A Generalized Label Shift Perspective for Cross-Domain Gaze Estimation
Authors: Hao-Ran Yang
Xiaohui Chen
Chuan-Xian Ren
Venue: NeurIPS 2025 poster ()
Keywords: Gaze estimation
Generalized label shift
Distribution estimation
Primary area: applications

Abstract: Aiming to generalize the well-trained gaze estimation model to new target domains, Cross-domain Gaze Estimation (CDGE) is developed for real-world application scenarios. Existing CDGE methods typically extract the domain-invariant features to mitigate domain shift in feature space, which is proved insufficient by Generalized Label Shift (GLS) theory. In this paper, we introduce a novel GLS perspective to CDGE and modelize the cross-domain problem by label and conditional shift problem. A GLS correction framework is presented and a feasible realization is proposed, in which a importance reweighting strategy based on truncated Gaussian distribution is introduced to overcome the continuity challenges in label shift correction. To embed the reweighted source distribution to conditional invariant learning, we further derive a probability-aware estimation of conditional operator discrepancy. Extensive experiments on standard CDGE tasks with different backbone models validate the superior generalization capability across domain and applicability on various models of proposed method.

A Generalized Label Shift Perspective for Cross-Domain Gaze Estimation
Hao-Ran Yang" Sun Yat-Sen University
Xiaohui Chen"
Sun Yat-Sen University
Guangzhou, China
Guangzhou, China
yanghr266mai12. sysu. edu.
chenxh278@mail2 sysu www.:ssssuuu...uoomm edu.
Chuan-Xian Ren
Sun Yat-Sen University Guangzhou, China
rchuanx@mail. sysu edu. on
Abstract
Aiming to generalize the well-trained gaze estimation model to new target domains, Cross-domain Gaze Estimation (CDGE) is developed for real-world application scenarios. Existing CDGE methods typically extract the domain-invariant features mitigate domain shift in feature space, which is proved insufficient by General- ized Label Shift (GLS) theory. In this paper, we introduce a novel GLS perspective CDGE and modelize the cross-domain problem by label and conditional shift problem. A GLS correction framework is presented and feasible realization is proposed. in which an importance reweighting strategy based on truncated Gaus- sian distribution is introduced to overcome the continuity challenges in label shift correction. To embed the reweighted source distribution conditional invariant estimation of conditional opera-
learning, we further derive a probability- tor discrepancy. Extensive experiments on standard CDGE tasks with different backbone models validate the superior generalization capability across domain and applicability on various models of proposed method.
Introduction
Gaze estimation (GE) is crutial for understanding human attention in many areas, such human-robot interaction [40], virtual reality (3 and medical analysis 24 Recently, leveraging the capabilities of deep models, appearance-basse gaze estimation [8
17 has attracted wide attention due its low device requirements and end-to-end workflows. However, the great performance of deep models relies on the identical assumption of training and testing data distributions, which is hard to fulfill in real-world applications. When being applied across data domain, models' performance usually degrade dramatically due the domain shift caused by the changes of subjects, environment, etc. Therefore, dealing with the cross cross-domain gaze estimation (CDGE) problem is crutial for expanding application prospects of gaze estimation.
25
Existing CDGE methods can be divided into two categories: Domain Generalization (DG) 5
32 4
and Unsupervised Domain Adaptation (UDA) 110 1. Technically, DG methods mainly focus on removing gaze-irrelevant factors from samples to obtain domain-invariant features that are generalizable unseen target domains. And UDA methods typically align the feature distributions
'Equal contribution. *Coresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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so the predictor trained on them can be generalized to the specific target domain. Therefore, existing methods can be essentially summarized the fomain-invarian representation learning methods.
In this paper, we introduce a generalized label shift (GLS) perspective to CDGE problem. Formally, the GLS consists of distribution shifts in both the label and conditional distribution. We point out that in CDGE problem the difference of gaze range and concentration area results in label shift, while the conditional shift often arises from the difference of data collection environment. Integrating these two shifts, the CDGE problem is characterized as a GLS problem. According to GLS theory [3 23). when the label shift exists, the invariant representation learning is insufficient to correct the domain shift. Therefore, in contrast to previous work, we introduce the GLS correction framework as new paradigm settle the CDGE problem. An intuitive illustration is provided in Fig. !
Existing GL.S correction methods are primarily developed for classification problems, in which the label variables are discrete and finite. The la- bel shift correction methods typically require it- eration over all classes to estimate the class-wins wwiee distribution proportions, which is impractical for regression problems with continuous label vari- ables, such as gaze variable. On the other hand, the conditional distribution alignment method proposed for regression problem such as Condi- tional Operator Discrepancy (COD), is operator- based method that entails the expectation oper- ation on label variable, which is vulnerable to the label shift. Therefore, we propose a GLS correction method that is feasible for gaze es- timation. Specifically, the label distribution is modeled as a bivariate truncated Gaussian dis- tribution, based on which the continuous impor- tance weight function is estimated to correct the label shift. Further, a mrobability-wwar estima- tion of COD is derived, which enables the opera- tor embedding of reweighted source distribution. Generally, contributions summarized as follows:
DG
Gaze irelevantfacton
Remove
Labeled Source data
Invariant representtion leaming
redict
Label shift
correction
Alignment
Gaze
Probabiity-awaare
Unlabeled
Target data
Figure 1: Illustration GLS correction framework. The DG and UDA methods typically learn the in- variant representation across domains. Differently, the proposed GLS correction framework consider both the label shift correction and the probability- aware invariant representation learning. Note that the color of the "Alignment" component is split into orange and black to indicate that both GLS and UDA perform distribution alignment.
A novel GLS perspective is introduced and the CDGE problem is characterized as GLS prob- lem. The insufficiency of existing CDGE methods for successful cross-domain generalization is inferred from the GLS theory. Then GLS correction framework is presented as a new paradigm settle the CDGE problem.
GLS correction method feasible for CDGE problem is further developed to overcome the chal- lenges arise from the continuity of gaze variable, in which a continuous importance reweighting strategy is proposed and a probability-aware estimation of COD derived.
Extensive validations are conducted on standard cross-domain tasks with diffeerent backbone models. The proposed GLS correction method achieve SOTA performance in comparison experiments, and significantly reduces the prediction error by 27.2%, 26.3%, 19.1% and 12. 1% on four backbone models. The evaluation results highlight its superior generalization capability cross-domain tasks and applicability across backbone models.
2 Preliminary
2.1 Problem Setup
Let X and Y be image and gaze variables defined on X' and V, === {(x",, yy"}} and D = =(x", {(x*,²")} be source and target domains with elements sampled from variables following different distributions, i.e. (x:", ))
Pt Pxy- For distribution P, the lowercase p is the probability density function. Denote g X - E 2 as the feature transformation, Z + Y as the gaze predictor, then a learning model can be regarded tuple (g. h). Let
Pxx and (xf yy") (X

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: y - R, be loss function, then the CDGE problems aim to learn the model that minimizes the prediction error target domain
arg min e ( g) Eryy [e(h(o(X)), Y)], g.h
(1)
with only source samples and unlabeled target samples only labeled source samples available.
2.2 Cross-domain Gaze Estimation (CDGE)
Existing CDGE methods be roughly categorized as DG and UDA methods:
DG methods. Puregaze [5] and GazeCon [39] tackle the CDGE problems by gaze feature purifi- cation [3, which eliminate the gaze-irrelevant factors to obtain gaze-specific feature. GFAL [38
utilizes the gaze frontalization process and CUDA-GHR [12 adopts gaze redirection 44 as auxiliary learning task improve the generalization capability of features. An attractive advantage of DG methods that they don't need target domain samples and the gaze features can be generalized to
unseen target domain. However, the improvement usually limited due to the absence of knowledge from target domain.
UDA methods. Given some unlabeled samples from target domain, the UDA methods enhance the model performance in target domain by distribution alignment. DAGEN [O), PnP-GA 20 and
PnP-GA+ [88] align the source and target distributions by different metrics while CRGA a contrastive learning paradigm. Recently, UnReGA DUCA 46 propose to extract domain-invarian representations by reducing the uncertainty within features. Although these DG and UDA methods differ technology, they essentially focus on domain-invariann representation learning.
34 and
HFC 19 minimize the discrepancy
2.3 Generalized Label Shift (GLS) Theory
A joint distribution Pxy can be factorized Pxy Pr. Then the discrepancy between y and Px is naturally deduced the following shifts:
pΦ + Pt,
(label shift)
Px # P
(conditional shift)
Most of the works concentrate correcting of them and assume the other shift has been corrected. Specifically, conditional shift correction methods [21 aim to learn conditional invariant transformation g X + ZZ such that 121r = P while label shift correction methods [[6] 42
estimate reweighting strategy for source label distribution such that P/- Pe - w Pý. By integrating the label and conditional shift, GL-S problem is developed as a general setting on the joint
distribution. Recent theoretical results [50 31 23 have verified the sufficiency and necessity of GLS
correction for domain shift minimization:
(Necessity) If the label shift exists, the invariant representation learning is insufficient to minimize the domain shift.
(Sufficiency) The domain shift are sufficiently bounded by the label shift and conditional shift between domains.
However, existing GLS methods 45
30
are developed for classification scenarios with finite
classes. The typical class probability ratio estimation (B5] and class- conditional alignment methods [7] are infeasible for regression problems with continuous and infinite label variable.
GLSGE: Perspective and Methodology
Here we provide brief overview of the following sections. In Sec. 3.1 we introduce a novel GLS perspective CDGE problem. In Sec. 3.22 we present the GLS correction framework and point out that existing DG and UDA methods can be viewed partial realizations of this framework. In Sec. 3.3 and Sec. 3.4, we provide feasible realization of the framework, in which a continuous importance reweighting strategy and probability estimation of COD are proposed to overcome the challenges arise from the continuity of gaze variable. In Sec. 3.5 we summarize the proposed GLS correction method.
3

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3.1 A GLS Perspective of CDGE problem
By the triangle inequality, the target prediction enror can be easily factorized as
e o) (hog) + ( e'(h g)l
Iddentty Illumination
Iarget enror
source error
domain shift
P Target
(0g.
(4)
Background
where the last term often regarded as the do- main shift. Therefore, solving Eq. (I) with only source domain samples and unlabeled target do- main samples available is reduced to minimize the source prediction error (h g) and the domain shift ( o g) e² g)l. As mentioned above, the domain shift bounded by the GL.S problem between domains. And we further point out that the CDGE problem can be indeed characterized as the GLS problem. An intuitive illustration of the characterization is shown in Fig. 2 The detailed analyses are follows:
(a) label shift
(b) conditional shift
Figure 2: A GLS perspective of CDGE prob- lem. (a) The label probability functions differ in functions support (colored area) and probability values (color degree), resulting the label shift between domains. (b) Conditional distributions of the same gaze label may differ due to factors like identity, background and illuminations.
Label shift. CDGE problem, the source domain usually larger than target domain, in meaning that the gaze distribution of source domain has larger support. Meanwhile, in different application scenarios, people may focus their attention in different directions, leading to variety of gaze concentration area. So the probability of the same gaze may also differ. Generally, the observation above can be described as
supp phy * supp p
(5)
P+(() + (t))), By E supp pty O supp py-
where supp Py is the support of the gaze distribution. These formula induce the label shift Eq. (2)
Conditional shift. The difference sample level more intuitive. In different scenarios the appearance image may differ significantly due to the background, illuminations, etc, leading to the discrepancy of conditional distributions, i.c..
P * p By supp Pt O supp py.
Summarizing Eq. (5) and Eq. (6) the CDGE problem can be modeled as GLS problem.
Remark 3.1. From probabilistic perspective, both factorizations of the joint distribution-iiee. or P( -are mathematically valid. However, we adopt because it provides more intuitive and interpretable modeling perspective for the CDGE task. Further discussion can be found in Appendix
P(X, Y) = P(X|Y)
the factorization P(X.Y) Y)
3.2 GLS correction framework for CDGE
Now we introduce the GLS correction framework for CDGE problem as follows:
Estimating the importance weight function w(y) and reweighted Py.. 2 w(y)PP by
= arg min Liab (PS., Pt)).
(7)
Learning conditional invariant transformation with reweighted source distribution:
g arg min Loond (P2 P PZ
(8)
Learning gaze predictor h on reweighted P by
= arg min Lare (h PS, Psw);
(9)
where the h4PE denotes the pushforward distribution.

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Probabilty-awwre learning ox
P
P
p
p ay
P av
pv
(c) Predictor learning
(0
(a) Importance reweighling
(b) Invariant repressnnattion
RKHS
learning
(0g
p'
pr zyv zy=-ys
p
pr zIY=ya
Target prediction
(a) Label Shiff Correction
(b) Conditional Alignment
(c) Task-specific Learning
Figure 3: Overview of the proposed GLS correction method. (a) Reweighting source label distribution with bivariate Gaussian distribution estimated by pseudo target labels. The reweighted label distri- bution is utilized in both conditional alignment and task-specific training. (b) Conditional invariant representation learning. Two sets of conditional distribution are map to RKHS and the discrepancy measured by the PCOD. (c) After label and conditional shift correction, i.e., the GLS correction, predictor trained on source domain can be generalized to target domain. Note that the text color reflects the component each learning objective is associated with.
The terms (,) :) denotes the learning Objectives in each step. By considering different steps and distance distributions, this framework connected with existing CDGE methods
DG methods neglects the label shift since the target domains are unknown the training stage, which means = 1. Intuitively, DG methods aim feature transformation g that only extract task- specifce factors from samples of any domain. In other words, the feature distributions of all domains are aligned to be the task-specific feature distribution. Formally, considering distribution set P' { y hizz in which all potential target distributions are involved, the generalization training process can be described
arg min Care P2, Pq) + Asup LDG P2). g.h
(10)
VEI
Intuitively, a feature transformation is well generalized to any unseen target domain only when the extracted feature contains all task speciici information and nothing else, which is extremely hard to obtain as the target distribution set IP' is unknown and unreachable.
UDA methods greatly improve the generalization capability with few unlabeled target domain samples available. However, existing UDA methods in CDGE only consider the domain-invariant feature learning and neglect the label shift problem. The training process can be abstracted as:
arg min Lure (h Pž, P.) + A LUDA P2,, P2), 9.h
(11)
where the LUDA refers to marginal alignment loss conditional alignment loss. The ideal (g. h) in UDA method is easier to attain than in DG as there is some unlabeled samples accessible.
In summary, previous works essentially connected with the invariant representation learning and omit the label shift correction. According to the GLS theory, such methods are insufficient for successful cross domain learning when the label shift exists. Therefore, beyond previous methods, propose to tackle CDGE problem based on the GLS correction framework. In the following sections, provide feasible realization of proposed framework. Specifically, (1) modelize the gaze distribution as bivariate truncated Gaussian distribution for label shift correction and (2) derive an probability-awar estimation of conditional distribution discrepancy.
3.3 Label Probability Distribution Estimation
Due to the continuity gaze variable, impractical to correct label shift by existing classification- based methods that iterate over all classes to compute the class-wise distribution proportions. In fact,
5

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even though the number of distinct label values is finite in training stage, the gaze variable still admits the continuous property, which means the discrete clustering still infeasible. Therefore, we propose a continuous importance reweighting method that is feasible for CDGE problem.
Note that unlike in general regression tasks, the gaze variable has specific limited range. In fact, the range of gaze can be considered as rectangular region (or spherical region, depending on the choice of coordinate system). From a mathematical perspective, the gaze distribution has a compact support. Besides, in real-world scenarios, people's gaze is primarily concentrated in a certain area, with probability becoming lower towards the edges of the support. With these priors, we modelize the label distribution as a bivariate truncated Gaussian distribution with probability density function:
Py (g) frGau (y: p, U, b)
laxb(y) fGau (3): pl, Σ)
(12)
(Gau(U2233 14, Σ)- FGau(V123//4 E) Ga (1213 (. Σ) + FGau (111; Σ)
where 11 = (y1, (y1,32)² is the gaze value, Laxb(y) is the characteristic function. Let v11 (a1, b1) .112 = (a1, b2)"
"or")",o 2211 (ag, (02.b1)² ,222 = b2)" denote the vertices of the rectangu- lar range (a1, 02) x (b1, b2).
(aj, 02) and = b2) denote the intervals. faauly; μ. 2) and FGau (3/ μ. ) denote the probability and cumulative density functions of bivariate Gaussian distribution with mean μ and covariance E. By estimating the statistics (μ, Σ) with target pseudo label j' h(g(z') }), the importance weight function takes the form
w(v) = .rGaaly: fi", 6', a.
(13)
Py.
Then for any y E supp pý, the reweighted probability can be directly approximated by
p (v) w(y)p (v) frGau fi" 0", a, b),
(14)
where the interval vectors a and b are determined by the confidence area of Gaussian distribution.
Note that the truncated Gaussian distribution is only alternative to model the gaze distribution. Other continuous distributions with compact support, such as exponential family distributions, uniform distributions, or even shallow neural networks, can also be utilized. The choice of py be adapted to the characteristies of the specific task or dataset, e.g., in driver gaze datasets where gaze points tend concentrate around few specific directions like forward, mirrors, truncated Gaussian mixture model may provide a more accurate fit.
3.4 Probability-Aware Conditional Alignment
Most of the existing conditional invariant learning methods are developed for classification problem and are trapped in similar dilemma with label shift correction methods in regression problem, i.c., the class-wise distribution alignment induces infinite matching problems for Pxly on every possible y due to the continuity of label variable. To overcome this inevitable obstacle, latest work [41] propose Conditional Operator Diserepancy (COD) that admits the metric property on conditional distributions via the kernel embedding theory to align all Px y as a whole Pxly:
Me dCoo ( Pa 12 - 42 1:ke + tr(C22 + CZZZY 2C%
(15)
where UZiY and C2ZlY are conditional operators in Reproducing Kernel Hilbert Space (RKHS).
Note that COD embeds the whole label distribution Py rather than embeds (v) one by one to the RKHS. And it contains the expectation operation on the label variable. Thus, the empirical estimation provided in [1 is not capable of reweighted label distribution embedding. For example, the empirical estimation of Er, [r(Y)] for any function r(y) is E, py (v)r()) 1 Eiez r(us), regardless of the prior of distribution Py. In contrast, probability estimation with importance weight function w(y)
Σ Err-(w)rou) = Σ [u())(()))()) = 1 -
(16)
REI
iEI
Such change in estimation differs the distribution embedding. As we aim to align Pyyy with the reweighted Ps P2 rather than Ps Y it is necessary to derive a probability-awar estimation of COD,

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denoted by PCOD. We carefully complete the derivation, which involves extensive probability and matrix analysis. Here we directly present the results of the derivation and provide the details of the derivation process in Appendix C
Let (x); and {(x} ) be sets of samples drawn i.i.d. from source and target
domain, z; = ), z = g(x)) ) be the feature extracted by 9. kz Z x Z + R. by y x y + R be the kernel functions on and The kernel matrix computed as (K )ij kz zj), and so
as K2 K. Ky and Ky. Let q = (qI..... 9n)² be the discretization vector of reweighted source label distribution my be the number of distinct source sample labels. Then
Ppcon "
=tr(K =I) - QKKy +1I)-1Q
+ +r(K K + ml)) K (K + =enl=11
(17)
-2tr(K Q + +=1=== K (KY + =En) + str [ (G ~ + =1
2
+etr GG +mI,,)
M 2 M
vn
where the I1-11, is the nuclear norm, M and M are defined by MM = H +cI,)= and M-M-7 B + Other notations used in the formula are summarized in Tab.
Note that in practice the target label y is re- placed by target pseudo label. When the domain difficult for the predic-
1, (l...., 1)x
L. d
shift is significant, tor to produce high-quality pseudo labels at the begining of training process, which may trap the model into sub-optimal solutions. For this reason, we add a marginal aligmment term to improve the reliability of pseudo labels as done in Then the probability-awad conditional alignment loss formulated as
H. In 1
G H,, KH,
(i B = (VQ - q/a")
H. Q qqb BB"
Gy By
Table 1: Part of notations used in PCOD
Lcond (P p% amary (Pž, P2),
(18)
where the dmary can be any marginal distribution discrepancy metric. In this paper we adopt the DAGE-GRAM (26] that proposed for regression problem.
As the PCOD is derived for the continuity and label shift of gaze variable, it is applicable to other tasks with similar distributional properties, providing more seneral-purpose solution within the GLS correction framework. Of course, beyond distributional aspects, gaze estimation has other unique properties, such rotational equivariance. Integrating these Nomain--specin priors into the GLS correction method is an interesting direction for future work.
3.5
GLS Correction Model for CDGE
Combining Sec.
3.3 and Sec. 3.4 we obtain practical realization of the GLS correction framework:
p (v)
frGau ft ff,.*' ot b). (g. h) arg min L + XLcond
(19)
g.h
where A is the trade-off parameter. The task task-specific loss Leve is the reweighted C1 loss:
D Σ py- (vi) yfllı.
(20)
An intuitive illustration is shown in Fig. 3 For convenience, We shortly name our works as GLSGE.
A form of algorithm and computational efficiency analysis are provided in Appendix A.2
Note that the introduced GLS correction framework not limited to gaze estimation and can be potentially extended to broad range of tasks with similar distributional properties such as pose estimation. We would also like to emphasize that our contribution not only in the formulation of a general framework, but also in the design of feasible and effective implementation for CDGE. We believe this step is crucial when adapting the general GLS framework to different tasks, and it often requires task-specific insights and custom solutions.

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Experiment
Setups. We conduct experiments
on four standard CDGE datasets:
DE DDM
DE
De
DG
UTH-XGaze (De)
47 Gaze360
Method
Avg
DD
+ Da
DD
(De) MPM.IIIaaaC M [49]
and EyeDiap (Dp)
18. The face
ResNet-11
8.05
9.03
7.41
8.83
8.33
image data are normalized according to the preparation process summarized in 77. Details about these datasets are provided in Appendix A.I Following previous works, we adopt Dr. Da as source domains and Drs, Dn as target domains, as the former two domains have wider gaze distributions than the latter. Then four CDGE tasks are established: DE DM. DE + D. Da + Das and DG - >DD During cross-domain learning, of the unlabeled target domain images for training and another 10% for validation, with the remaining 80%
PureGaze
9.14
8.37
9.28
9.32
9.03
GazeCon
6.50
7.44
7.55
9.03
7.63
GFAL
5.72
6.97
7.18
7.38
6.81
AGG
7.10
7.07
7.87
7.93
7.49
FSCI
5.79
6.96
7.06
7.99
6.95
CGaG
6.47
7.03
7.50
8.67
7.42
DAGEN
5.73
6.77
7.38
8.00
6.97
PnP-GA
5.53
5.87
6,18
7.92
6.38
GSA-Gaze
6.45
7.25
6.37
8.95
7.26
HPC
5.35
6.24
7.18
8.61
6.84
DCUA
7.31
5.95
5.59
6.4
6.31
10%
PnP-GA+
5.34
5.73
6.10
7.62
6.20
5.31
6.21
5,43
7.30
6.06
GLSGE
10.02
1:0.04
=0.03
1:0.02
1:0.03
used for testing. It means that 4500 images in DM and 1667 images in
Table 2: Comparision with ResNet-18 18 as backbone model.
DD are used for training in each task. Further discussion about the influence of target training data
size is provided in Appendix A.3 while implementation details are provided in Appendix A.4 Each evaluation experiment is repeated five times and the average results are reported.
4.1 Comparison with SOTA methods
Comparison Methods. We compare GLSGE with (1) DG methods: PureGaze GFAL 38
AGG (2) FSCI [15], CGaG 377 and (2) UDA methods: DAGEN [IO], PnP-GA (20), GSA-Gazz 11
HFC 19), DCUA 46). PnP-GA+ [8] Following previous works, the evaluation are based on
ResNet-18 and ResNet-50. Angular error in degrees are reported as evaluation metric for all tasks.
Results with ResNet-18. In Tab. 2 we compare GLSGE with SOTA meth- ods that use ResNet-18 as backbone model. Note that the enhancement ap- proaches such as data augmentation that are widely used for improving per- formance are not applied in GLSGE. Nevertheless, GLSGE still exhibits superior performance among SOTA methods in average results and attains best performance Dr > DM and DG + DM tasks.
DE
DE
De
DG
Method
Avg
+ D
+ DD
+ DD
+ DD
ResNet-50
8.03
8.06
7.75
8.79
8.16
PureGaze
7.08
7.48
7.62
7.70
7.47
GSA-Gaze
7.62
8.14
9.83
10.02
8.90
GazeCon
8.35
8.8
8.24
8.83
8.56
AGG
5.91
6.75
9.20
11.36
8.31
FSCI
5.47
6.19
10.20
7.14
PnP-GA
6.58
6.79
5,70
7.14
6.55
HFC
7.68
9.54
8.86
7.66
8.44
DCUA
7.58
5.65
5.61
6.11
6.24
Results with ResNet-50. As shown in Tab. GLSGE achieves compara- ble performance in most of tasks and outperforms existing methods in av- erage result. In particular, GLSGE largely reduces the prediction error in task DG + Dar (from 5.61 to 5.27 compared with the second best) and achieves the second best performance tasks DE -> DM and DE - DD. Such evaluation results firmly verify the effectiveness of the proposed method.
PaP-GA+
6.49
6.61
5.64
7.09
6.46
5.54
6.10
5.27
7.14
6,01
GLSGE
40.03
Table 3: Comparision with ResNet-50 as backbone model.

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Method
Dr: - Du
DE
Dn
Da - D
De Dr
Avg
Res-18
8.05
9.03
7.41
8.83
8.33
Res-18 GL.GGE
5.31
34.0%
6.21 31.2%
5,43
26.7%
7.30
17.3%
6.06 v 27.2%
Res-50
8.03
8.06
7.75
8.79
8.16
Res-50 + GLSGE
5.54 31.0%
6.10
24.3%
5.27
32.0%
7.14 v 18.8%
6.01 v 26.3%
GazeTR
8.69
10.94
7.11
9.20
8.99
GazeTR + GLSGE
5.90
32.1%
9.07 v 17.1%
5,36 v 24.6%
8.75 V 4.9%
7.27 v 19.1%
FSCI
5.79
7.06
7.99
6.95
FSCI + GLSGE
5.38 v 7.1%
6.19 v 11.1%
6.08 w 13.9%
6,80 V 14.9%
6.11 v 12.1%
Table 4: Evaluation results of plugging GLSGE different models and SOTA method. Angular error in degrees and error reduction percentages with GLSGE plugged are reported.
4.2 Applicability Analysis
As GLSGE is general framework for CDGE problem, we futher investigate its applicability across different models. Apart from ResNet-18 and ResNet-50, we plug GLSGE into (1) GazeTR that employs vision transformer (ViT) backbone and (2) FSCL, the newly proposed SOTA DG method. As shown in Tab. 4 for the most widely used ResNet models, GLSGE significantly boosts the performance by over 25% in average results. In particular, GLSGE reduces the prediction error by 34.0% and 31 31.2% in tasks DE - DM and DE - DD compared with the ResNet- 18 baseline. For the ViT-based model GazeTR, GLSGE largely reduces the prediction error by 32.1% in task Dg + DM and 19.1% in average. Moreover, although FSCI is already the SOTA DG method, GLSGE further reduces the prediction error by 12.1 % in average. Note that the prediction error of FSCI- GLSGE is larger than ResNet-18 + GLSGE. The loss of task-specific factors during the DG training process might be the reason. Above all, the great improvement across models further supports the broad applicability of GLSGE and its potential value to the area as a plug-play method.
4.3 Ablation Analysis
The ablation experiment conducted on tasks DE + DD and DG - DM with ResNet- 18 as backbone model. The major components of proposed method are evaluated, i.e., the label shift correction and the conditional distribution alignment. The evaluation re- sults are shown in Tab. 5 (1) From the _nth rows, can be observed that both label shift correction and conditional alignment largely reduce the prediction error. particular, the PCOD is more effective than the original COD method, while both of them are negatively affected by the label shift problem. (2) As shown in the 5th and 6th rows, the combination of label and contional shift correction, i.c. the GLS correction, gains prominent improvement in both tasks, which exhibits the mutually beneficial relation between the modules. And the comparison between the 5th and 6th rows further verifies the reasonability of the probability- estimation for conditional distribution discrepancy.
Objectives
DB
DD
De - DM
Lire
9.03
7.41
L
6.70
6.37
Cave doon
7.42
6.58
Care + dPcoD
7.15
6.44
C dCOD
6.32
5.76
C + dPCOD
6.21
5.43
Table 5: Ablation study.
4.4 Visualization Analysis of GLS Correction
To gain insights into how GLSGE improves model's generalizability in target domain, a visualization analysis on DE + DM is provided in Fig. 4 (a)-(c): Before GLS correction, the source label distribution has much wider range than that of target distribution and the feature distributions differ severly, which implies the significant label and conditional shift between domains. The prediction error is large when the predictor trained in Dg directly deployed in Das (d)-(f): After GLS correction, the reweighted source label distribution overlaps with target label distribution to large extent, while the conditional distributions are aligned and the features are matched according to gaze values. The changes of color shows clear gradient of features so the labeling rules are similar for both domains. As shown in (f), the prediction distribution brought much closer to the ground-truth label distribution by GLS correction.

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(dM))(NN))
source trae
target true label
larged pesudo label
reweighted source lahel
For (lb) and (e)
(a) true label
(b) t-SNE before correction
(c) prediction before correction
0.9
source fisobure
lorget footure
(d) reweighted label
(e) t:SN after correction
prediction after correction
Figure 4: Visualization of GLS correction process. For scatter plot (a)(c)d)(t), label variables are all denoted by 'o' regardless of value and are distinguished from other domains by colors. contrast, for I-SNE figure (b) and (c), label values are denoted by color gradients and source domain features are denoted by "'' while target features are denoted by '+'
4.5 Hyper- paraaee
The sensitivity of pappeeearraeeerrs in Eq. (19 is analyzed on DE + D.- The variation of confidence alters the trun- cated area Eq. 12 while the decides the participation of PCOD in the traming process. The results shown in validate the robustness of proposed GLSGE method. With standard deviation of only 0.05, the deviation is slight in a wide range of parameters variation. Besides, the GLSGE is stable in random experiments, which also demonstrates the reliability of GL.S correction.
Avg: 5.40
Sid: 0.05
Min: 5.49
Max: 5.31
Fig. 5
4
confidence
Figure 5: Prediction error under different settings of syyeer-praametes.
5 Conclusion
In this work, introduce novel GLS perspective to CDGE and point the label shift and condi- tional shift phenomena in cross-domain learning to characterize CDGE as a GLS problem. A GLS correction framework is then presented as new paradigm for solving the CDGE problem due the insufficiency of existing CDGE methods for successful cross-domain learning that are inferred from the GLS theory. To overcome the challenges arise from the continuity gaze variable, we introduce the truncated Gaussian distribution for label importance reweighting and derive a probability- estimation of COD for conditional invariant learning. Numerical evaluation experiments on standard CDGE tasks with different backbone models validate the superior generalizabillty across domains and applicability on various models of proposed method.
Limitations and future work. (1) As domain shift correction is the core of GLSGE, target domain samples are required in training stage. While this enhances the model's generalization capability, its applicability to other unseen domains is limited. (2) The adopted conditional alignment method involves the computation of kernel matrices, which is computationally intensive. To address this issue, previous works such as random features can be employed to effectively reduce the computational load. More fundamentally, the proposed method is feasible implementation of the GLSGE framework. Therefore, more efficient conditional distribution metrics suitable for continuous variables can also be applied within this framework, which will be a worthwhile direction for future exploration. (3) The provided label distribution correction method is general but simple implementation. In real- world application, the label distribution should be estimated based on the prior knowledge of problem. For instance, in the task of driver's gaze estimation, the label distribution can be modeled as a Gaussian mixture model. In summary, the effectiveness and efficiency of the proposed general GLSGE framework can be further enhanced by problem-speciffe prior in application scenarios.
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Acknowledgements
This work is supported in part by National Key R&D Program of China (2024YFA101 1900), Na- tional Natural Science Foundation of China (Grant No. 62376291), Guangdong Basic and Applied Basic Research Foundation (2023B1515020004) Science and Technology Program of Guangzhou (2024A0464413) and the Fundamental Research Funds for the Central Universities, Sun Yat-sen University (24xkjc013).
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Notations
Notation
Description
De, Dr X. v.
Source and target domain. Input, output and feature space. Image, gaze and feature variables. Image, gaze and feature samples. Vectorized source images, gazes and features samples. Vectorized target images, gazes and features samples. Number of samples in source/target domain. Feature extractor and gaze estimator.
X, X,Y.Z
3, 1. [(x) , [ (1), 2
h)
Push-forward operator of h.
(xr((, v), (x,
Joint prohability of (X; Y)) in source and target domain. Marginal prohability of X in source and target domain.
(T), Px (a)
(3). (g)
Marginal probability of Y in source and target domain.
(zlv), Px (
Conditional probability of X given Y in source and target domain.
P (z P (zly)
Conditional probability of Z given Y in source and target domain.
(yla), Py Pyxx (vl)
Conditional probability of Y given X in source and target domain. Probability measure space. Loss function.
(())
"0.e e0 w(y)
Source and target domain prediction Label importance weighting function. Learning objective.
frGau fGsu
Probability density function of truncated Gaussian distribution. Probability density function of Gaussian distribution.
FGou g"
Cumulative density function of Gaussian distribution.
Pseudo label target gaze.
p. L
Mean and covariance Gaussian distribution.
a Ex
Estimated mean and covariance target pseudo label. Interval vectors truncated Gaussian distribution.
a (a1, 02),
(b1, b2)
U11. 012, V21, 122
Vertices of the rectangular range (a1, a2) x (b1, b2).
U 26
Conditional mean embedding of P and P
22 C2
Conditional covariance operators of P2 and P P2
Cz
Cross conditional covariance operator between P and Pajy- Regularization parameter.
Trade-off parameter.
kz. by
Kernel functions Z and
K% Ktt K
Kernel matrices on for source-source, target-target and target-source samples. Kernel matrices on Y for source-source, target-target and target-source samples. All-ones vector and identity matrix of size n.
Ky KÝ, K
I.. I.
H.,
Centering matrix of size n. Centered kernel matrix. Diagonal matrix discretization vector q.
Q
H
Centering matrix of Matrix defined by H, BB".
B
Gy
Centered reweighted kernel matrix on y
Table 6: Notations used in this paper.
A Experimental details
A.1 Datasets
ETH-XGaze (De) 147 is collected under varying lighting conditions with custom hardware. It
contains over one million high-resolution images different gaze directions under extreme head poses. Following the original paper,
756,540 images from 80 participants the training
Gaze360 (Da) 13 collected both indoor and outdoor environments with a 360 camera. contains data from 238 participants. We use only 84,902 images of frontal faces the training set.
21

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is collected during daily usage of laptops, including data from 15 participants. Following the standard evaluation protocol, 3,000 images from each subject are used for evaluation.
MPIIFaceGaze (DM) 49
EyeDiap (Dp) collected in laboratory environment with screen and floating ball gaze largets, which includes multiple video clips from 16 participants. We use 16,674 images from the screen target sessions of 14 participants for evaluation.
A.2 Algorithm
Algorithm 1: Optimization of GLSGE
Input: Source data y() and unlabeled target data {x]] source pretrained model
[g, h);
for N1 steps do
Estimate the reweighted label probability function pp (3) by Eq. 14
for N2 epoches do
Compute the conditional alignment loss Eq.
Compute the task-specific loss Eq. (20)
Optimize (g, h) by objective Eq. 19
end end
Output: Adapted target model (g" h²).
Emprically, N2 is set to and N₁ depends on the size of target dataset. The primary computational
load comes from the calculation of the exponential functions in Eg. (14 and Eq. (17). In commonly
used scientific computing packages, this calculation process 28 can be employed to reduce the computational load.
A.3 Influence of target training data size
As mentioned in Sec. we used 4500 images in DM and 1667 images in DD for cross-domain learning each task. Here we provide the number of target domain samples used by various DA methods in our comparisons:
Method
DAGEN
PnP-GA
GSA-Gaze
HFC
DCUA
PnP-GA+
Num.
500
<100
1000
100
100
<100
Table 7: Number of target domain samples used by various DA methods
While the amount of unlabeled target-domain data may be larger than that used in some DG or UDA methods, we note the following:
The unlabeled data is inexpensive and easy collect real-world scenarios (e.g., by simply capturing images with a camera), whereas labeled data requires careful calibration and anno- tation. Therefore, using slightly more unlabeled samples does not significantly increase practical costs. Besides, their effective contribution to training is much lower than that of labeled data, so small differences in sample size have limited influence on performance.
Many existing approaches implicitly increase the diversity and amount of target-domain data via aggressive data augmentation, style transfer, synthetic sample generation. Note that DG methods do not rely on target domain samples, but often heavily utilize data augmentation or auxiliary tasks to simulate domain shifts implicitly. While such methods implicitly enhance the effective size of training data, our method does not employ such augmentation strategies.
To address possible concern that performance gain might come from using more data, we conducted an experiment using only 100 unlabeled target samples, and report results below: Although performance degrades slightly with fewer samples, our method still achieves SOTA performance. It's also important to note that PnP-GA+, while using fewer samples, relies heavily extensive data augmentations and
22

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DR
DR
De
De
Method
Avg
D
D
DM
-> D
ResNet-18
8.05
9.03
7.41
8.83
8.33
GLSGE nt 1000
5.31
6.21
5.43
7.30
6.06
GLSGE n'n = 100
5.47
6,38
5.65
7.41
6.23
PnP-GA+
5.34
5.73
6.10
7.62
6.20
Table Comparision using only 100 unlabeled target samples with ResNet-18 as backbone model.
up to 10 auxiliary models. In contrast, our method uses no data augmentation auxiliary modules, making our results more interpretable and directly attributable the core methodology.
Finally, we briefly explain why our method remains effective even with fewer target samples. It's note that unlabeled data provides distributional information rather than supervision. In our method, label shift correction does not involve learning process and requires only moderate number of samples for direct estimation. Conditional shift correction relies on feature alignment, which is more sample-sensitive and accounts for most of the performance drop. Nonetheless, even with limited number samples, our method achieves competitive results.
A.4 Implementation details
For experiments using ResNet-18 18 and ResNet-50 as backbones models, the ResNet model pre-trained on ImageNet is assembled with two-layers MLP shallow feature extractor and linear layer as the gaze predictor. The deep backbone module of ResNet is frozen in cross-domain training process. use the Adam optimizer with the learning rate of 3e and cosine annealing scheduler to decrease the learning rate in the training process. The batch size is set to be 100. As the domain shift distinct at the beginning, we alternately correct the label shift and the conditional shift to produce better pscudo label. The confidence that decides the truncated area in label shift correction process is emprically set to 0.7 for all tasks.
For experiments using GazeTR as backbone models, the hybrid Gaze'TR is pretrained in Dg. All training parameters are consistent with the settings original paper ( For experiments using FSCI as backbone models, we directly download the pretrained models open-sourced by the authors. Similarly, we froze their deep backbone module in cross-domain training process and assemble shallow MLP model as in ResNet models.
NVIDIA RTX 4080 GPU is used for the experiments. The mean results five times random experiments are reported.
B Further discussions on factorization of joint distribution
In 3.1 we mentioned the factorization of joint distribution Px,y (z, y) and its implications for CDGE. Here provide more detailed explanation. Specifically adopted the factorization Px Px, (x (xly (y because:
=
P(Y) reflects the distribution of gaze directions in the population, which is naturally subject to domain-specifie priors (e.g., gaze behavior differs between indoor vs. outdoor settings). So the label shift naturally arises in CDGE.
P(X represents how visual observations (images) are generated given a particular gaze di- rection, capturing domain-specifie appearance variations (e.g., lighting, head pose, background), which significantly affects the predictor behavior.
In contrast, the alternative factorization ((X))) presents several limitations in our setting:
Aligning the marginal distribution P(X) is less informative, as it mixes samples from all labels and may not preserve task-relevant structure.
Domain shift P(Y)X), i.e., the posterior distribution, lacks clear, observable semantic meaning and is harder to control or interpret in practice.
23

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Therefore, we choose the P(X formulation as it better reflects the generative structure of the data gaze estimation and provides clearer insight into the nature of domain shift in this task.
C PCOD
We consider the estimation of PCOD as two parts: first-order and second order statistics terms
LcoDl x Px 1x) | - 1x llnn tx(Cxxxxv+++ CX 20 xx
(21)
lirst-odder
second- order
Let (xf.y)) and (xf, y:) be two set of samples drawn i..d. from source and target domain. For simplicity, n, and ny are both set to n. In kernel method, samples are mapped to RKHS H.x + My by the implicit feature map (Ø) Denote the feature map matrices by ($",, Vra) and (Φ', V the explicit kernel matrices K =x Ki
xf " are computed as (KX); kx (x] xf).
(ky)) ky (yf. respectively. And as K K and Ky.
Let I, be the n-dimensional vector with all elements equal and I, diag (1,) be the dimensional identity matrix, q = (41,-.., "n,) be the discretization of the target probability density function py (y) and Q diag(q) be the diagonal matrix induced by q. The empirical centering centering matrix depending on q can be defined as H, Q qq" Then the centralized kernel matrix is defined as Gy H Ky H., and Gy = T Ky B, where B = (vQ qrq") satisfies Hg = BB".
matrix can be defined as H, I, 1 1, 1 And the probability-
12
C.1 First-order term of PCOD
With above notations, the prohability-awan covariance operator ČxY" can be estimated as Exy" = "QU² Then the conditional mean operator can be estimated as
Úxp = -xr== +cI)
+QV DQQUQQQU (VQU +eI)-1
(22)
+Q(KrQ + - TT
the last equation is deduced by the following equations:
V (QQU + +I] = "QU +
(23)
v T QQQ + u
and
4 TTPQUT +eI)
y QQ + =l) LT
(24)
The the HS-norm of the condition mean operator can be calculated as
[ 1lss = - (Ky Q + - Q T- = tr
(25)
QKy + +cI) Q].
By replacing Q as the empirical I the estimation for target domain is derived
| = tr ( (K + =nnn) -' V'KK + -=nI)
(26)
(K* (KY +
(KY + +=nl)
The inner product term is then calculated as
Mixi==uxxp + Q(K*Q + (K + -
(27)
tr [K
Enl) +
Then the first part of PCOD is derived:
&
tr (K - -
+ tr (K
(28)
- 2tr + -
24

---

C.2
Second-order term of PCOD
For the second-order terms, the cross conditional operator can be estimated
Cx = Êxx - Cxr (Cry- + =I) Cy
- x 2H 4 W v0 +
(29)
@B In - 1h-BB'ya B"y (VB - B B"
By SWM formula [36 the inverse term can be calculated as
(Crr- + =T) (el VB((B))))
LD (BIn + "PKyB)-'B"u
1 -B BB'KyB-B'KKrB' + (BKBB)-''K,Bl Bd'
+B
I,
1
I -Gy - Gy + Gy-) BT
E
(30)
For the term I, - (Gy- - Gy- (sl, + Gy Gy=)) we perform eigenvalue decomposition on
Gyw such that Gy UDUT where U is orthogonal and D is diagonal with d: 2 0. Then we can simplify the term as
In Gy - + GGy
I UDUT - UDDU"
L [UDUT UD(EL, - DU
(31)
I - -D- + D) DD) U"
UD'U",
where D' is diagonal and
d? >
dgf d/ (d:
(32)
dg +
dg +
Finally, it is derived as
In (Gy (sI, Gy Gyy) e(Gy + elm)
(33)
Then the conditional covariance operator is simplified as
Čx IBI, (Gy
Gy - "5
BB T
(34)
4B: + =I,) 1 -BB BB
+B=(GGy- + =I,) - B TT.
where L = E - .nn the cross conditional covariance operator is then calculated as
est CXX
CiXIx V
B'BL-BTT* (', V' B
(35)
Vn. Vv /D T. D'M( PM TT
VT (VV B.M'M''''' th' 3'M) B.M-M'' क. T'M)
25

---

So the trace term is
1
un( x((xxx)-)==
trV (V D''MM " 4'M)" 4' -- M
Vn
MIVMMMD (D'M-M ''
Vn
(36)
" (M
| |MKKMM"
where the M and M are defined by H, = MM" and BL"B" = M Summarizing the results on conditional operator. the second part of PCOD is then derived:
de dCKB- ctr (Gix ( + +II)) + etr (G'x + =nd,)) vn
|M |M
(37)
26


Reviews:
[{"review_id": "1Lu0si4wDs", "reviewer": "Reviewer_W2yU", "summary": "This paper proposes to introduce GLS into the CDGE task, and designs a design that simultaneously utilizes GLS, UDA, and DG techniques to improve the performance of CDGE. The paper introduces a lot of theoretical analysis and derivations to introduce GLS into CDGE tasks.", "questions": "Alignment in Fig.1 and Probability-aware learning with ... in Fig.3 are in different colors, kind of confuse me, does this have a special meaning?", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\n1.  The paper has a detailed ablation study, feature visualization and hyper-param visualization.\n2.  The paper includes detailed propagation, which makes the paper more theoretically sound.\n\n**Weaknesses**\n\n1.  There are too many notations in the paper, which makes the paper not easy to follow.\n2.  The results from other domains to EyeDiap are not good.\n3.  Although the paper puts the enhancement of GLSGE by Res-18, Res50 and GazeTR, the enhancement is not easy to see compared with the enhancement of other methods on these backbones, if we can put the comparison of different methods on GazeTR+X (X is other methods), we can understand this paper better.\n4.  Although the paper puts a lot of analysis of the introduced GLS theory, the results on the EyeDiap dataset are not so good, which makes me concern whether the theory is suitable enough for this estimation task.\n\n**Minors**\n\nMinors.\n\n1.  L8: a → an\n2.  Equations 2, 3, 8, 12, 19 lack the necessary punctuation", "was_revised": true, "final_justification": "After carefully reviewing the authors' response and the comments from other reviewers, I find that some of my concerns have been addressed. Therefore, I am willing to slightly increase my score.", "rebuttal": "Thank you for taking the time to review our paper and provide professional and constructive reviews. We are encouraged by the positive comments on the extensive experiments and the theoretical support. All suggestions will be carefully incorporated into final version.\n\n**_Q: There are too many notations in the paper._**\n\nWe acknowledge the reviewer’s concern about the number of notations potentially impacting readability. While a relatively large set of symbols is necessary to ensure rigorous and precise method description, we took care to **define each notation clearly upon its first appearance.** We agree that streamlining the notation can further enhance accessibility. For the final version, we will make a concerted effort to simplify the notation without sacrificing rigor or clarity. For example, to **provide a comprehensive list of all notations in the appendix.**\n\n**_Q: Results on the EyeDiap dataset._**\n\nWe acknowledge the reviewer's concern about GLS performance on EyeDiap. While the absolute performance is not SOTA, we emphasize two key points:\n\n--**Validity.** GLS demonstrates significant improvement over the backbone model (e.g., error reduction of 31.2% in $\\mathcal{D}_E \\to \\mathcal{D}_D$ with backbone Res-18), indicating its effectiveness for the EyeDiap dataset.\n\n--**Implementation Flexibility.** The current implementation of GLSGE is only **an instantiation of the general framework** (e.g., truncated 2D Gaussian for label distribution). As our core goal is to present the GLS perspective for CDGE, we intentionally **avoided too much dataset-specific tuning in evaluation.** As shown below, \nsimply tuning the truncated 2D Gaussian to linear combination of three uniform distributions can improve the performance in EyeDiap with Res-18, making it comparable to SOTA methods.\n\n|      \t  | $\\mathcal{D}_E \\to \\mathcal{D}_D$ | $\\mathcal{D}_G \\to \\mathcal{D}_D$ |\n|------------|------------|------|\n|GLSGE w/ Gau. | 6.21 | 7.30 |\n| GLSGE w/ Uni. | 6.02 | 6.58 |\n|DCUA  | 5.95 | 6.4 |\n\nThese observations indicate that **GLSGE has greater potential than what is shown in the current version.** We will emphasize this point more clearly in the final version of the paper to better convey the strength and flexibility of the framework.\n\n**_Q: Comparison based on GazeTR._**\n\nThank you for your valuable suggestion. In **Tab. 2 & 3**, we compare GLSGE with other methods using ResNet-18 and ResNet-50 as backbones. The results show that GLSGE brings more significant improvements to these backbones compared to other methods, which demonstrates the effectiveness and general applicability of our approach.\n\nAs for GazeTR, due to the lack of publicly available GazeTR-based CDGE methods, we present the results in Table 3 in the form of plug-play studies. The results still indicate clear performance gains, suggesting that GLSGE is compatible with transformer-based architectures as well.\n\nIn the final version, we plan to reproduce and incorporate some representative plug-and-play methods with the GazeTR backbone (e.g., GazeTR+PnP-GA), to provide a more comprehensive comparison. We believe this will further enhance the clarity and completeness of our experimental analysis.\n\n**_Q: Colors of \"Alignment\" in Fig. 1 and \"Probability-aware learning with \\omega\" in Fig. 3._**\n\nThank you for pointing this out. The color choices in Fig. 1 and 3 were intentionally designed to provide more intuitive visual cues, though we understand they may have caused confusion.\n\nIn Fig. 1, the orange solid line represents the GLS method, and the black solid line represents the UDA method. **The color of the \"Alignment\" component is split into orange and black to indicate that both GLS and UDA perform distribution alignment.**\n\nIn Fig. 3, the text color reflects the component each learning objective is associated with: red for importance reweighting, blue for invariant representation learning, and purple for predictor learning. Since the weight function ω(y) is learned from the importance reweighting module, we use red for the term ω(y). At the same time, ω(y) also plays a role in both invariant representation learning and predictor learning. Thus, **the term probability-aware learning with ω is colored in a combination of blue and purple to reflect its joint influence.**\n\nWe will clarify this design in the caption of the figures in the final version to avoid misunderstanding.\n\n**_Q: Typo._**\n\nWe sincerely appreciate the reviewer's careful reading. All reported typos will be corrected in the final manuscript, and we will conduct a full proofread to ensure textual accuracy throughout the paper."}, {"review_id": "EEhk2zFyqZ", "reviewer": "Reviewer_bWiS", "summary": "This paper focuses on Cross-domain Gaze Estimation (CDGE) task, which helps apply gaze estimation to various environments. This work introduce Generalized Label Shift (GLS) theory to gaze estimation, consisting of shifts in both the label and conditional distributions. Evaluations are conducted on different datasets and network structures, showing that the proposed method has achieved state-of-the-art performance.", "questions": "Please refer to the weaknesses.", "limitations": "NA", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength:\n1. The tasking definition is clear and easy to understand its value.\n2. The authors clearly outline limitations in existing GLS correction methods for gaze estimation and propose appropriate solutions.\n3. Comprehensive experiments have been conducted, which helps confirm the method's effectiveness.\n\nWeakness:\n1. The rationale for modeling the label distribution as a bivariate truncated Gaussian is unclear and needs stronger justification.\n2. The relationship between sections in Sec. 3 lacks clarity. Specifically, Sec. 3.2 appears to describe previous methods, making it more suitable for inclusion in the related work section. From my point of view, it would be useful to have an overview at the beginning of Sec. 3.\n3. It should address how GLS correction methods have benefited other tasks and clarify their value specifically in gaze estimation. Additionally, an overview of the current state of gaze estimation research should be provided to better contextualize the contribution.", "was_revised": true, "final_justification": "Thanks for the responses. After reviewing the rebuttal and considering the comments from the other reviewers, I have decided to maintain my original rating as borderline accept.", "rebuttal": "Thank you for taking the time to review our paper and provide professional and constructive reviews. We are encouraged by the positive comments on the clear paper writing, limitations outlining and extensive experiment. All suggestions will be carefully incorporated into final version.\n\n**_Q: Why modeling the label distribution as a bivariate truncated Gaussian._**\n\nThank you for the insightful question.\nAt its core, we aim to model a continuous function $f(y)$ that explicitly represents either the reweighting function ω(y) or the target-domain label distribution $p^t(y)$. The function $f$ should satisfy the following properties:\n\n-- Since gaze is a continuous variable, $f(y)$ should be defined over a continuous space so that **for any $x∈D$, we can compute $y=f(x)$.**\n\n-- Because the gaze range is bounded, the support of $f$ should also be compact. This can be naturally enforced by using truncated distributions.\n\nUnder these constraints, there are many possible modeling choices beyond a Gaussian—such as exponential family distributions, uniform distributions, or even shallow neural networks. The choice of $f$ can be adapted to the characteristics of the specific task or dataset, which we believe is a promising direction for future work.\n\nIn our current paper, as stated in **lines 175–180**, we adopt a bivariate truncated Gaussian for $f$ due to its simplicity, analytic tractability, and because it aligns reasonably well with the empirical observation in the target domain.\n\nHowever, we acknowledge that in other settings—e.g., driver gaze datasets where gaze points tend to concentrate around a few specific directions (e.g., forward, mirrors)—a **truncated Gaussian mixture model** may provide a more accurate fit. Incorporating such prior knowledge into the choice of $f$ is indeed a meaningful extension, and we plan to explore this in future work.\n\nDue to the time constraints of the rebuttal phase, we include here a reference result using a linear combination of three uniform distributions for $f$ as a comparison:\n\n|      \t  | $\\mathcal{D}_E \\to \\mathcal{D}_D$ | $\\mathcal{D}_G \\to \\mathcal{D}_D$ |\n|------------|------------|------|\n|GLSGE w/ Gau. | 6.21 | 7.30 |\n| GLSGE w/ Uni. | 6.02 | 6.58 |\n|DCUA  | 5.95 | 6.4 |\n\nThese observations indicate that GLSGE has greater potential than what is shown in the current version.\n\n**_Q: Relationship between sections in Sec. 3._**\n\nWe appreciate the reviewer’s comments regarding the organization of Sec. 3 and the suggestion to clarify the structure and flow. Here we provide a brief overview to clarify the writing logic of Sec. 3:\n\n-- Sec. 3.1: We reformulate the CDGE problem through a GLS perspective, identifying label shift and conditional shift as core challenges.\n\n-- Sec. 3.2: We introduce the GLS correction framework as a principled solution to CDGE. We then point out (in **lines 162–168**) that **existing DG and UDA methods can be viewed as partial realizations of this framework**—specifically, they focus on invariant representation learning while ignoring label shift correction. This perspective highlights why such methods may fail under significant label shift.\n\n-- Sec. 3.3-3.4: We propose two key components of our method tailored to the CDGE problem: Label Probability Distribution Estimation (Sec. 3.3), corresponding to Eq. (7), and Probability-Aware Conditional Alignment (Sec. 3.4), corresponding to Eq. (8). \n\n-- Sec. 3.5: We present a concrete implementation of the full GLS correction framework, combining the Eq. (20).\n\nWe agree that the discussion of DG and UDA methods in Sec. 3.2 may initially appear to overlap with related work. However, we believe that **revisiting these methods after introducing our framework allows us to reinterpret existing approaches from a unified theoretical perspective, thereby revealing their inherent limitations and motivating our design choices more clearly.**\n\nWe appreciate the suggestion, and in the final version we will improve the section transitions and **provide an overview at the beginning of Sec. 3** to better guide the reader through the structure.\n\n**_Q: How GLS correction methods have benefited other tasks and their value specifically in gaze estimation_**\n\nThank you for raising this important point.\nThe GLS framework was originally proposed in the study of domain shift to address scenarios where both label shift and conditional shift are present. We have provided a detailed discussion of related GLS theory and methods in **Sec. 2, lines 96–111**, where we note that most existing GLS approaches are designed for discrete classification tasks with a finite label space.\n\nIn contrast, our work is among the first to explore the application of the GLS framework to CDGE problem with continuous labels. In Sec. 3.1, we analyze the domain gap in CDGE and argue that it naturally involves both label and conditional shifts. Therefore, leveraging the GLS perspective in this context is a natural and theoretically grounded choice. Moreover, this perspective allows us to reinterpret and unify prior CDGE methods and motivates a new class of solutions beyond existing paradigms.\n\n**_Q: An overview of the current state of gaze estimation research._**\n\nWe appreciate the reviewer’s suggestion. An overview of the current state of gaze estimation research is provided in **Sec. 2, lines 82–95**. In that section, we summarize recent advances in CDGE, including both DG and UDA methods. Specifically, in **line 94-95**, we point out that they essentially focus on domain-invariant representation learning although differ in technology.  \n\nIn the final version, we will consider making this overview more explicit to better contextualize our contributions."}, {"review_id": "ajZndfgJ8B", "reviewer": "Reviewer_ZkHw", "summary": "This paper addresses the cross-domain gaze estimation (CDGE) problem by introducing a novel perspective based on Generalized Label Shift (GLS) theory, modeling the domain shift as both label and conditional shifts. It proposes an importance reweighting strategy using a truncated Gaussian distribution to effectively handle the continuity challenges in label shift correction. Extensive experiments demonstrate that the method achieves superior generalization across domains and is applicable to various backbone models.", "questions": "Check Cons above", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Pros:\n1. The paper proposes a new perspective for the domain adaptation in gaze estimation which is interesting.\n2. This paper is theoretically sound.\n3. Experiments are extensive and support the conclusion well.\n\nCons:\n1. The proposed method seems effective but complicated. I am curious about the time cost for the adaptation process.\n2. Is it too simple to modelize the gaze label distribution as a bivariate truncated Gaussian distribution? Do you have better choices?\n3. PXY = PX|Y*PY or PY|X*PX. Does your approach also work for PY|X*PX? Why not choose PY|X*PX?\n4. Figure 1 is not clear for illustrating the difference among different methods. I advise to including three subfigures to show the main ideas of the three methods.\n5. How many images from the target domain are used for domain adaptation? It should be included in Table 2 and 3.", "was_revised": true, "final_justification": "After checking the reviews and rebuttals, I think the authors have addressed most concerns well. Even though I still have a small concern about the fairness of the comparison, I accept the explanation in the rebuttal. I will keep my original rating of  Borderline accept.", "rebuttal": "Thank you for taking the time to review our paper and provide professional and constructive reviews. We are encouraged by the positive comments on the clear paper writing, limitations outlining and extensive experiment. All suggestions will be carefully incorporated into final version.\n\n**_Q: Better choice to modelize the gaze label distribution._**\n\nAt its core, we aim to model a continuous function $f(y)$ that explicitly represents either the reweighting function ω(y) or the target-domain label distribution $p^t(y)$. The function $f$ should satisfy the following properties:\n\n-- Since gaze is a continuous variable, $f(y)$ should be defined over a continuous space so that for any $x∈D$, we can compute $y=f(x)$.\n\n-- Because the gaze range is bounded, the support of $f$ should also be compact. This can be naturally enforced by using truncated distributions.\n\n**Under these constraints, there are many possible modeling choices beyond a Gaussian—such as exponential family distributions, uniform distributions, or even shallow neural networks.** The choice of $f$ can be adapted to the characteristics of the specific task or dataset, which we believe is a promising direction for future work.\n\nIn our current paper, as stated in **lines 175–180**, we adopt a bivariate truncated Gaussian for $f$ due to its simplicity, analytic tractability, and because it aligns reasonably well with the empirical observation in the target domain.\n\nHowever, we acknowledge that in other settings—e.g., driver gaze datasets where gaze points tend to concentrate around a few specific directions (e.g., forward, mirrors)—a **truncated Gaussian mixture model** may provide a more accurate fit. Incorporating such prior knowledge into the choice of $f$ is indeed a meaningful extension, and we plan to explore this in future work.\n\nDue to the time constraints of the rebuttal phase, we include here a reference result using a linear combination of three uniform distributions for $f$ as a comparison:\n\n|      \t  | $\\mathcal{D}_E \\to \\mathcal{D}_D$ | $\\mathcal{D}_G \\to \\mathcal{D}_D$ |\n|------------|------------|------|\n|GLSGE w/ Gau. | 6.21 | 7.30 |\n| GLSGE w/ Uni. | 6.02 | 6.58 |\n|DCUA  | 5.95 | 6.4 |\n\n**_Q: Decomposition of the joint distribution._**\n\nWe thank the reviewer for this insightful question. From a probabilistic perspective, both factorizations of the joint distribution—i.e., P(X,Y)=P(X|Y)P(Y) or P(Y|X)P(X) —are mathematically valid. **However, we adopt the factorization P(X,Y)=P(X|Y)P(Y) because it provides a more intuitive and interpretable modeling perspective for the CDGE task.** Specifically:\n\n-- P(Y) reflects the distribution of gaze directions in the population, which is naturally subject to domain-specific priors (e.g., gaze behavior differs between indoor vs. outdoor settings). So the label shift naturally arises in CDGE.  \n\n-- P(X|Y) represents how visual observations (images) are generated given a particular gaze direction, capturing domain-specific appearance variations (e.g., lighting, head pose, background), which significantly affects the predictor behavior.\n\nIn contrast, the alternative factorization P(Y|X)P(X) presents several limitations in our setting:\n\n-- Aligning the marginal distribution P(X) is less informative, as it mixes samples from all labels and may not preserve task-relevant structure.\n\n-- Domain shift in P(Y|X), i.e., the posterior distribution, lacks clear, observable semantic meaning and is harder to control or interpret in practice.\n\nTherefore, we choose the P(X|Y)P(Y) formulation as it better reflects the generative structure of the data in gaze estimation and provides clearer insight into the nature of domain shift in this task.\n\n**_Q: The difference among different methods in Fig. 1._**\n\nThank you for the helpful suggestion. We agree that presenting three separate subfigures could make the differences among the methods more visually explicit. However, due to **space constraints** and our intent to **emphasize the conceptual connections between the approaches**, we chose to present all three within a single integrated diagram in Fig. 1.\n\nTo help distinguish the methods clearly, we adopted a combination of colors and line styles, as described in the figure caption:\n\n-- The black dashed line represents DG methods, which perform invariant representation learning using only labeled source-domain data.\n\n-- The black solid line denotes UDA methods, which perform invariant representation learning using both labeled source and unlabeled target data.\n\n-- The orange solid line represents our proposed GLS correction method, which not only performs invariant representation learning, but also explicitly corrects for label shift, and incorporates probability-aware modeling into both the representation and predictor learning.\n\nWe acknowledge that the current figure may not fully convey these distinctions at a glance. In the final version, we will enhance the caption with a more detailed explanation and consider making visual refinements (e.g., clearer legends, annotations) to improve clarity. If space permits, we will also explore the possibility of using subfigures, as the reviewer suggests.\n\n**_Q: Number of target domain images used for domain adaptation._**\n\nDuring cross-domain learning, we use **10%** of the unlabeled target domain images for training and another **10%** for validation, with the remaining **80%** used for testing. Concretely, this corresponds to 4,500 unlabeled images in the DM dataset, and 1,667 unlabeled images in the DD dataset.\n\nWhile the amount of unlabeled target-domain data we use may be larger than that used in some DG or UDA methods, we note the following:\n\n-- **the unlabeled data is inexpensive and easy to collect in real-world scenarios** (e.g., by simply capturing images with a camera).\n \n-- **Many existing approaches implicitly increase the diversity and amount of target-domain data via aggressive data augmentation, style transfer, or synthetic sample generation.** While such methods implicitly enhance the effective size of training data, our method does not employ such augmentation strategies.\n\nWe believe our experimental setting remains fair and reflects a practical and realistic scenario for cross-domain learning, and it highlights the benefit of our approach in leveraging readily available unlabeled data. We will clarify this setting more explicitly in the final version of the paper."}, {"review_id": "JRYbFjxD1o", "reviewer": "Reviewer_nmUr", "summary": "This paper proposes a method for cross-domain gaze estimation (CDGE), addressing key limitations of existing domain adaptation methods for label shift and conditional shift. It introduces the Generalized Label Shift (GLS) theory as the solution. Experiments between several datasets demonstrate the superior generalizability compared to existing methods.", "questions": "See the weaknesses", "limitations": "yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- Extensive experiments demonstrate its superior performance on various datasets.\n- Sufficient theoretical support provided.\n\nWeaknesses:\n- While the abstract and introduction clearly state the contributions, the motivation for addressing Generalized Label Shift (GLS) in gaze estimation could be better justified. The authors should explicitly explain why existing domain-invariant methods fail in gaze estimation compared to other tasks. Current explanations mainly focus on the general difference between the proposed method and previous works, but are not related to task-specific problems. \n\n- It should also be explained why label shift and conditional shift are significant and inherent problems for gaze estimation and differ from in other tasks. In the descriptions of L128-137, the involved problems can also appear in other tasks like object detection and categorization. How Sec. 3.4 solves the problems specifically in gaze estimation is also unclear, which is more like a general solution. It's better to make the relationship between the task-specific problem analysis and the proposed solution more clearly. \n\n- Typos. e.g. L18, \"appereance-based\"", "was_revised": true, "final_justification": "According to the rebuttal, my main concerns were addressed. The remaining problem is that the proposed method in this work seems to be a general solution for the label and conditional shift problems for continuous labels, which is loosely coupled with the task of gaze estimation. It's better to include a discussion about its relationship to other tasks with continuous labels and the potential to apply the method to other fields to extend the impact of this work. Given the strong performance and the theoretical support, I'll keep my positive rating of borderline accept.", "rebuttal": "Thank you for taking the time to review our paper and provide professional and constructive reviews. We are encouraged by the positive comments on the extensive experiments and the theoretical support. All suggestions will be carefully incorporated into final version.\n\n**_Q: Why existing domain-invariant methods fail in gaze estimation compared to other tasks_**\n\nExisting domain-invariant methods primarily address the covariate or conditional shift, assuming that the label distribution remains the same across domains. However, in CDGE problem, we observe that both label and conditional shift naturally occur, as discussed in Sec. 3.1, and illustrated in Fig. 2. Moreover, in **lines 162–165** of Sec. 3.2, we explicitly state:\n\n**\"Previous works are essentially connected with invariant representation learning and omit the label shift correction. According to the GLS theory, such methods are insufficient for successful cross-domain learning when label shift exists.\"**\n\nTherefore, domain-invariant methods are insufficient for the CDGE problem. Our work addresses this limitation by introducing a GLS correction framework that jointly models and corrects the label and conditional shift.\n\n**_Q: Why label shift and conditional shift are significant and inherent problems for gaze estimation and differ from in other tasks._**\n\n(1) Why label and conditional shift are significant and inherent in gaze estimation:\n\nAs described in Eq. (4), the core objective in CDGE is to minimize the target error by reducing the source error while correcting the domain shift. In **lines 96–108**, we introduce the GLS theory, which rigorously establishes the sufficiency and necessity of correcting both label and conditional shift for cross-domain learning. We further demonstrate in **lines 128–137** and **Fig. 2** that both forms of shift are naturally present in CDGE.\n\nTherefore, the label and conditional shift are inherent and significant in CDGE. We will highlight this conclusion more explicitly at the end of Sec. 3.1 in the final version.\n\n(2) Why this differs from other tasks:\n\nAs discussed in **lines 96–111**, the GLS framework is general and applicable to various domain adaptation problems, including object detection and classification. However, existing GLS methods are primarily designed for classification tasks with finite discrete labels.\n\nIn contrast, gaze estimation involves continuous-valued labels, making label and conditional shift correction significantly more challenging. This distinction is elaborated in **lines 170–174 and 190–193**, where we explain the difficulty of modeling continuous label distributions and designing appropriate conditional alignment strategies.\n\nDue to these challenges, existing GLS methods are not directly applicable to CDGE. In response, our work (Sec. 3.3 to 3.5) proposes a feasible GLS correction implementation which explicitly integrates the task-specific prior of gaze estimation.\n\n**_Q: How Sec. 3.4 solves the problems specifically in gaze estimation._**\n\nSec. 3.4 introduces the PCOD method, which is motivated by the distributional properties of gaze estimation.\n\nAs discussed in **lines 190–193**, most existing conditional alignment methods are designed for classification tasks with discrete labels, making them not applicable for gaze estimation with continuous labels. Then we consider the COD method as a starting point, which is more suitable for regression settings.\n\nFurthermore, in **lines 198–205**, we identify a critical limitation of COD under label shift: COD fails to embed conditional distribution with respect to a reweighted label distribution. To address this, we derive the Probability-aware COD (PCOD), which incorporates the learned reweighting function ω(y) into the alignment process, making it suitable for CDGE problem.\n\nTherefore, PCOD is applicable to other tasks with similar distributional properties, providing a more general-purpose solution within the GLS correction framework. Of course, beyond distributional aspects, gaze estimation has other unique properties, such as rotational equivariance. Integrating these domain-specific priors into the GLS correction method is an interesting direction for future work.\n\n**_Q: Typo._**\n\nWe sincerely appreciate the reviewer's careful reading. All reported typos will be corrected in the final manuscript, and we will conduct a full proofread to ensure textual accuracy throughout the paper."}]

Author rebuttal:

Meta-review: The paper proposes a novel Generalized Label Shift framework for cross-domain gaze estimation. The authors effectively addressed reviewer concerns regarding method rationale, experimental design (including performance on EyeDiap and the use of target data), and presentation clarity, through detailed explanations and new experiments. All reviewers acknowledged that their concerns were adequately resolved and maintained a positive, borderline accept score. Therefore, I recommend acceptance.
Decision: Accept (poster)