of 100. Each experiment is repeated five times, and the mean and standard deviation are reported in Table Despite its lower theoretical FLOOPs, PDINet runs slower in practice, primarily due to the highly optimized low-level implementations of ResNet-18, We expect this gap can be narrowed through further engineering and implementation optimization.
E.3 Application on keypoint detection
To further demonstrate the practical benefit of PDINet, we conduct an additional experiment on keypoint detection, task that inherently involves projective distortions and thus provides natural
VII

---

Table 4: Memory usage and FLOPs of ResNet-18 and PDINet at different input resolutions.
Memory (MB)
FLOPs
Input Size
ResNet-18
PDINet
ResNet-18
PDINet
32 32
0.53
0.93
3.72 x 10
3.65
10
64 x 64
2.10
3.73
1.49 x 10a
1.46
10
128 x 128
8.38
14.91
5.94 x 10a
5.84 x 10
256 x 256
33.50
59.63
2.38 x 10
2.33
10
512 x 512
134.00
238.50
9.51 x 10
9.34 x 10
Table 5: Runtime comparison 1000 RGB images (224 x 224) with batch size 100.
Model
ResNet-18
PDINet
Runtime (s) 3.61 40.35 6.84
setting to evaluate the effectiveness of built-in projective equivariance. We integrate PDINet as the backbone into the REKD [Lee et al. [022] framework for keypoint detection. To reduce confounding factors, we simplify the pipeline by removing the orientation estimation branch (which depends on steerable filters) and retain only the keypoint detection component. A three-layer PDINet is used and compared against a CNN baseline with the same architecture. The number of channels adjusted to keep the parameter counts on the same order of magnitude, with PDINet using fewer parameters overall.
We evaluate both models on the viewpoint split of the HPatches dataset, which includes 59 scenes. Each scene contains reference image and five target images captured from different viewpoints. resulting in projective distortions between image pairs. We follow the setup of Lee et al. [2022] for data construction and loss formulation, and train each model for 20 epochs using the Adam W optimizer with a cosine learning rate schedule (initial learning rate 0.01). A downsampling pyramid with scaling factor 1.2 is applied during training, and a symmetric pyramid with scaling factor v2 (plus an identity branch) is used inference, with two levels of down- down- sampling and up-sampling.
We report the Repeatability metric, which measures the consistency of keypoint detection under viewpoint changes. A higher value indicates better robustness. As shown in Table PDINNet achieves higher repeatability than the baseline while using fewer parameters, suggesting that built-in projective equivariance enhances geometric consistency in keypoint detection. This supplementary experiment complements our main results and demonstrates the broader applicability PDINet to real-world tasks.
Table 6: Keypoint detection results on the HPatches viewpoint split.
Model
Repeatability
# Params
CNN
42.1 45.2
8.3K
PDINet
3.2K
F
Discussion
While PDINet consistently outperforms the baselines in our main experiments, some misclassifications still occur. We believe these errors are largely attributable the nature of synthetically generated data. Synthetic projective transformations involve interpolation and padding, which can introduce aliasing, distortions, and unnatural edges, especially under strong shearing, scaling or non-orthogonal rotations. Such artifacts can interfere with the model's ability to maintain equivariance and thus degrade performance.
addition, since PDINet relies on differential invariants that depend on discrete approximations of derivatives, image resolution may also affect performance. For example, PDINet performs better on Irojeenmee than on Proj-STIL-10, may partially attributed to the higher resolution enabling more accurate derivative estimation.
VIII

---

We expect that applying PDINet to real-worid data at higher resolutions with naturally occurring projective distortions would help mitigate these artifacts and better demonstrate its full potential. In practical scenarios, such as multiview settings involving planar objects, transformations between different viewpoints are well modeled by projective mappings, making PDINet naturally suited for these cases. While exact equivariance may not strictly hold for non-planar or 3D objects, the projective inductive bias still contributes to improved robustness by approximately preserving geometric structure under near projective transformations, Exploring such extensions, including more general 3D settings, is promising direction for future work.
Another avenue for exploration is to extend our method to other transformation groups. According to the general theory of differential invariants [Olver] 1993). such invariants exist for any regular Lie group action on a smooth manifold, for example, S2 C) acting on the sphere, which corresponds Möbius transformations. the appropriate differential invariants and designing corresponding equivariant architectures. While identifying explicit, concise, low-order, and numerically stable invariant forms for different groups (e.g., the Möbius group) is highly non-trivial, our experience suggests that recognizing relative invariants as atomic components can greatly facilitate the construction and simplification of complete and practical invariant basis.
principle, our framework can adapted to these settings by deriving
IX


Reviews:
[{"review_id": "84wxbg7d97", "reviewer": "Reviewer_gqa7", "summary": "This paper intrdocues PDINet, a network architecture that is equivariant to homographies. The key idea is to construct differential invariants of the projective group using the method of moving frames. Specifically, the authors derive a set of second-order differential invariants for multi-channel input functions, thus supporting RGB images and intermediate CNN features. These invariants are combined with small MLPs to produce learnable equivariant layers, which can be used as drop-in replacements for standard convolutional layers.", "questions": "* Can the authors report accuracy on the unwarped test?\n* Can the authors add a baseline where ResNet-18 is trained with random projective data augmentations?\n* can the authors discuss runtime, memory and training?\n* what would happen if the network is applied to a near projection mapping? say, in a multiview setting?", "limitations": "the paper should put more emphasis on scalability and computational overhead", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**  \n(+) It's really refreshing to see a apper tacking equivariance of projective transformations. I think it's a great research topic with potentially valuable outcomes in practice\n(+) Thue usage of second-order differential invariants derived from the moving frame method is elegant and novel. Removing the need to sample/discretize the group \n(+) As far as i could tell, the paper is mathematically sound\n(+) improvement over non equivariant baseline is significant\n\n**Weaknesses**  \n(-) the evaluation design is lacking. Usually, it is common to show the vanilla (non equivariant network) performance with augmentatinos (see works like vector neurons, or G-CNNs). Also, reporting the network results on a non transformed data would give a clue regarding the results themselves -- is 51% on STL-10 considered strong?\n(-) i'm also missing some analysis --  what are failure cases? do they make sense?\n(-) the baseline homConv is only showed with OOM -- is there a setup where its performance could be evaluated and conpared? \n(-) i'm missing a discussion of computational cost, stability is also of interest given the usage of second order derivatives \n(-) classificaiton is generally not a problem where homographies are of interest. It is more suitable for point correspondences -- if the authors would have showen this it would really strenghten the paper in my view", "was_revised": true, "final_justification": "I appreciate the authors detailed response. I was mainly impressed by their effort in swiftly implementing the point correspondence experiment which as I argued and the authors acknowledged is the more common use case for this approach. The results look promising and will likely set an important baseline for future research. I'm happy to maintain my acceptance score for this work.", "rebuttal": "### Weaknesses\n\n> **W1:** the evaluation design is lacking. Usually, it is common to show the vanilla (non equivariant network) performance with augmentatinos (see works like vector neurons, or G-CNNs). Also, reporting the network results on a non transformed data would give a clue regarding the results themselves -- is 51\\% on STL-10 considered strong? \n\n**R1:** \nThank you for the suggestion.\nWe agree that comparisons with data augmentation baselines and evaluations on non-transformed data are common in other equivariant network studies.\nHowever, our work adopts the out-of-distribution (OOD) evaluation setup used by homConv [1], which is the most relevant prior work targeting projective equivariance and serves as our primary point of comparison.\nAs prompted by your suggestion, we have added a direct numerical comparison with homConv in Tables 1 and 2 (please refer to R3 for details).\n\nThis evaluation setup is designed to test whether a model can generalize to projectively transformed inputs without access to the specific distribution of test-time transformations.\nWhile our model incorporates prior knowledge of the symmetry group (i.e., the projective group), it does not rely on assumptions about the specific range or distribution of transformation parameters.\nIn contrast, data augmentation approaches require explicitly specifying and sampling from these parameter spaces, which are often unknown or difficult to specify in practice.\n\nOur focus is on assessing the model’s ability to generalize under complex geometric transformations via built-in symmetry.\nResults on non-transformed data, where projective distortions are rare, are therefore not the primary concern, as the goal is not to optimize in-distribution performance but to test generalization under transformation shifts.\nWe show that PDINet outperforms a ResNet with the same architecture in this setting, and scales more effectively than homConv, highlighting its advantages in both robustness and scalability.\n\nWe believe this evaluation protocol remains consistent with prior work [1] and is well-suited for assessing the intended capabilities of projective equivariant networks.\n\n> **W2:** i'm also missing some analysis -- what are failure cases? do they make sense? \n\n**R2:** \nIn our current experiments, we think the main sources of failure cases can be attributed to the following factors.\n\nTransformation-Induced Artifacts:\nSynthetic projective transformations involve interpolation and padding, which can introduce aliasing, distortions, and unnatural edges, especially under strong shearing, scaling or non-orthogonal rotations.\nUnder certain transformations, these effects are especially pronounced and can degrade performance by interfering with the model’s ability to maintain equivariance.\n\nPixel Discretization Errors:\nOur model leverages differential invariants, depending on discrete approximations of derivatives.\nAs a result, image resolution may have some impact on performance.\nPDINet performs better on Proj-Imagenette than on Proj-STL-10, may partially attributed to the higher resolution enabling more accurate derivative estimation.\n\nWe expect that applying PDINet to real-world data at higher resolutions with naturally occurring projective distortions would help mitigate these artifacts and better demonstrate its full potential.\n\n> **W3:** the baseline homConv is only showed with OOM -- is there a setup where its performance could be evaluated and conpared? \n\n**R3:** \nTo enable a direct numerical comparison with homConv, we follow the original network structure used in [1] and reduce the number of samples to avoid out-of-memory (OOM) issues.\nThe experimental results are shown in Tables 1 and 2.\nDue to its restriction on network depth, homConv struggles to perform well on image datasets with higher resolutions.\n\n**Table 1:** Test accuracy (\\%) on Proj-STL-10 after training on STL-10.\n| Model                      | Accuracy  | # Params  |\n|----------------------------|----------------|------------------|\n| ResNet-18                  | $39.73 \\pm 0.33$ | $11.18$ M  |\n| homConv                    | $21.26$          | $376$ K  |\n| PDINet (ours)              | $51.12 \\pm 0.47$ | $8.90$ M  |\n\n**Table 2:** Test accuracy (\\%) on Proj-Imagenette after training on Imagenette.\n| Model                      | Accuracy  | # Params  |\n|----------------------------|----------------|------------------|\n| ResNet-18                  | $65.64 \\pm 0.39$ | $11.18$ M  |\n| homConv                    | $26.57$          | $376$ K  |\n| PDINet (ours)              | $71.30 \\pm 0.45$ | $8.90$ M  |\n\n> **W4:** i'm missing a discussion of computational cost, stability is also of interest given the usage of second order derivatives \n\n**R4:** \nOur projective equivariant layer scales linearly in both time and space complexity with respect to the input size.\nSpecifically, its computation include estimating derivatives, computing and combining differential invariants, each with linear complexity.\nWe use torchstat to numerically measure the memory usage and the FLOPs of a single layer required for model inference.\nAs shown in Table 3, both the memory usage and the FLOPs scale linearly with the input size.\n\nPDINet consistently shows stable training behavior.\nWe plan to include convergence plots in future revisions.\n\n**Table 3**: The FLOPs and memory occupation of a projective equivariant layer.\n| Input Size     | Memory (MB) | FLOPs          |\n|----------------|-------------|----------------|\n|$32 \\times 32$  | $0.02$      |$5.53\\times10^4$|\n|$64 \\times 64$  | $0.09$      |$2.21\\times10^5$|\n|$128\\times128$  | $0.38$      |$8.84\\times10^5$|\n|$256\\times256$  | $1.50$      |$3.54\\times10^6$|\n|$512\\times512$  | $6.00$      |$1.42\\times10^7$|\n\n> **W5:** classificaiton is generally not a problem where homographies are of interest. It is more suitable for point correspondences -- if the authors would have showen this it would really strenghten the paper in my view\n\n**R5:** \nThank you for the insightful comment.\nWe agree that classification may not be the task most inherently associated with projective transformations.\nHowever, it remains a common and practical benchmark for evaluating equivariant architectures, as also adopted by homConv [1].\nSince object categories should remain unchanged under projective distortions, classification still serves as a meaningful testbed for assessing a model's ability to generalize under such transformations.\n\nThat said, we fully agree that tasks such as point correspondences may better reflect the utility of projective equivariance in real-world geometric transformation.\nMotivated by your suggestion, we conduct a preliminary experiment to evaluate the effectiveness of our proposed PDINet in the context of keypoint matching.\nSpecifically, we integrate PDINet as the backbone into the REKD [2] framework for keypoint detection.\nTo reduce confounding factors, we simplify the pipeline by removing the orientation estimation branch and retain only the keypoint detection component.\nWe use a three-layer PDINet and compare it against a CNN baseline with the same architecture. \nThe number of channels is adjusted to keep the parameter counts on the same order of magnitude, with PDINet using fewer parameters overall.\n\nWe evaluate both models on the viewpoint split of the HPatches dataset, which includes 59 scenes, each with a reference image and five target images captured from different viewpoints, inducing projective distortions.\nBoth models are trained for 20 epochs using the AdamW optimizer with a cosine learning rate schedule (initial learning rate 1e-2).\nFollowing [2], we adopt the same data construction and loss, and apply a downsampling pyramid with scaling factor $1.2$ during training.\nAt inference, we use a symmetric downsampling/upsampling pyramid with scale factor $\\sqrt{2}$ and an identity branch.\nThe number of down-sampling and up-sampling are both set to 2. \n\nWe report the Repeatability metric, which measures the consistency of keypoint detection under viewpoint changes.\nA higher value indicates better robustness.\nAs shown in Table 4, PDINet achieves higher repeatability than the baseline while using fewer parameters, suggesting that built-in projective equivariance enhances geometric consistency in keypoint detection.\n\nWe appreciate your suggestion, which has helped us demonstrate the broader applicability of PDINet beyond classification.\nWe believe this direction is promising and worth further exploration in future work.\n\n**Table 4**: Keypoint detection results on the HPatches viewpoint split.\n| Model    | Repeatability ↑ | # Params |\n| -------- | --------------- | ------------ |\n| CNN      | 42.1            | 8.3 K         |\n| PDINet   | 45.2            | 3.2 K         |\n\n### Questions\n\nResponses to Q1 and Q2 can be found in our reply to W1.\n\nResponses to Q3 and limitations can be found in our reply to W4.\n\n> **Q4:** what would happen if the network is applied to a near projection mapping? say, in a multiview setting?\n\n**A4:**\nThank you for the question.\nIn multiview settings involving planar objects, transformations between two viewpoints are related by projective mappings, making PDINet naturally suitable for such scenarios.\nFor instance, the HPatches dataset used in our keypoint matching experiment (Table 4) contains planar scenes captured from different angles, and PDINet performs well under these projective variations.\n\nWhile exact equivariance may not strictly hold in cases involving non-planar objects, we believe that the projective inductive bias still contributes to improved robustness by approximately preserving geometric structure under near projective transformations.\nExploring such extensions, including more general 3D settings, is a promising direction for future work.\n\n[1] MacDonald et al. Enabling equivariance for arbitrary Lie groups (2022)\n\n[2] Lee et al. Self-Supervised Equivariant Learning for Oriented Keypoint Detection (2022)"}, {"review_id": "cpecjqJLca", "reviewer": "Reviewer_6v2q", "summary": "This paper introduces a deep neural network equivariant to projective transformations. The core contribution is the theoretical derivation of a complete and concise set of second-order fundamental differential invariants for the projective group. These invariants are derived using moving frames with a novel cross-section for multi-dimensional functions (e.g., multi-channel feature maps), which allows the use of second-order derivatives, unlike previous approaches for scalar functions that required more complex higher-order derivatives. The resulting neural network is sampling-free and thus does not have the scalability problem as previous work.", "questions": "Please address the mentioned weaknesses. \n\nBesides, how are first and second-order partial derivatives computed in practice within the network? Does the pixel discretization affect the accuracy of the computed invariants and the overall performance of the network?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:\n\nThe paper could be the first work that tackles the problem of developing neural networks with projective equivariance, which is common in camera geometry, in a scalable way. The theoretical framework based on differential invariants is novel and solid. The framework leads to a scalable network implementation, which is impressive. \n\nThe paper is well-structured and well-written. The paper requires a substantial mathematical background to follow, but the authors did a good job explaining the background and the derivations. \n\nWeaknesses:\n\nLimited Experimental Comparisons: The primary baseline is a standard ResNet-18, which is not designed for geometric robustness. While the OOM issue with homConv is a valid point about scalability, the lack of any direct numerical comparison to it is a weakness. It would strengthen the paper to include a comparison on a shallower network where homConv could run. Furthermore, a comparison to a strong data augmentation baseline (i.e., ResNet-18 trained on projectively augmented data) is missing. Such a comparison helps disentangle the benefits of the equivariant architecture from the benefits of simply exposing the model to the test distribution's transformations during training. \n\nComputational Cost Analysis: The paper argues that its invariants are more concise and computationally efficient than alternatives. However, it lacks a practical analysis of the computational overhead. How does it compare with the original ResNet-18 in terms of training/inference time, memory consumption, and FLOPs?", "was_revised": true, "final_justification": "The paper's theoretical and technical contributions are outstanding. After providing the extra information and experimental results, I recommend accepting this paper.", "rebuttal": "### Weakness\n\n\n> **W1:** Limited Experimental Comparisons: The primary baseline is a standard ResNet-18, which is not designed for geometric robustness. While the OOM issue with homConv is a valid point about scalability, the lack of any direct numerical comparison to it is a weakness. It would strengthen the paper to include a comparison on a shallower network where homConv could run. Furthermore, a comparison to a strong data augmentation baseline (i.e., ResNet-18 trained on projectively augmented data) is missing. Such a comparison helps disentangle the benefits of the equivariant architecture from the benefits of simply exposing the model to the test distribution's transformations during training.\n\n**R1:** \nThank you for the valuable suggestions.\n\nTo enable a direct numerical comparison with homConv, we follow the shallow network structure used in [1] and reduce the number of samples used to approximate convolution and pooling, thus avoiding out-of-memory (OOM) issues.\nThe experimental results are shown in Tables 1 and 2.\nDue to its restriction on network depth, homConv struggles to perform well on image datasets with higher resolutions.\n\nWe also agree that comparisons with data augmentation baselines are common in other equivariant network studies.\nHowever, our work adopts the out-of-distribution (OOD) evaluation setup used by homConv [1], which is the most relevant prior work targeting projective equivariance and serves as our primary point of comparison.\nThis evaluation setup is designed to test whether a model can generalize to projectively transformed inputs without access to the specific distribution of test-time transformations.\nWhile our model incorporates prior knowledge of the symmetry group (i.e., the projective group), it does not rely on assumptions about the specific range or distribution of transformation parameters.\nIn contrast, data augmentation approaches require explicitly specifying and sampling from these parameter spaces, which are often unknown or difficult to specify in practice.\nWe believe this evaluation setup remains consistent with prior work [1] and is well-suited for assessing the intended capabilities of projective equivariant networks.\n\n\n**Table 1:** Test accuracy (\\%) on Proj-STL-10 after training on STL-10.\n| Model                      | Accuracy  | # Params  |\n|----------------------------|----------------|------------------|\n| ResNet-18                  | $39.73 \\pm 0.33$ | $11.18$ M  |\n| homConv                    | $21.26$          | $376$ K  |\n| PDINet (ours)              | $51.12 \\pm 0.47$ | $8.90$ M  |\n\n**Table 2:** Test accuracy (\\%) on Proj-Imagenette after training on Imagenette.\n| Model                      | Accuracy  | # Params  |\n|----------------------------|----------------|------------------|\n| ResNet-18                  | $65.64 \\pm 0.39$ | $11.18$ M  |\n| homConv                    | $26.57$          | $376$ K  |\n| PDINet (ours)              | $71.30 \\pm 0.45$ | $8.90$ M  |\n\n\n\n\n\n\n> **W2:** Computational Cost Analysis: The paper argues that its invariants are more concise and computationally efficient than alternatives. However, it lacks a practical analysis of the computational overhead. How does it compare with the original ResNet-18 in terms of training/inference time, memory consumption, and FLOPs?\n\n**R2:** \nThe conciseness of our invariants is relative to the invariants derived using the method in [2], which involves third-order derivatives.\nFor example, the projective invariants for scalar functions in [2] contain dozens of terms with degrees as high as 12.\nExtending such a method to the multi-channel setting would result in significantly more complex expressions with hundreds of terms.\nIn contrast, our derived invariants only involve up to second-order derivatives, with polynomial degrees no greater than 3 and fewer than 8 terms.\nThis conciseness facilitates both theoretical analysis and practical applications.\n\nRegarding computational overhead, we thank the reviewer for the suggestion, as it highlights the scalability of our model to higher-resolution inputs.\nOur projective equivariant layer exhibits linear growth in both time and space complexity with respect to the input size.\nSpecifically, the computation includes estimating derivatives, computing differential invariants, and combining differential invariants, each of which has linear complexity.\nTo quantify the overhead, we compare PDINet and ResNet-18 in terms of memory usage and FLOPs using \\emph{torchstat}, and report the results in Table 3.\nAs shown, both models exhibit linear scaling with respect to input resolution.\nPDINet has slightly higher memory usage than ResNet-18, but requires fewer FLOPs.\nIn practice, PDINet's inference speed is slower than ResNet-18, despite the lower FLOPs.\nThis is primarily because convolutional layers in ResNet-18 benefit from highly optimized low-level libraries, whereas our custom layers currently do not.\nWe will try to develop a CUDA-accelerated implementation in future work to further improve training and inference efficiency.\n\n**Table 3**: The FLOPs and memory occupation of ResNet-18 and PDINet.\n| Input Size      | ResNet-18               | PDINet                |\n|-----------------|------------------------|-----------------------|\n|                | **Memory (MB)**        | **Memory (MB)**       |\n| $32\\times 32$   | 0.53                   | 0.93                  |\n| $64\\times 64$   | 2.10                   | 3.73                  |\n| $128\\times 128$ | 8.38                   | 14.91                 |\n| $256\\times 256$ | 33.50                  | 59.63                 |\n| $512\\times 512$ | 134.00                 | 238.50                |\n|                | **FLOPs**              | **FLOPs**             |\n| $32\\times 32$   | $3.72\\times 10^7$      | $3.65\\times 10^7$     |\n| $64\\times 64$   | $1.49\\times 10^8$      | $1.46\\times 10^8$     |\n| $128\\times 128$ | $5.94\\times 10^8$      | $5.84\\times 10^8$     |\n| $256\\times 256$ | $2.38\\times 10^9$      | $2.33\\times 10^9$     |\n| $512\\times 512$ | $9.51\\times 10^9$      | $9.34\\times 10^9$     |\n\n\n\n\n### Questions\n\n> **Q1:** Besides, how are first and second-order partial derivatives computed in practice within the network? Does the pixel discretization affect the accuracy of the computed invariants and the overall performance of the network?\n\n**A1:** \nAs described in Appendix D, we estimate partial derivatives using Gaussian derivatives.\nFor example, the partial derivative with respect to \\( x \\) is computed via convolution as:\n\\[ \\frac{\\partial f}{\\partial x} \\approx f \\ast \\frac{\\partial G_\\sigma}{\\partial x} \\],\nwhere \\( G_\\sigma \\) is a Gaussian kernel with zero mean and standard deviation \\( \\sigma \\).\n\nPixel discretization does affect the accuracy of derivative computations, \nwhich in turn influence the computation of differential invariants and, consequently, the overall equivariance of the network.\nTo quantify this effect, we verify equivariance by following the approach in [1] to test the equivariance error, which is defined as:\n\\[ \\text{Error} = \\frac{\\| g \\cdot \\psi(u) - \\psi(g \\cdot u) \\|^2}{\\| g \\cdot \\psi(u) \\|^2} \\],\nwhere \\( g \\) is a random projective transformation, \\( u \\) is the input image, and \\( \\psi \\) is the equivariant layer.\nWe computed this error on Imagenette test images resized to multiple resolutions, as shown in Table 4.\nThe results indicate that the equivariance error remains small across resolutions and consistently decreases as image resolution increases, which is expected due to more accurate derivative approximation at higher resolutions.\n\n**Table 4**: Equivariance error.\n| image size         | $16 \\times 16$   | $32 \\times 32$   | $64 \\times 64$   | $128 \\times 128$ | $256 \\times 256$ |\n|--------------------|------------------|----------------|------------------|------------------|------------------|\n| equivariance error (\\%) | 0.34             | 0.14           | 0.08             | 0.04             | 0.02             |\n\n\n\n\n[1] MacDonald et al. Enabling equivariance for arbitrary Lie groups (2022)\n\n[2] Peter J. Olver. Projective invariants of images (2023)"}, {"review_id": "30wBuMunGN", "reviewer": "Reviewer_VbGP", "summary": "This paper demonstrates a novel method to design a projective equivariant neural network. It is based on finding the projective differential invariants explicitly and use these features to perform learning. The experimental result demonstrates its effectiveness and improvements.", "questions": "1. What is the computational complexity/cost of these operations? \n2. Would the authors extend experiment/baseline study? How to verify the exact equivariance experimentally? \n3. What type of changes would happen when we act a random projective group action to an image? Would the authors be able to show a few example visualizations in the main paper (or the appendix)?  \n4. Could the authors be sure to include more equivariant network literature?", "limitations": "Yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. Projective equivariance is an important type of equivariance, and there have not been many works addressing this. Having a projective equivariant design can be important for general vision problems. The significance and originality of this paper are good.\n2. This paper is theoretically strong, and the reviewer believes the derivation of differential invariants is correct.\n3. The experimental results demonstrate the initial performance and improvement of this method.\n\nWeaknesses:\n\n1. This paper is not formulated in its best. The derivation is important but could be moved to the appendix. The reviewer suggests that the authors move Figure 1 earlier in the paper.\n2. The paper demonstrates a promising strategy to enable projective equivariance, but the claim is not fully verified. The authors should find a way to visualize and verify that the map is equivariant. It is hard to fully understand what causes the improvement.\n3. The OOM issue might be resolved either by using a larger device or by reducing the sample size (if the reviewer understands the OOM error correctly).\n4. The experimental study and baseline comparison are relatively weak.", "was_revised": true, "final_justification": "With added experiments the reviewer is clear that this is a valid contribution in this community. As a research who had a few works on this domain, my comment can be a bit conservative, but the reviewer believes that this 'signature-like' method is valid and can be useful.", "rebuttal": "### Weakness\n\n> **W1:** This paper is not formulated in its best. The derivation is important but could be moved to the appendix. The reviewer suggests that the authors move Figure 1 earlier in the paper.\n\n**R1:** \nThank you for the suggestion.\nIn the next version, we will reduce the detailed derivations in the main text and move Figure 1 earlier in the paper.\n\n\n\n> **W2:** The paper demonstrates a promising strategy to enable projective equivariance, but the claim is not fully verified. The authors should find a way to visualize and verify that the map is equivariant. It is hard to fully understand what causes the improvement.\n\n**R2:** \nTheoretical verification: The projective equivariance of our operators is rigorously guaranteed by the fundamental properties of differential invariants, as shown by the equation:\n\\[\\hat{I}(g \\cdot u)(x) = I(x, g \\cdot u) = I(g^{-1} \\cdot x, u) = \\hat{I}(u)(g^{-1} \\cdot x) = (g \\cdot \\hat{I}(u))(x)\\].\n\nExperimental validation: To quantitatively verify equivariance, we follow the approach in [1] to test the equivariance error, which is defined as:\n\\[ \\text{Error} = \\frac{\\| g \\cdot \\psi(u) - \\psi(g \\cdot u) \\|^2}{\\| g \\cdot \\psi(u) \\|^2} \\],\nwhere \\( g \\) is a random projective transformation, \\( u \\) is the input image, and \\( \\psi \\) is the equivariant layer.\nWe computed this error on Imagenette test images resized to multiple resolutions (Table 1). \nResults demonstrate a monotonic decrease in equivariance error with increasing image size, which is primarily due to the more accurate estimation of derivatives at higher resolutions.\n\nVisualization: We acknowledge the value of qualitative demonstrations.\nIn the next version, we will add visualizations comparing features $g \\cdot \\psi(u) $and $\\psi(g \\cdot u)$ to illustrate equivariance more intuitively.\n\n\n**Table 1**: Equivariance error.\n| image size         | $16 \\times 16$   | $32 \\times 32$   | $64 \\times 64$   | $128 \\times 128$ | $256 \\times 256$ |\n|--------------------|------------------|----------------|------------------|------------------|------------------|\n| equivariance error (\\%) | 0.34             | 0.14           | 0.08             | 0.04             | 0.02             |\n\n\n\n\n\n\n> **W3:** The OOM issue might be resolved either by using a larger device or by reducing the sample size (if the reviewer understands the OOM error correctly).\n\n**R3:** \nTo enable a direct numerical comparison with homConv, we follow the original network structure used in [1] and reduce the number of samples to avoid out-of-memory (OOM) issues.\nThe experimental results are shown in Tables 2 and 3.\nDue to its restriction on network depth, homConv struggles to perform well on image datasets with higher resolutions.\n\n**Table 2:** Test accuracy (\\%) on Proj-STL-10 after training on STL-10.\n| Model                      | Accuracy  | # Params  |\n|----------------------------|----------------|------------------|\n| ResNet-18                  | $39.73 \\pm 0.33$ | $11.18$ M  |\n| homConv                    | $21.26$          | $376$ K  |\n| PDINet (ours)              | $51.12 \\pm 0.47$ | $8.90$ M  |\n\n**Table 3:** Test accuracy (\\%) on Proj-Imagenette after training on Imagenette.\n| Model                      | Accuracy  | # Params  |\n|----------------------------|----------------|------------------|\n| ResNet-18                  | $65.64 \\pm 0.39$ | $11.18$ M  |\n| homConv                    | $26.57$          | $376$ K  |\n| PDINet (ours)              | $71.30 \\pm 0.45$ | $8.90$ M  |\n\n\n\n> **W4:** The experimental study and baseline comparison are relatively weak.\n\n**R4:** \nThank you for your feedback.\nProjective equivariance remains a challenging problem, and to our knowledge, homConv [1] is the first and, so far, the only existing equivariant model designed for projective symmetry on the 2D plane.\nTherefore, we adopt it as our primary point of comparison.\nFollowing your suggestion, we have added a direct numerical comparison with homConv.\nOur experimental study is consistent with that of homConv and follows the out-of-distribution (OOD) evaluation protocol, which tests a model’s ability to generalize to projectively transformed inputs that are not seen during training.\nUnder this evaluation setting, PDINet consistently outperforms a ResNet with the same architecture and scales more effectively than homConv, highlighting its advantages in both robustness and scalability.\n\nIn addition, responding to suggestions from other reviewers, we have further extended our experimental evaluation to demonstrate the practical benefit of PDINet in the context of keypoint matching.\nThis task inherently involves projective distortions and is a natural fit evaluate the benefits of built-in projective equivariance.\nSpecifically, we integrate PDINet as the backbone into the REKD [2] framework for keypoint detection.\nTo reduce confounding factors, we simplify the pipeline by removing the orientation estimation branch (which depends on steerable filters) and retain only the keypoint detection component.\nWe use a three-layer PDINet and compare it against a CNN baseline of the same architecture. \nThe number of channels is adjusted to keep the parameter counts on the same order of magnitude, with PDINet using fewer parameters overall.\n\nWe evaluate both models on the viewpoint split of the HPatches dataset, which includes 59 scenes.\nEach scene contains a reference image and five target images captured from different viewpoints, resulting in projective distortions between image pairs.\nBoth models are trained for 20 epochs using the AdamW optimizer with a cosine learning rate schedule with initial learning rate 1e-2.\nFollowing the setup in [2], we use the same training data construction and loss function, and apply a downsampling pyramid with scaling factor $1.2$ during training.\nAt inference, we employ a symmetric downsampling and upsampling pyramid with scaling factor $\\sqrt{2}$, alongside an unscaled identity branch.\nThe number of down-sampling and up-sampling are both set to two. \n\nWe report the Repeatability metric, which measures the consistency of keypoint detection under viewpoint changes.\nA higher value indicates better robustness.\nAs shown in Table 4, PDINet achieves higher repeatability than the baseline while using fewer parameters, suggesting that built-in projective equivariance enhances geometric consistency in keypoint detection.\nThis experiment complements our main results and demonstrates the broader applicability of PDINet to real-world tasks.\n\n**Table 4**: Keypoint detection results on the HPatches viewpoint split.\n| Model    | Repeatability ↑ | # Params |\n| -------- | --------------- | ------------ |\n| CNN      | 42.1            | 8.3 K         |\n| PDINet   | 45.2            | 3.2 K         |\n\n\n\n\n### Questions\n\n\n\n\n> **Q1:** What is the computational complexity/cost of these operations?\n\n**A1:** \nOur projective equivariant layer exhibits linear growth in both time and space complexity with respect to the input size.\nSpecifically, the computation include estimating derivatives, computing differential invariants, and combining differential invariants, each of which has linear complexity.\nWe numerically measure the computational cost of a single projective equivariant layer and report the results in Table 5.\nWe use \\emph{torchstat} to calculate the memory usage and the FLOPs required for model inference.\nAs shown in Table 5, both the memory usage and the FLOPs scale linearly with the input size.\n\n**Table 5**: The FLOPs and memory occupation of a projective equivariant layer.\n| Input Size     | Memory (MB) | FLOPs          |\n|----------------|-------------|----------------|\n|$32 \\times 32$  | $0.02$      |$5.53\\times10^4$|\n|$64 \\times 64$  | $0.09$      |$2.21\\times10^5$|\n|$128\\times128$  | $0.38$      |$8.84\\times10^5$|\n|$256\\times256$  | $1.50$      |$3.54\\times10^6$|\n|$512\\times512$  | $6.00$      |$1.42\\times10^7$|\n\n\n\n\n\n\n> **Q2:** Would the authors extend experiment/baseline study? How to verify the exact equivariance experimentally?\n\n**A2:** \nPlease refer to our responses to W2, W3 and W4.\n\n\n\n\n\n\n> **Q3:** What type of changes would happen when we act a random projective group action to an image? Would the authors be able to show a few example visualizations in the main paper (or the appendix)?\n\n**A3:** \nA projective transformation preserves straight lines but introduces perspective distortion, such as changes in relative scale, loss of parallelism, and nonlinear deformation of shapes.\nFor example, a square may appear as a trapezoid or an arbitrary quadrilateral under a projective transformation.\nThese transformations naturally arise in multi-view settings, where a planar object is projected onto different image planes due to changes in viewpoint or camera pose.\nAppendix A includes a conceptual illustration of a projective transformation, showing how such distortions occur when projecting a planar object onto another plane.\nFor concrete examples, Figure 1 in [1] shows how projective transformations affect images in practice.\nWe will also include additional visualization examples in the next revision to help readers better understand the visual effects of projective transformations on natural images.\n\n\n\n\n\n> **Q4:** Could the authors be sure to include more equivariant network literature?\n\n**A4:**\nThank you for the suggestion.\nWe include a section of related works on equivariant networks in Appendix B.2, covering both discrete and continuous group equivariance approaches.\nIn addition, in response to Reviewer rwYg’s helpful comment, we will expand the related work section to include the study in [3], which explores equivariant networks under the action of SL(2, C) on the Riemann sphere —- a different type of projective group action. \n\n\n[1] MacDonald et al. Enabling equivariance for arbitrary Lie groups (2022)\n\n[2] Lee et al. Self-Supervised Equivariant Learning for Oriented Keypoint Detection (2022)\n\n[3] Mitchel et al. Möbius convolutions for spherical CNNs (2022)"}, {"review_id": "V9JbZ6CgBp", "reviewer": "Reviewer_rwYg", "summary": "This paper presents a novel architecture equivariant under projective transformations (which are defined as the action of SL(3, R) on the plane via fractional linear transformations) based on differential features that are provably invariant. The paper focuses on the theoretical exposition of the recovery of the proposed family of differential invariants, though empirical evaluations are included. The appendix focuses on practical considerations regarding the implementation.", "questions": "See above.", "limitations": "See above -- in particular, whether the generalization to other projective groups (such as SL(2, C) ) is not discussed. \n\nThis concern has now been satisfactorily addressed in the authors' rebuttal.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Overall, the authors method is relatively original, an extensive theoretical investigation is provided, and a simple neural module is proposed which serves as a building block and is empirically shown to be robust performance under the desired perturbations. On these criterion I feel this paper meets the bar for acceptance. \n\nA few key weaknesses are as follows:\n\nAs stated, it is not clear the authors proposed method generalizes to other projective groups, the most obvious similar case being the action of SL(2, C) on the (Riemann) sphere. (This action has been previously studied in https://arxiv.org/pdf/2201.12212, which is closely related to the authors work -- appearing even to define a class of SL(2, C) moving frames to facilitate equivariance, see section 5.1 -- but is not discussed in the related work section and should be). \n\nMost importantly, one of the authors main claims is that the proposed method is more efficient than previous approaches and is able to scale to higher resolution. This obviously appears to be true, and is demonstrated explicitly in experiments. However, the key practical benefit of this is that this should allow the model to actually be applied to real-world data. Unfortunately, such an experiment is lacking and in fact there is no evidence that projective equivariance is useful in practical problems other than being theoretically interesting. To be fair, such a motivation is not included in the work of MacLaughlan et al. nor any other paper I know of which seeks to take more challenging forms of equivariance. However, it begs the question of whether the authors work is necessary or useful from a practical perspective because the practical usefulness of projective equivariance is not justified in the literature.  \n\nSpecifically, the efficiency of the method means it can actually be applied to real-world data, and showing it can be effective in those cases, and outperform existing methods, is extremely important and what this paper is missing. If the authors can show something convincing along these lines, I will increase my rating. Essentially, all the ingredients for success are here, but it's up to the authors to be more ambitious and creative in their applications. Simply saying that the network holds promise for vision applications in the conclusion is not enough and is potentially misleading.", "was_revised": true, "final_justification": "This paper presents an original and scalable solution to handle projective equivariance.  Both the method's theoretical underpinnings and results are compelling, and the core module proposed by the authors is simple and can be used as a drop in layer in standard image networks. \n\nI think this paper meets the bar for a clear accept and I will be happy to argue for this paper.", "rebuttal": "### Weakness\n\n> **W1:** As stated, it is not clear the authors proposed method generalizes to other projective groups, the most obvious similar case being the action of SL(2, C) on the (Riemann) sphere. (This action has been previously studied in https://arxiv.org/pdf/2201.12212, which is closely related to the authors work -- appearing even to define a class of SL(2, C) moving frames to facilitate equivariance, see section 5.1 -- but is not discussed in the related work section and should be).\n\n**R1:** \nWe thank the reviewer for highlighting this important work.\nWe agree that the connection between our method and Möbius Convolutions (MC) [1] deserves explicit discussion.\nMC [1] achieve equivariance to spherical Möbius transformations and has demonstrated strong performance in geometry and spherical image processing tasks, particularly for triangular mesh datasets augmented with deformations.\nBoth MC and our method aim to build equivariant models under certain projective group actions on specific geometric domains.\nHowever, the groups considered and the underlying geometric settings differ substantially.\nOur work considers SL(3, R) acting on the 2D projective plane: $x'=\\frac{\\alpha x+\\beta y+\\gamma}{\\rho x+\\sigma y+\\tau},y'=\\frac{\\gamma x+\\mu y+\\nu}{\\rho x+\\sigma y+\\tau}$, where $\n\\begin{pmatrix}\n\\alpha&\\beta&\\gamma\\\\\\\\\n\\gamma&\\mu&\\nu\\\\\\\\\n\\rho&\\sigma&\\tau\n\\end{pmatrix}\\in R^{3\\times 3},x\\in R,y\\in R$,\nwhereas MC considers SL(2, C): $z'=\\frac{az+b}{cz+d}$ acting on the Riemann sphere through complex-valued Möbius transformations:\n$\\begin{pmatrix}\na&b\\\\\\\\\nc&d\n\\end{pmatrix}\\in C^{2\\times 2},z\\in C$.\nThus, while both fall under the umbrella of projective geometry, the symmetry groups and domains of action, planar vs. spherical, are distinct, as are the types of input data they address (planar images vs. spherical signals).\nAdditionally, the \"frame operator\" in MC is used to define how filters transform at each point on the sphere to satisfy Möbius equivariance, whereas the \"moving frame\" concept we introduce is used to compute differential invariants of the projective group.\n\nRegarding generalizability, Theorem 2.17 in [2] guarantees the existence of differential invariants for any regular Lie group action on a smooth manifold with finite-dimensional orbits.\nThis applies not only to SL(3, R) acting on functions over the 2D plane, but also to other projective groups such as SL(2, C) acting on functions over the sphere.\nTherefore, in principle, our method is extendable to these settings.\nActually, the spherical setting also aligns with prior works such as [3], which explore equivariant partial differential operators (PDOs) on spherical domains.\nTheir results indicate that both derivative estimation and invariant construction are practically feasible in such contexts.\nHowever, while the existence of differential invariants is well established, identifying explicit, concise, low-order, and numerically stable invariant forms for different groups (e.g., Möbius) is highly non-trivial.\nIn addition, how to design architectures that effectively leverage such invariants across different domains remains to be explored.\n\nWe will include the above discussion in the revised related work section.\nThank you again for bringing up this important point.\n\n\n\n\n\n\n> **W2:** Most importantly, one of the authors main claims is that the proposed method is more efficient than previous approaches and is able to scale to higher resolution. This obviously appears to be true, and is demonstrated explicitly in experiments. However, the key practical benefit of this is that this should allow the model to actually be applied to real-world data. Unfortunately, such an experiment is lacking and in fact there is no evidence that projective equivariance is useful in practical problems other than being theoretically interesting. To be fair, such a motivation is not included in the work of MacLaughlan et al. nor any other paper I know of which seeks to take more challenging forms of equivariance. However, it begs the question of whether the authors work is necessary or useful from a practical perspective because the practical usefulness of projective equivariance is not justified in the literature.\n\n> Specifically, the efficiency of the method means it can actually be applied to real-world data, and showing it can be effective in those cases, and outperform existing methods, is extremely important and what this paper is missing. If the authors can show something convincing along these lines, I will increase my rating. Essentially, all the ingredients for success are here, but it's up to the authors to be more ambitious and creative in their applications. Simply saying that the network holds promise for vision applications in the conclusion is not enough and is potentially misleading.\n\n**R2:** \nThank you for the detailed and thoughtful feedback.\n\nProjective transformations are fundamental in computer vision, as they describe common phenomena such as perspective distortion and viewpoint variation.\nIncorporating projective equivariance into network design offers a principled way to embed such geometric priors, enabling models to generalize under these transformations.\nWe believe this makes projective equivariance a meaningful and worthwhile direction to explore, both theoretically and practically.\n\nWe fully agree that demonstrating real-world utility is essential for establishing the practical value of this approach.\nHowever, progress in this challenging area has been gradual.\nThe first projective equivariant model, homConv [4], is limited to shallow networks and low-resolution grayscale inputs (e.g., MNIST) due to exponential memory requirements.\nOur work advances the field by enabling deeper networks and supporting high-resolution RGB images, allowing evaluation on realistic datasets including STL-10 and Imagenette.\nWhile further applications may require task-specific adaptations, we believe our contributions significantly improve scalability and bring projective equivariant networks closer to real-world deployment.\n\nMotivated by your suggestion, we go a step further and conduct a preliminary experiment demonstrating the practical benefit of PDINet in the context of a real-world task: keypoint matching.\nThis task inherently involves projective distortions and is a natural fit evaluate the benefits of built-in projective equivariance.\nSpecifically, we integrate PDINet as the backbone into the REKD [5] framework for keypoint detection.\nTo reduce confounding factors, we simplify the pipeline by removing the orientation estimation branch (which depends on steerable filters) and retain only the keypoint detection component.\nWe use a three-layer PDINet and compare it against a CNN baseline of the same architecture. \nThe number of channels is adjusted to keep the parameter counts on the same order of magnitude, with PDINet using fewer parameters overall.\n\nWe evaluate both models on the viewpoint split of the HPatches dataset, which includes 59 scenes.\nEach scene contains a reference image and five target images captured from different viewpoints, resulting in projective distortions between image pairs.\nBoth models are trained for 20 epochs using the AdamW optimizer with a cosine learning rate schedule with initial learning rate 1e-2.\nFollowing the setup in [5], we use the same training data construction and loss function, and apply a downsampling pyramid with scaling factor $1.2$ during training.\nAt inference, we employ a symmetric downsampling and upsampling pyramid with scaling factor $\\sqrt{2}$, alongside an unscaled identity branch.\nThe number of down-sampling and up-sampling are both set to two. \n\nWe report the Repeatability metric, which measures the consistency of keypoint detection under viewpoint changes.\nA higher value indicates better robustness.\nAs shown in Table 1, PDINet achieves higher repeatability than the baseline while using fewer parameters, suggesting that built-in projective equivariance enhances geometric consistency in keypoint detection.\n\nWe sincerely appreciate your suggestion, which help us expand our experimental scope beyond classification.\nWe believe this initial result demonstrates the feasibility and promise of applying PDINet to real-world vision tasks, and we look forward to exploring additional applications in future work.\n\n**Table 1**: Keypoint detection results on the HPatches viewpoint split.\n| Model    | Repeatability ↑ | # Params |\n| -------- | --------------- | ------------ |\n| CNN      | 42.1            | 8.3 K         |\n| PDINet   | 45.2            | 3.2 K         |\n\n\n\n### Limitations\n\n> **L1:** in particular, whether the generalization to other projective groups (such as SL(2, C) ) is not discussed.\n\n**R1:** \nPlease refer to our response to W1.\n\n\n[1] Mitchel et al. Möbius convolutions for spherical CNNs (2022)\n\n[2] Peter J. Olver. Applications of Lie Groups to Differential Equations (1993)\n\n[3] Shen et al. PDO-eS2CNNs: Partial differential operator based equivariant spherical CNNs (2021)\n\n[4] MacDonald et al. Enabling equivariance for arbitrary Lie groups (2022)\n\n[5] Lee et al. Self-Supervised Equivariant Learning for Oriented Keypoint Detection (2022)"}]

Author rebuttal:

Meta-review: [Summary]

This paper proposes a network architecture with projective equivariance (equivariant to homographies). The main contributions of this paper are two-fold: (i) theoretical contribution: a set of second-order fundamental differential invariants for the projective group; (ii) practical network design: sampling-free (enabling scalability), easy drop-in replacement for standard convolution layers, and showing good empirical results.


[Strengths]

- The paper tackles a problem that is very underexplored, and proposes an idea with high originality.
- Theoretically, the paper has solid theoretical derivations and proposes a novel network design based on that.
- Empirically, the proposed framework has good scalability and robustness, with good experimental results.
- The paper is well-written.


[Weaknesses]

The reviewers speak very highly of the work. Most of the questions and concerns are well-addressed by the authors during the rebuttal period. If I have to point out some weaknesses, it could only be that: at the current stage, the problem and framework are not yet fully developed to make real-world impacts. But I think all research problems need to go through this beginning stage.

Also, as a lot of additional experiments and clarifications are added in the discussion period, I hope the authors can add them to the paper in their final revisions.


[Reasons for decision]

This paper received very high comments from the reviewers: proposing almost the first framework that can truly tackle a difficult problem, with both theoretical and practical contributions. The reviewers would also like to strongly champion this work.

As I also work in this field, I understand that having a work like this is non-trivial: many theoretically sound works struggle in experiments, while many works with good performances lack theoretical justifications. The small concerns/questions are well-addressed by the authors during the rebuttal period, with promising experimental results and detailed analysis.

Therefore, I recommend acceptance, and I think it deserves at least a spotlight. Either spotlight or oral makes sense to me. (I'm a bit unsure about the criteria between spotlight and oral.) I personally would like to recommend oral, as I prefer this type of risk-taking works, even if its real-world applications are not fully developed.


[Discussion summary]

Initially, the paper received an average score of borderline acceptance. All reviewers appreciate the originality of the work. Certain concerns were around: (i) lack of clarity on some technical details; (ii) the experimental comparisons/analysis were relatively weak; (iii) the task chosen in the experiments (classification) was not fully addressing the strength of the proposed method; (iv) not demonstrating enough capability/potential of real-world applications.

During the discussion period, the authors have added a number of experiments and clarifications to address the reviewers' comments. The reviewers have raised their scores to all acceptance, with multiple reviews strongly championing this work. I also appreciate all their efforts made in the discussions.


[SAC Update for final oral-spotlight calibration]

This submission explores how to build projective invariants, with applications to computer vision. Its main technical contribution is a novel architecture based on so-called differential invariants, which leverage relationships between group actions across several derivatives of the input field. 
Overall, this is a strong submission that tackles an under-explored problem, and where consensus quickly emerged amongst reviewers and AC. While highly promising, the method currently applies only to specific projective settings, somewhat limiting its significance. All in all, we recommend a spotlight presentation.
Decision: Accept (spotlight)