Geodesic Optimization for Predictive Shift Adaptation on EEG data
Authors: Apolline Mellot
Antoine Collas
Sylvain Chevallier
Alexandre Gramfort
Denis Alexander Engemann
Venue: NeurIPS 2024 spotlight (2024)
Keywords: EEG
brain age
Neurosciences
Riemannian geometry
Domain Adaptation
Mixed-effects models
Primary area: neuroscience_and_cognitive_science
TL;DR: This paper proposes Geodesic Optimization for Predictive Shift Adaptation to address multi-source domain adaptation where source domains have distinct y distributions in the context of brain age prediction from EEG covariance matrices.

Abstract: Electroencephalography (EEG) data is often collected from diverse contexts involving different populations and EEG devices. This variability can induce distribution shifts in the data $X$ and in the biomedical variables of interest $y$, thus limiting the application of supervised machine learning (ML) algorithms. While domain adaptation (DA) methods have been developed to mitigate the impact of these shifts, such methods struggle when distribution shifts occur simultaneously in $X$ and $y$. As state-of-the-art ML models for EEG represent the data by spatial covariance matrices, which lie on the Riemannian manifold of Symmetric Positive Definite (SPD) matrices, it is appealing to study DA techniques operating on the SPD manifold. This paper proposes a novel method termed Geodesic Optimization for Predictive Shift Adaptation (GOPSA) to address test-time multi-source DA for situations in which source domains have distinct $y$ distributions. GOPSA exploits the geodesic structure of the Riemannian manifold to jointly learn a domain-specific re-centering operator representing site-specific intercepts and the regression model. We performed empirical benchmarks on the cross-site generalization of age-prediction models with resting-state EEG data from a large multi-national dataset (HarMNqEEG), which included $14$ recording sites and more than $1500$ human participants. Compared to state-of-the-art methods, our results showed that GOPSA achieved significantly higher performance on three regression metrics ($R^2$, MAE, and Spearman's $\rho$) for several source-target site combinations, highlighting its effectiveness in tackling multi-source DA with predictive shifts in EEG data analysis. Our method has the potential to combine the advantages of mixed-effects modeling with machine learning for biomedical applications of EEG, such as multicenter clinical trials.

Geodesic Optimization for Predictive Shift Adaptation on EEG data
Apolline Mellot; Antoine Collas" Inria, CEA, Université Paris-Saclay
Sylvain Chevallier
TAU Inria, LISN-CNRS, University Paris-Saclay, France. sylvain. chevallier universite-paris-aacll
Palaiseau, France
apolline www.mmmiiiimiia........mmmmm notoine.collassnrii..
Alexandre Gramfort
Denis Engemann
Inria, CEA, Université Paris-Saclay
Roche Pharma Research and Early Development,
Palaiseau, France
Neuroscience and Rare Diseases,
al alexandre, gramf ort@inria..aa...
Roche Innovation Center Basel,
F. Hoffmann La Roche Ltd., Basel, Switzerland. denis engemann@roche.
Abstract
Eleetroencephalography (EEG) data is often collected from diverse contexts involy- ing different populations and EEG devices. This variability can induce distribution shifts in the data X and in the biomedical variables of interest y. thus limiting the application of supervised machine learning (ML) algorithms. While domain adaptation (DA) methods have been developed to mitigate the impact of these shifts, such methods struggle when distribution shifts occur simultancously As sate-offthhe-rt ML models for EEG represent the data by spatial covariance ma- trices, which on the Riemannian manifold of Symmetric Positive Definite (SPD) matrices, it is appealing study DA techniques operating the SPD manifold. This paper proposes a novel method termed Geodesic Optimization for Predictive Shift Adaptation (GOPSA) address test-time multi-source DA for situations in which source domains have distinct distributions. GOPSA exploits the geodesic structure of the Riemannian manifold to jointly learn domain-specifee re-centering operator representing site-specific intercepts and the regression model. We per- formed empirical benchmarks on the cross-site generalization of age-prediction models with resting-state EEG data from a large multi-national dataset (HarMN- qEEG), which included 14 recording sites and more than 1500 human participants. Compared to sfate-of-the-at methods, our results showed that GOPSA achieved significantly higher performance three regression metrics (R², MAE, and Spear- man's 's p) for several nource-target site combinations, highlighting its effectiveness in tackling multi-source DA with predictive shifts EEG data analysis. Our method has the potential to combine the advantages of mixed-effects modeling with machine learning for biomedical applications of EEG, such as multicenter clinical trials.
X and y.
Introduction
Machine learning (ML) has enabled advances in the analysis of complex biological signals, such as magneto- and electroenccphallgrrapyy (M/EEG), in diverse applications including biomarker
Equual contribution.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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X:
Mean log-power
y/: Age distribution
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Barbados
0-
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Colombia
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Figure 1: Joint shift in X and y distributions on the HarMIqEEGG dataset [33]. Subset of mean PSDs (A) and age distributions (B) from three recording sites used for the empirical benchmarks.
exploration [58, 20, 60] or developing Brain-Computer Interface (BCI) [57, 15, 1, 2]. However, major challenge in applying ML to these signals arises from their inherent variability, a problem commonly referred to as dataset shift [12]. In the case of M/EEG data this variability can be caused by differences recording devices (clectrode positions and amplifier configurations), recording protocols, population demographics, and inter-subject variability [36, 14, 23, 29]. Notably, shifts can occur not only in the data but also the biomedical variable we aim to predict, further complicating the use of ML algorithms.
Riemannian geometry has significantly advanced EEG data analysis by enabling the use of spatial covariance matrices as EEG descriptors [5, 4, 6, 38, 31, 35, 33, 48, 17, 56]. In [4, 6]. the authors introduced a classification framework for BCI based on the Riemannian geometry of covariance matrices. These methods classify EEG signals directly on the tangent space using the Riemannian manifold of symmetric positive definite (SPD) matrices (S*+)) )) effectively capturing spatial infor- mation. More recently, [47, 48, 7] extended this framework to regression problems from M/EEG data in the context of biomarker exploration. Furthermore, [47, 48] proved that Riemannian metrics lead to regression models with statistical guarantees in line with log-linear brain dynamics [10] and are, therefore, well-suited for neuroscience applications. Across various biomarker-exploration tasks and datasets, recent work has shown that Riemannian M/EEG representations offer parameter-sparse alternatives to Riemammmnnnn deep learning architectures [13, 40, 17].
Domain adaptation addresses the challenges posed by differences in data distributions between source and target domains, e.g., when data are recorded with different cameras in computer vision [55], different writing styles in natural language processing [32], or varying sensor setups in time series analysis [22]. In particular, DA considers target shift where the shift is in the outcome variable y. For classification it means source and target data share the same labels but in different proportions [34]. Target shift is also frequent in the context multicenter neuroscience studies, as the studied popula- tion of one site may vary significantly from the studied population of another site (cf. Figure 1). To tackle various sources of variability in neurophysiological data like EEG, there is need for a DA approach that can deal with joint shift in X and y.
Related work In [62], the authors addressed DA for EEG-based BCI using re-centering affine transformation of covariance matrices (Section 2) align data from different sessions human participants, improving classification accuracies. Yair et al. [59] extended this with parallel transport showing its effectiveness in EEG analysis, whereas, Peng al [43] introduced a domain-specific regularizer based on the Riemannian mean. Notably, this parallel transport approach reduces to [62] when the common reference point is the identity. In deep learning context, Kobler al. [31] proposed to do per-domain online re-centering which can be seen as domain specific Riemannian batch norm. Going beyond re-centering, Riemannian Procrustes Analysis (RPA) [45] was proposed for EEG transfer learning, using three steps: mean alignment, dispersion matching, and rotation correction. However, the rotation step unsuitable for regression problems and RPA adapts only single source to a target domain. Recently, [36] demonstrated the benefits of re-centering for regression problems, showing improvements in handling task variations MEG and enhancing across-dataset inference in EEG.

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On the other hand, mixed -ereedfffeets models (or multilevel models) have been successfully used to tackle data shifts in X and y [16. 25]. In biomedical data, mixed-effects models are crucial due to the presence of common effects, such as disease status and age. Riemannian mixed-effects models have been used to analyze observations on Riemannian manifolds, accommodating individual trajectories with mixed effects at both group and individual levels [30, 49, 50]. These models adapt base point on the manifold for each data point and utilize parallel transport for this adaptation, which is necessary for accurate trajectory modeling. However, they differ significantly from the problem we address this work. Notably, the input data X are covariates (e.g., age or disease status) which belong to Euclidean space and the variables y to predict belong to the manifold (e.g.. MRI diffusion tensors on / which is the opposite of the paper's s studied problem. This distinction is critical as it highlights that while both methods use the geometry of Riemannian manifolds, the nature of the predicted variables and the type of data used differ from existing Riemannian mixed-effects models.
Contributions In this work, we address the challenging problem of multi-source domain adaptation with predictive shifts on the SPD manifold, focusing on distribution shifts in both the input data X and the variable predict y. We propose a novel method called Geodesic Optimization for Predictive Shift Adaptation (GOPSA). It enables mixed-effects modeling by jointly learning parallel transport along a geodesic for each domain and global regression model common to all domains, with the assumption that the mean Fy of the target domain is known. GOPSA aims to advance the state of the art by: (i) addressing shifts both covariance matrices and the outcome variable y, (ii) being tailored for regression problems, and (iii) being a multi-source test-time domain adaptation method, meaning that once trained on source data, can generalize to any target domain without requiring access to source data or retraining new model.
We first introduce in Section how to do regression from covariance matrices on the Riemannian manifold of S We also interpret classical learning methods on 7 from heterogencous domains as parallel transports combined with Riemannian logarithmic mappings. This leads us to GOPSA in Section 3, which learns to parallel transport each domain, with algorithms at train and test times. Finally, in Section 4, we apply GOPSA well as different baselines on simulated data and the HarMNqEEG dataset.
Notations Vectors and matrices are represented by small and large cap boldface letters respectively (e.g., x, X). The set {1,...,K) K} is denoted by []. $j+ and Sa represent the sets of d x d symmetric positive definite and symmetric matrices. uvec : Sa + IR 4/44//22 vectorizes the upper triangular part of a symmetric matrix. Frobenius and 2-norms are denoted by ||-I] and 11-112, respectively. - defines the left part of the equation as the right part. The transpose and Euclidean inner product operations are represented by .T. 1, [l..... _]J denotes an n-dimensional vector of ones. For loss function L, VL and L' denote its gradient and derivative. We consider K labeled source domains, each consisting of N& covariance matrices, along with their corresponding outcome N.
values, denoted by {(E,,,, :), i= The target domain is (E, NT with the average outcome ğT.
2 Regression modeling from covariance matrices using Riemannian geometry
Riemannian geometry of + The covariance matrices belong to the set of d d symmetric positive definite matrices denoted S [51, 44]. The latter is open in the set of d x d symmetric matrices denoted Sy, and thus + is smooth manifold [9]. A vector space is defined at each Σ E s called the tangent space, denoted T and is equal to Sy, the ambient space. Equipped with a smooth inner product at every tangent space, a smooth manifold ecomes a Riemannian manifold. To do so, we make use of the affine invariant Riemannian metric [51, 44). Given r, I' e T this metric is (T, I')E
tr ( (E-Trr-
Riemannian mean The Riemannian distance (or geodesic distance) associated with the affine invariant metric is on E, E') & |log (Σ 3-'/E''> ->>>>>/>)>> with log s - Sa being the matrix logarithm (see Appendix A. for a definition). This distance is used to compute the Riemannian mean Σ defined for set [Σ, y C S as Σ é arg min zest Σn d ),
This mean is efficiently computed with a Riemannian gradient descent [44, 63].
3

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Riemannian logarithmic mapping The idea of the covariance- based approach is to define non- linear feature transformations into vectors that can be used input for classical linear machine learning models. To do so, the Riemannian logarithmic mapping of E' at Σ is defined as
logz (')= E log Σ -E''''"" Σ E Tz
(I)
Thus, matrices in the Riemannian manifold $ are transformed into tangent vectors.
Parallel transport A classical practice to align distributions is parallel transport of covariance matrices from their mean to the identity and then apply the logarithmic mapping (1). Parallel transport along a curve allows to move SPD matrices from one point on the curve to another point on the curve while keeping the inner product between the logarithmic mappings with any other vector transported along the same curve constant. The following lemma gives the parallel transport of E' from E to I along the geodesic between these two points (See proof in Appendix A.2).
Lemma 2.1 (Parallel transport to the identity). Given Σ, E' E Est+, the parallel transport of Σ' along the geodesic from E to the identity Ia al E [0, 0,11 is
PT Σ, a)
A Σ 4/3 "'E' %/2
Learning on $ The logarithmic mapping (1) at the identity is simply the matrix logarithm.
Thus, a classical non-linear feature extraction [6, 36, 8] of dataset {E of Riemannian mean Σ combines parallel transport and logarithmic mapping at the identity,
Σ Σ - uvec (logr, (PT E, E, = uvec (log
Σ
1/2 EE
1/2 ))
R did+
(2)
where uvec vectorizes the upper triangular part with off-diagonal elements multiplied by v² to preserve the norm. Correcting dataset shifts by re-centering all source datasets [62], corresponds to parallel transporting data M of each domain [1, K] from its Riemannian mean Eg to the identity,
ER Σx) = uvec (log Σ
This method the go-to approach for reducing shifts of the covariance matrix distributions coming from different domains and has been applied successfully for brain-computer interfaces [45, 59] and age prediction from M/EEG data [36].
3 Learning to recenter from highly shifted y distributions with GOPSA
In this section, we develop a novel multi-source domain adaptation method, called Geodesic Opti- mization for Predictive Shift Adaptation (COPSA), that operates on the S manifold and is capable of handling vastly different distributions of y. Our approach implements Riemannian mixed-effects model, which consists of two components: single parameter estimating geodesic intercept specific to each domain and a set of parameters shared across domains.
At train-time, GOPSA jointly learns the parallel transport of each of the K source domains and the regression model shared across domains. At test-time, we assume having access to the target mean response value ğт and predict on the unlabeled target domain of covariance matrices (ET,)) Nr GOPSA focuses solely on learning the parallel transport of the target domain so that the mean prediction, using the regression model learned at train-time, matches ğT.
Parallel transport along the geodesic In Section 2, we presented how domain adaptation is performed on 8 In particular, (3) presents how account for data shifts of each domain. However, this operator can only work if the variability between domains is considered as noise. As explained earlier, we are interested in shifts in both features and the response variable. Thus, (3) discards shift coming from the response variable and hence harms the performance of the predictive model. Based on the Lemma 2. we propose to parallel transport features to any point on the geodesic between domain nomain--pecific specific Riemannian mean Σ and the identity. Indeed, COPSA parallel transports Σg, on this geodesic with a E [0, 1] and then applies the Riemannian logarithmic mapping (1) at the identity,
Eg Ek,a) A uvec (logı, (PT (EK,;; Ex, a))) = uvec (log Σx
This allows each domain undergo parallel transport to certain degree, effectively moving it toward the identity.

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Algorithm 1: Train-Time GOPSA
Algorithm 2: Test-Time GOPSA
Input: For all & E [1. K]] {EEki Vk.1) 1 initialization Ys. step-sizes (Erre))
1.Nr mean outcome value ØT, Input: =
initialization of yT. trained Ridge coeff. (73) step-sizes (6
for k === -1 K do
Σg f- Riemannian mean of 1
Σg t Riemannian mean of TT
end
t s+1
t f-1
while not converged do
while not converged do
(T(T)) + Compute features with (9)
Zs(rs) Compute features with (7)
(TT k Compute loss derivative of (10)
(1s()s) < Compute Ridge coeff. with (8)
(Ts(ss) < Compute loss gradient of (8)
YT t Y (xr)
78 4 7/8 - (VVsss)))
t + ++1
( (-1+1
end
end
yT - Z (3)(7)))
return (7s
return ÿT
3.1
Train-time
GOPSA aims to learn simultaneously features from (4) and regression model. To do so, solve the following optimization problem
N2 (yk. B5 ( Ex, xx))
minimize BsER (4+
with as = [ai...... [x)] [T This cost function is decomposed into three key aspects. First, covariance matrices undergo parallel transported using Lemma 2.1 to account for shifts between domains. Second, they are vectorized, and a linear regression predicts the output variable from these vectorized features. Third, the coefficients the linear regression 3s and the as are learned jointly so that the predictor is adapted to the parallel transport and reciprocally. Besides, to enforce the constraint on ags. we re- ee-parmmeterize it using the sigmoid function, which defines bijection between IR and (0, 1). thereby ensuring that the resulting as values lie within the desired range: ag ( & (1 + exp(-7x)) Thus, the constrained problem (5) can be formulated as the following unconstrained optimization problem
K N ΣΣ
minimize
ykki
( Eg, 2
(6)
BEER BSER"(Tty d1
7
Let us define the matrix Zs(r) E Nax with Ns = Ek No. as the concatenation of the source data, where each row corresponds a feature vector:
with 7s = [,....,, 7k] JT.
(>)) = [ (EI,1. En)o(t)),.... (K,,KK, EK [(xk))]]]]
(7)
In the same manner, the source labels are concatenated to ys = [y1.1..... Nx.N=] E R's,, Given a fixed ys. the problem (6) is solved with the ordinary least squares estimator. In practice, choose to regularize the estimation of the linear regression with a Ridge penalty. Thus, (6) is rewritten as
r' A arg min YERK subject
[()(T) e lys 2s(7)83(>11
(8)
Bb(r) Zs(>) (AIN Zs(7)
where (r) d(/+ are the Ridge estimated coefficients given fixed and À > 0 is the regu- larization hyperparameter. The problem (8) is efficiently solved with any gradient- solver [39].
The train-tima of GOPSA is summarized in Algorithm 1. by calculating the Riemannian mean of covariance matrices for each domain It then iteratively optimizes the parameters "ss by computing the feature matrix (7), determining Ridge regression coefficients (8), and updating Ys using a gradient descent step on the loss function (8) until conver- gence. The output result is the optimized Ridge regression coefficients. For clarity of presentation.
The proposed training algorithm begins
5

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Algorithm employs gradient descent. In practice, we use L-BFGS and obtain the gradient using automatic differentiation through the Ridge solution that is plugged into the loss in (8).
3.2 Test-time
At test-time, we now have a fitted linear model on source data with coefficients S (7')- The goal is to adapt new target domain (ET,)) Nr
i=1 for which the average outcome ÜT is assumed to be known. First, let us define the matrix (r) E ERR N as the concatenation of the target data
ZT(1) = [ TT,, ET,
(ET,Ny: Σ,, (+))]
Then, COPSA adapts to this new target domain by minimizing the error between By and its estimation computed with the fitted linear model. This minimization performed with respect "T that parametrizes the parallel transport of the target domain, i.e.,
{
cr()) - VT
1 NT
arg min
IN, ZT (T()))))))3
(10)
PER
Finally, the predictions on the target domain are
ÿr Z7(77)83(78) R.T.
(11)
The test-time procedure of GOPSA is summarized in Algorithm 2. The algorithm begins by calculating the Riemannian mean of the target covariance matrices (E It then iteratively optimizes the parameter "rr by computing the feature matrix (9), the derivative of the loss function (10), and updating Yr using gradient descent step until convergence. The algorithm determines the estimated target outcomes, yr. by using the optimized o on the feature matrix, combined with the pre-trained regression coefficients (7S). The output result is the predicted target outcomes ÿr- It should be noted that, once again, for clarity of presentation, Algorithm 2 employs a gradient descent, but other derivative-based optimization methods can be used.
4 Empirical benchmarks
In this section, we built empirical benchmark to evaluate the performance of GOPSA. We first present the simulated data that used to illustrate the relevance of our method when there is joint distribution shift of the data and the labels. Then, we present the EEG dataset that we used to evaluate the performance of GOPSA with real data from different recording sites. Finally, we present the baseline methods that are compared with GOPSA.
Simulated data To generate simulated data, used the generative model described in [47, 48, 36).
The data follow the classical instantaneous mixing model:
(t) (t;(t)
(12)
where x; (t) E IRd are the observed series, (t) Rd are the underlying signal of the neural generators and A is the mixing matrix whose columns are the observed spatial patterns of the neural generators, Furthermore, we assume that follow a log-linear model:
y/ Bo
Σ Be log(pe))
(13)
where > 0 is the variance of the ε-th element of the underlying signal s;(t) as introduced [47, 48, 36]. From this, we generate domains (source and target) by applying shifts X and y. To do so, we introduced per-domain shift in the data distribution by applying an affine transformation to the covariance matrices Σ; 2 (t)x,()),,(1) T
Σ, - B
(14)
with By E S and & 2 0 controlling the amplitude the shift. Then, we shifted the label distribution by modifying the variance of the underlying signal pe:
Pes P
1+45
(15)

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with ç 2 0 still controlling the amplitude of the shift. Thus, the distribution of y is shifted per domain because of the log- log-linear relationship of (13). should be noted that ,3 is kept constant across domains.
HarMNgEEG dataset The HarMNqEEG dataset [33] was used for our numerical experiments. This dataset includes EEG recordings collected from 1564 participants across 14 different study sites, distributed across countries. In our analysis, we consider each study site as a distinct domain. Appendix A.5 provides detailed demographic information. The EEG data were recorded with the same montage of 19 channels of the 10/20 International Electrodes Positioning System. The dataset provides cre-codpeed cross-spectral tensors for each participant rather than raw data, and anonymized metadata including the age and the sex of the participants. More precisely, the shared data consists of cross-spectra' matrices with frequency range of 1.17 Hz to 19.14 Hz, sampled at a resolution of 0.39 Hz. A standardized recording protocol was enforced to ensure the consistency across EEG recording of the dataset. In addition to recording constraints, this protocol included artifact cleaning procedures. The cross-spectrum were computed using Bartlett's method (See Appendix A.3). Our pre-processing steps were guided by the pre-processing pipeline outlined in [33]. First, we performed a common average reference (CAR) on all cross-spectrum (See Appendix A.3) as different EEG references were used across domains. Subsequently, we extracted the real part of the cross cross-spectral tensor to obtain co- speseettru tensors containing rrequency-spectie estimates along the frequency spectrum. Due to the linear dependence between channels introduced by the CAR, the covariance matrices are rank deficient. To address this, we applied a shrinkage implemented a global-scale
covariance
regularization with coefficient of 10 5 to the data. Additionally, factor (GSF) correction, which compensates for amplitude variations between EEG recordings by scaling the covariance matrices with subject-specifie factor [24, 33] (See Appendix A.3). Following these pre- rre-processing steps, we obtained a set of 49 covariance matrices for each EEG recording, with each matrix corresponding to a specific frequency bin of the EEG signal. This pre-processed co-spectrum served as the input data for our domain adaptation study.
Performance evaluation and hyperparameter selection To evaluate the performance of the compared methods, we conducted experiments across several combinations of source and target sites We selected source domains such that the union distribution of their predictive variable y encompasses a broad age range. All remaining sites were assigned as target domains. For each surce-targer combination performed a stratified shuffle split approach with 100 repetitions the target data. Stratification was based on the recording sites to ensure that each split contained a balanced proportion of participants from each site. The regularization parameter À in Ridge regression was selected with nested cross-validas validation (grid search) over logarithmic grid of values from 10-1 to 10 To evaluate the benefit of GOPSA, we compared it against four baselines. Detailed mathematical formulations of these baselines can be found in Appendix A.4. For each baseline method, the regression task was performed with Ridge regression.
Domain-aware dummy model (DO Dummy) As GOPSA requires access to the mean Th of each domain, we used domain- dummy model predicting always the mean Th of each domain.
No re-center/ No domain adaptation (No DA) This second baseline method involves applying the regression pipeline outlined in [47, 48] without any re-centering. In this setup, all covariance matrices are projected to the tangent space at the source geometric mean Σ computed from all source points, no matter their recording sites.
Re-center to a common reference point (Re- (Re-center and Re- As introduced in Section 2, common transfer learning approach is Riemannian re-centering of all domains to common point on the manifold [62, 36). This baseline thus correspond to re-centering each domain k; source and target, independently by whitening them by their respective geometric mean Σg. An extension of this approach is to perform a Riemannian re-scaling of all domains to a common dispersion, as presented in [45, 36].
Domain-aware intercept (DO Intercept) This method consists in fitting one intercept Pu per domain. In practice since we assume to know J/,, we correct the predicted values so that their mean is equal to .J. This approach is in line with defining mixed-effects models on the Riemannian manifold [33].
Deep learning (GREEN) The GREEN model [40] is a deep-learning architecture tailored for EEG applications like age prediction. Since the HarMNqEEG dataset consists of covariance matrices, we
7

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A
B
C
Methods
Shift in X
Shift in
Shift in (X, y)
1.0
DO Dumny
No DA
& 0.5
GREEN
Re-center
0.0
Re-scale
No shift
Max. shift
No shift
Max. shift No shift
Max. shift
DO Intercept
COPSA
Figure 2: R² scores for different methods simulated data. Performance is measured across 5 source domains and target domain, with shifts controlled by E (0 to maximum). Data are generated 100 times, with 5 sensors and 300 covariance matrices per domain. The target domain is randomly selected between the 6 domains generated as presented in Section 4, with the remaining domains used sources. (A) A shift is applied the covariance matrices following (14). (B) A shift is applied on the variances following (15). (C) Both shifts from (14) and (15) are applied simultaneously.
used the 'G2' variant of GREEN, which starts at the covariance matrices level and includes pooling layers. This variant is designed for SPD matrices, making it SPD network [26]. Although GREEN has been evaluated on multiple datasets for various predictive tasks, it has not yet been applied in a domain adaptation context and does not include an adaptation layer.
We applied the domain- aomain-adaptation methods independently to each of the 49 frequency bins, resulting in 49 geometric means per domain, except for GREEN, which processes all frequency bands simul- tancously. シт of each domain was estimated on target splits (50% of the data) that do not overlap with the evaluation target splits (50% remaining). Statistical inference for model comparisons was implemented with a corrected t-test following [37]. Experiments with 100 repetitions and all site combinations have been
standard Slurm cluster for 12 hours with 250 CPU cores.
5 Results
Simulated data Figure 2 presents the results of simulated experiments where shifts are applied on either X, y, both (X, y) presented in Section 4. All methods were evaluated in three simulation scenarios: shift in X only, shift in only, and joint shift in X and y. The intensity of the shift was controlled by & in all scenarios. If there is no shift in X, observe that No DA perfectly estimates the y because the log-linear model is easily estimated across domains even when the distribution changes (Figure B). The performance of No DA however drops when shift in X is introduced (Figure 2 A and C). Re-center and Re-scale led to the same results as no scaling shift was applied in the simulation. Both were able correct the shift in X, but performed poorly when a shift in y was added (Figure 2 B and C). GREEN notably showed consistant performance across all scenarios, and was relatively resistant both types of shifts given it is not designed for domain adaptation. DO Intercept and GOPSA showed the best performance across all scenarios, with an ad advantage for GOPSA. The interest of GDPSA is to estimate this log-linear model with shifts in (X, y) per domain (Figure 2 C) which other methods were not able to do. These experiments demonstrate the efficiency of the proposed method in estimating shifts in X between domains even in the presence of shift in y- contrary to the baseline methods. Theoretically, based on the generative model of the simulated data, the data X and outcome y are linked by a log-linear relationship. This implies that, knowing the shift in X for the target domain, predictions can be made even when y distributions do not overlap between the source and target. Since GOPSA estimates the target shift in X by minimizing (ū mcan )).)) is capable of handling such scenarios.
HarMNqEEG data We computed benchmarks for five combinations of source sites and we displayed the results for the three metrics selected for performance evaluation, each colored box representing one method (Figure 3). A min-max normalization was applied each site combinations separately across methods. We first conducted model comparisons in terms of absolute performance across all baselines (A). No DA, without domain specific re-centering, performed worse than DO Dummy in terms of R2 score and MAE. Re-center and Re-scale led to lower performances across all metrics, which can be expected as the Riemannian mean is correlated with age in our problem setting Figure 1. Eventhough its architecture does not include an adaptation layer, GREEN
8

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A
DO Dunmy No DA CREEN Re-center er Re-acale- DO Intercept GCPSA
0.25
0.5
0.75
0.25
0.5
0.75
0.25
0.5
0.
Normalized Spearman's s p
Normalized R score t
Normalized MAE
B
Difference: GOPSA DO Intercept
Ba,Cho, G,5 0.000
0'372
0.31G
Bh,Ch 0.000
p
0.000
0.000
Ba,Co,G- 0.001
0.136
0.051
C,03,M.R,5 0.361
0.473
0.385
Ba,Be,Che
Co,Cu80.G.R p
0.000
p
0.000
0.006
0.2
.011
0.1 0.2
0.2
-0.1
0.1
0.2 0.2
-0.1
0.1
Normalized Spearman's 's p Normalized R score
Normalized MAE
Figure 3: Normalized performance of the different methods on several source-target combi- nations for three metries: Spearman's p t (left), R² score f (middle) and Mean Absolute Error I (right). As large variability in the score values was present between the site combinations, applied min-max normalization per combination to set the minimum score across all methods to O and the maximum score to 1. (A) Boxplot of the concatenated results for the three normalized scores. One point corresponds to one split of one site combination. (B) Boxplots of the difference between the normalized scores of GOPSA and DO Intercept. row corresponds to one site combination, one point corresponds to one split. For each plot, the associated results of Nadeau's 's & Bengio's 's corrected t-test [37] are displayed. A p-vuuue lower than 0.05 indicates a significant difference between the two methods. Ba: Barbados, Be: Bern, Chb: CHBMP (Cuba), Co: Columbia, Cho: Chongqing, Cu03: Cuba2003, Cu90: Cuba90, G: Germany, M: Malaysia, R: Russia, Switzerland
reached better performance than the previous methods mentionned, but lacked consistency across site combinations and metrics with large variance especially for the R² score and MAE. For all scores, DO Intercept and GOPSA reached the best average performance with lower variance. A version of Figure 3 A without normalization is presented in Appendix A.7. As DO intercept and GOPSA showed overlapping performance distributions, we investigated their paired split-wise (non- rescaled) score differences (B). The site-specific differences of GOPSA scores minus DO Intercept are displayed with their associated p-values. For one site combination (Ba,Be, Che.CCho,o,CCun Cu,,,, DO Intercept cept yielded higher R² scores, and no significant difference was found between the two methods for Ba, Go. Similarly, no significant difference was observed on Spearman's 's p results for Cu03.M.R.S. Overall, GOPSA significantly outperformed DO Intercept in five site combinations for MAE, four for Spearman's 's p and three for R2 score. Detailed results for each source-target target combination are presented in Appendix A.6 for Spearman's 's p. R² score, and MAE. The bottom rows correspond the mean performance of each method of all site combinations, and their average standard deviation (see Appendix A.8 for associated boxplots). We expected GOPSA to outperform the baseline methods (e.g. DO Intercept) whenever joint (X, y) shifts occur. In our experimental benchmark, GOPSA significantly outperformed the baseline methods in some site combinations, but not all. This allows us to assume that not all site combinations show joint shifts.
Model inspection Next, we investigated the impact of the different re-centering approaches on the data Figure 4. Power spectrum densities (PSDs) were computed as the mean across sensors of the diagonals of the covariance matrices Riemannian mean for each site combination after No DA, Re- ee-eetter and GOPSA (A). PSDs for No DA display the initial variability between sites without recentering (cf. Figure 1). Re Re-center resulted in flat PSDs because all data were re-centered to the identity. PSDs produced by GOPSA are flattened and more similar across sites compared to No DA without removing too much information, unlike the un-effective Re-center method (cf. Figure 3). The alpha values are inspected as function of the site mean age (B). Re-center leads to alpha values all equal to one as all sites are re-centered to the identity. For GOPSA, we observed linear

---

NO DA
Recenter
GOPSA
A
B
Barbados (ax00.9)
Switzerland (a=0.7)
PSD (dB)
10
Source
1.0
sites
0-
10
R 0.99
NewYork (a=0.9)
Colombia (a=0.82)
0.8
PSD (dB)
10
Method GOPSA
Site
Target sites
Source
0.6
Recenter
Target
-10-
10
20
10
Frequency (Hz)
Frequency (Hz)
g: Mean Age
Figure 4: Model inspection of GOPSA versus No DA and Re-center. Power Spectral Densities (PSDs) and a values were computed on the source sites Barbados, Chongqing, Germany, and Switzerland. The remaining sites were used target domains. (A) Mean PSDs computed across sensors for No DA, Recenter and GOPSA on two source (Barbados and Switzerland) and two target (New York and Columbia) sites. (B) values versus site's mean age for Re-center and GOPSA. One point corresponds to one site. The coefficient of determination is reported for the GOPSA method.
relationship between alpha and the sites' mean age (Rt 0.99). This is direct consequence of the optimization process GOPSA, which thus can be regarded a geodesic intercept in a mixed-effects model. Overall, GOPSA effectively re-centered sites with younger participants closer to the identity matrix, Re-centering sites around a common point helped reduce the shift in X, while not placing all sites at the exact same reference point helped manage the shift in y, hence preserving the statistical associations between X and y.
Conclusion
We proposed novel multi-source domain adaptation approach that adapts shifts in X and y simulta- neously by learning jointly a domain specific re-centering operator and the regression model. GOPSA was specifically developed handle joint shifts in the data distribution and the outcome distribution, as illustrated by the simulations in Figure 2. GOPSA is a test-time method that does not require to retrain model when a new domain presented. GOPSA achieved ofate-of-the-art performance on the HarMNçEEG [33] dataset with EEG from 14 recording sites and over 1500 participants. Our benchmarks showed a significant gain in performance for three different metrics majority of site combinations compared to baseline methods. GOPSA can thus be used by researchers as a decision rule to infer the presence of joint shifts and, hence, serve as a tool for data exploration and model interpre- tation. While we focused on shallow regression models, the implementation of GOPSA using Py PyTorc readily supports its inclusion in more complex Riemannian deep learning models [26. 56, 11, 40, 31]. This direction seems promising given our observation that GREEN simple deep net combining Riemannian computation with a fully connected layer - already possessed some intrinsic robustness to data shifts. This may point at the capacity of the fully-connected layer to provide additional non-linear transformations that can accommodate the data-generating scenario which continuous log-linear generators modified in discrete manner by site factors. More generally, it emphasizes the potential of complex nonlinear methods for domain adaptation, in line with a recent study on the same dataset reporting positive generalization results using kernel method [28]. Furthermore, although this work specifically addresses age prediction, the methodology is applicable to a broader range of regression analyses. While GOPSA necessitates knowledge estimability of the average y per domain, this requirement aligns with that of mixed-effects models [16. 25, 61]. which are exten- sively employed in biomedical statisties. By combining mixed-effects modeling with Riemannian geometry for EEG, GOPSA opens up various applications at the interface between machine learning and biostatistics, such as, biomarker exploration in large multicenter clinical trials [46, 52, 53].
10

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Acknowledgment
This work was supported by grants ANR-20-CHIA-0016 and ANR-20-IADI-000 AG while at Inria, UNR-20-THIA-NN
to AM, ANR-22-CE33-0015-4 and ANR-17-CONV-0003 operated by LISN to SC and ANR 1ER--2--PESN-OOO
to AC under the France 2030 program, all managed by the Agence Nationale de la Recherche (ANR) Competing interests: DE is a full-time employee of F. Hoffmann-La Roche Ltd.
Numerical computation was enabled by the scientific Python ecosystem: Matplotlib [27], Scikit- learn [42], Numpy [21]. Scipy [54], PyTorch [41] PyRiemann [3], MNE [19] and SKADA [18].
This work was conducted at Inria, AG is presently employed by Meta Platforms. All the datasets used for this work were accessed and processed on the Inria compute infrastructure.
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14

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A Appendix
A.1 Matrix operations
Given E E + and its Singular Value Decomposition (SVD) Σ U ding(A).....,a)) A.) the
matrix logarithm of is
log( Σ U
(16)
Σ to the power is
E = U diag(A))
(17)
A.2 Proof of Lemma 2.1
First, we recall that the geodesic associated with the affine invariant metric from E to E'
E E D//
- 1 for [0,
(18)
Hence, Eig la = D1-a From [59], the parallel transport of E' from Ej to Mn is
EE' with E Σ Σ ΣンΣ
ΣT
1/2
(19)
Hence, the parallel transport of E' from E
to D is EΣ'
E with
E Σ
Σ
D 1 - 1/2) 1ー
(20)
Σ Σ -/2 9/2
which concludes the proof.
A.3 Cross-spectrum computation and preprocessing
Bartlett estimator From [33], the features provided in the HarMNqEEG dataset have been com- puted using the Bartlett's estimator by averaging more than 20 consecutive and non-overlapping segments. Thus, data consist of cross-spectral matrices with a frequency range of [min = 1.17 Hz to fmx = 19.18 14 Hz, sampled at resolution of Aw = 0.39 Hz. These cross-spectral matrices are denoted (w) E H where is the site, the participant and (Smins m + ,.,..., mmax).
Common average reference (CAR) their original montages with CAR:
The cross-spectrum matrices (i(w) were re-referenced from
Sw,(w) A HS)
(21)
where H 2 Ia- 1 T/d.
Co-spectrum matrices re-scaled with individual scalar G.,, that is calculated as the geometric mean of their power spectrum across sensors and frequencies:
Global Scale Factor (GSF)
N
C,,t
exp
log
(Sax)0mii (Aw) e.c.
(22)
N
where No Lum 1. The GSF correction is then applied to the co-spectrum (the real part of Au
the cross-spectrum) for all frequencies w:
Ee,(w) e R (Sk:(w) /Cn.s
(23)
The (w) s are the features used the Section 4.
15

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A.4 Baselines
The four baseline methods used in this work are detailed in the following. For every methods we have access to K labeled source domains, each including N& covariance matrices and their corresponding N, variables of interest (Ei,i. i1 The method DO Dummy and the mixed-effects model baseline DO Intercept both access the mean value VT of the target domain variable to predict. We remind that the dataset used in the empirical benchmarks is constituted of several frequency bands, each method is applied to each frequency band independently and then computed vectors are concatenated.
Domain-aware dummy model (DO Dummy) The DO Dummy simply returns the mean value ÄT for every predictions of given target domain.
No re-center No domain adaptation (No DA) The covariance matrices are used as input of the regression pipeline without any re-centering. First, the geometric mean of the source matrices is computed:
N₁ Σ
K
Es & arg min EESt* k=1
ÓR En.))
(24)
Then, source feature vectors extracted with:
E Es) uvec (log Egg 1/2
Eu v2) 2)
IR 1//
(25)
Finally, the target feature vectors extracted from the target data (ET,)) with:
( Es) = uvec (log
ES 1/2 ET ET ) R duri/
(26)
Re-center a common reference point (Re-center) Domains are re-centered to a common reference point, here we decided to use the identity. First, the geometric mean of each domain is computed:
No (E, Ex.:)
Σ
arg min Dest+
(27)
Then, feature vectors extracted using the specific geometric mean of cach domain:
ER,i }
uvec
(log
Σx
1/2
V))
R d(d-1)/s
(28)
Ny are also re-centered to the identity using their Covariance matrices of the target domain (Et,)) geometric mean:
Σf A arg min
Σ òR ET,)
(29)
DES
E En) = uvec (log (Σ ΣバΣ
E R 4/441//2
(30)
For both No DA and Re-center, the regression task was performed using a Ridge regression, which included an intercept:
BS Fi.s arg min
(yn.s B3 zk,i Bo)² + 11
(31)
BER d(a+1)y2
B0ER
where z is computed with (25) (28). The predicted values were computed
TT, = (B5) 2T. + 9b.s
(32)
where 27,; , computed with (26) or (30).
Domain-aware intercept (DO Intercept) In addition to the K labeled source domains, we assume have access to the mean of the variable to predict of the target domain ØT. At train-time, we fit Ridge regression with specific intercept for each domain
No
B
arg min
(yk.i BT (TO(ER,. Es) -xx)² 1111
(33)
B
16

---

Then, at test-time, we fit new intercept BU, using the target features:
aT Σs) uvec (log
Es
-/2 DT I ..'' )) E R 1uaiii//,
(34)
The fitted intercept is and the predictions are
NT
BO = VT
Σ
(35)
T T6( (ET,s, Es)
(35)
NT
i=1
iT = (B3s) To o (ET,i, Es) + BO,T
(36)
17

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A.5 HarMNqEEG dataset
Barbados
Bern
CHBMP
Chengdu
Chongqing
Colombia
Cuba2003
Cuba2004
Density
Cuba90
Germany
Malaysia
NewYork
Russia
Switzerland
50
75
50
75
Age
Figure 5: Age distribution of the 14 sites of the HarMNqEEG dataset [33]. The distributions are represented with kernel density estimate. The y-scales are not shared for visualization purpose.
18

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A.6 Table of performance scores.
Table I:
Performance scores for different combinations of source sites. The remaining sites were used target domains.
Spearman'ss 's p
Sites source
Dummy
DA
GREEN
Re-center
DO Intercept
GOPSA
Ca,
0.51 + 0.04
0.63 + 0.02
0.69 + 0.03
0.52 + 0.02
0.75 + 0.02
0.78 + 0.02
Be,Chb,S
0.58 + 0.02
0.73 + 0.01
0.75 0.02
0.43 0.02
0.68 + 0.02
0.72 - 0.02
Ba,Co,G
0.62 + 0.02
0.64 + 0.02
0.72 0.02
0.42 0.02
0.71 + 0.02
0.74 + 0.02
C,,,,M,R,S
0.62 + 0.03
0.63 + 0.01
0.70 0.05
0.46 + 0.02
0.76 + 0.02
0.75 0.04
Ba.Be, Cho.
Co,Cu90,G.R
0.77 + 0.02
0.79 + 0.01
0.82 + 0.01
0.44 + 0.03
0.85 + 0.01
0.87 + 0.01
Mean
0.62 + 0.03
0.68 + 0.01
0.74 0.03
0.45 + 0.02
0.75 + 0.02
0.77 + 0.02
R2 score
Sites source
Dummy
DA
Re-center
DO Intercept
GOPSA
Ba,Cho,G,S
0.21 + 0.02
0.06 + 0.06
0..6 0.33
-0.32 + 0.10
0.57 + 0.03
0.58 + 0.05
Be,Chb,S
0.25 + 0.02
-0.07 + 0.08
0.39 0.30
-1.36 0.13
0.43 0.03
0.49 - 0.03
Ba,Co,G
0.48 0.03
0.26 0.03
0.47 0.09
0.10 0.03
0.60 0.03
0.64 0.03
Cu03.M.R,S
0.26 0.02
0.26 0.04
0.48
-0.30 +
0.51 0.02
0.51 0.09
Ba.Be,Cho,
Co,Cu90,G.R
0.60 = 0.03
0.54 + 0.02
0.62 0.10
0.14 0.02
0.76 = 0.02
0.75 0.02
Mean
0.36 + 0.03
0.21 0.05
0.44 0.19
-0.35 0.07
0.57 0.02
0.59 + 0.04
MAE +
Sites source
DD Dummy
No DA
GREEN
Re-center
DO Intercept
GOPSA
Ba.Cho,G.S
9.25 + 0.16
12.00 + 0.21
9.08
14.69 0.24
7.69 0.19
7.61 + 0.25
Be,Chb.S
9.48 + 0.14
11.83 11.83 1: 0.37
8.67
2.30
22.48
0.20
Ba.Co,G
9.42 + 0.14
13.83 + 0.46
9.44 0 0.77
15.25 4
1 0.15
8.50 0.20
Cu03.M.R.S
9.64 + 0.17
10.98 + 0.22
8.53 1.12
16.50 0.25
8.94 1994:00
8.75 0.72
Ba,Be,Cho,
Co,Cu90,G.R
10.37 + 0.23
11.40 + 0.31
9.53
8.40
0.24
Mean
9.63 + 0.17
12.01 + 0.31
9.05 1.45
16.97 + 0.33
8.64 + 0.19
8.37 + 0.32
19

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A.7
Figure 3 without normalization
Without Re- center and scascale:
Dumny
No DA
CREEK
DO Intercept
GOPSA
0.50
0.75
1.00
-166
0.8
0.0
0.8
15
19
Spearman's
R score f
MAE (years)
Figure 6: Performance of four methods on several source-target combinations for three metries: Spearman's p (left), R² score (middle) and Mean Absolute Error (right). Re-center removed from the plot to better visualize the other methods. hox represents the concatenated results across all site combinations. One point corresponds to split of one site combination.
With Re-ten and Rcalc
Dumny
No DA
CREEX
Re-center
Re-scale
DO Intercept
COPSA
0.3
0.6
0.9
-2
10
13
16
19 22
Spearman's t
R2 score
MAE (years) 4
Figure 7: Performance of all methods on several source-target combinations for three metries: Spearman's 's t (left), R. score f (middle) and Mean Absolute Error (right). A box represents the concatenated results across all site combinations. One point corresponds to one split of one site combination.
A.8 Boxplots of each source-target sites for the three metries
The following figures represent the performance scores that are displayed in subsection A.6.
20

---

Ba,Cho,
Be,
Ca,Co,G
0.8 DO Dunmy
No DA
GREEN
center
Re-scale
DO Intercept
COPSA
0.4
0.6 Spearman's
0.8
0.4
0.6
0.8
0.4
0.6
0.8
Spearman's 1
Spearman's p
Cu03,M,RS
Co,
DO Dummy
No DA
GREEN
Re-center
Re-acale
DO Intercept
GUPSA
0.4
0.6
0.8
0.4
0.6
Spearman's
Spearman's f
Figure 8: Spearman's t for every site combination. One panel corresponds to the results of one site combination. One point corresponds to one split.
Ba,Cho,G,
Be, Chb,5
Ba,Co,
DO Dummy
No DA
GREEN
Re-center
Re-acale
DO Intercept
GOPSA
0.00
0.25
0.50
0.75
Rd score
R2 score 1
R score t
C.03.M.R.5
Ba. Be, Cho, Cu.90,,
DO Dummy
No Re-center
Re-scale
DO Intercept
GOPSA
0.0
0.5
0.00
0.25
0.50
0.75
R2 score
R²
score
Figure 9: R² score 1 for every site combination. One panel corresponds to the results of one site combination. One point corresponds one split.
21

---

Ba, G,o
Be, Shb,5
Ba,Co,G
DO Dunmy
No DA
GREEN
Re-center
Re-scale
DO Intercept
GOPSA
10
12
24
16
MAE (years)
MAE (years)
MAE (years)
Cu03, M,R.S
Ba, Be, Cho, Cu,C0,G
DO Dummy
GREEN
Re- Re-center
Re-scale
DO Intercept
GOPSA
10
13
16
13
16
MAE (years)
MAE (years)
Figure 10: Mean Absolute Error J. for every site combination. One panel corresponds to the results of one site combination. One point correspond one split.
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Reviews:
[{"review_id": "7MDSeP4LBh", "reviewer": "Reviewer_85Rg", "summary": "This study presents an approach to learning robust models under joint shifts in X and y (an important issue in healthcare). Focusing on EEG signals, a covariance-matrix-based learning framework is developed to address this challenge. Empirical results on EEG-specific benchmarks demonstrate its overall efficacy.", "questions": "- Q: Is the ComBat harmonization tool (used in imaging and genetics to correct batch effects) a relevant solution within the same problem setting? \"ComBat is a batch adjustment method that removes additive and multiplicative differences between sites due to the use of different scanning devices\" - https://cran.r-project.org/web/packages/combat.enigma/index.html.\n- Q: What happens when source and target distributions for the response variable don't overlap or only partially? (regarding the choice made in line 238).\n- Q: What happens if we only have a bad/noisy estimate of the target domain's mean value (in practice)? Is there any intuition on how things would break?", "limitations": "See weaknesses.", "rating": 8, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "- Focus on a highly relevant and unaddressed problem in EEG data analysis.\n- The proposed solution is innovative (mixed effects modeling w/ representation learning) and scalable (it does not need model retraining or access to source data).\n- Clear technical exposition/prose.", "weaknesses": "- Lack of discussion/comments on: 1) other relevant solutions for the same problem setting since joint shifts in X and y are an issue for many other biosignal modalities, 2) expected failure modes and future directions based on this approach.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer 85Rg,\n\nThank you for your positive evaluation of our submission. We took into account your concerns, and answered your questions point by point in the following:\n\n**Addressing Weaknesses and Questions:**\n1. **Other similar problem settings and Combat harmonization algorithm:** \\\nThe studied problem setting is indeed present in other biomedical applications. For example, you mentioned ComBat which was first developed for gene expression analysis. Other linear mixed-effect models are also usually applied to biomedical data in this context. However, they are often used to harmonize data to perform statistical analysis and thus are not adapted for predictive applications of machine learning. Indeed, to the best of our knowledge, mixed-effects models, like ComBat, assume to have access to the biomedical outcome (y) of source and target domains to compute corrections. In the studied setting, we assume to only have access to the y mean of each target site which is a much harder problem. Furthermore, the proposed benchmark includes the DO Intercept method, which is well suited for machine learning applications, as an alternative to linear mixed-effect models.\n2. **Overlap of y distributions between source and target domains:**\\\nThe second question raises an important point. Theoretically, based on the generative model of the simulated data (as described in the caption of Figure 1 in the rebuttal PDF), the data and outcome (y) are linked by a log-linear relationship. This implies that, knowing the shift in X for the target domain, predictions can be made even when y distributions do not overlap between the source and target. Since GOPSA estimates the target shift in X by minimizing $(\\bar{y} - \\hat{y}_i)^2$, it is capable of handling such scenarios.\n3. **Noisy estimate of the target $\\bar{y}$ :**\\\nFirst, we want to point out to the reviewer that we changed how the target $\\bar{y}$ is estimated in our empirical benchmark on real data.  Indeed, we estimate target $\\bar{y}$ on splits (50% of the data) that do not overlap with the evaluation target splits (50% of the data), rather than assuming target $\\bar{y}$ to be known. This leads to noisier target $\\bar{y}$ estimates. We noticed that the results are similar to the submission’s one. We still note that the performances of 3 splits (out of 100) in site combination 4 (Cu03, M, R, S) have deteriorated. Figure 2 of the rebuttal PDF presents the updated results.\\\nOur intuition is that the noisier the target $\\bar{y}$ estimation is, the worse the performance will get to the point of suppressing the interest in applying a domain-specific adaptation. A future perspective of this work is to study the point at which the adaptation does more harm than good.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "umrzzKRom1", "reviewer": "Reviewer_N9Jm", "summary": "The authors proposed GOPSA a new approach for alignment of EEG datasets from multiple subjects and sites. The method respects  the Riemannian manifold of the covariance matrices and learns the parallel transport length parameter simultaneously with the regression model used for solving the downstream task. While simple  and intuitive the approach is well justified, rooted in the previous research and  adequately evaluated using a large public EEG dataset and the associated age prediction task.", "questions": "1. Could the authors provide some interpretation of the age prediction model? What frequency bands appear pivotal? Which electrodes?\n2. Could the authors add the statistical test to Table 1?   \n3. Could the authors discuss and compare their approach against the technique used in several DL studies applied to the multisubject datasets when a specific subject adaptation trainable layer is used to interface  his or her data with the \"oracle\" classification engine?\n4. The authors use covariance matrices as features, however it seems that the transport learnt can be applied to the actual multichannel data directly - could the authors discuss this possibility in the light of forming large aligned datasets in the channel x time form rather than their covariance matrices?", "limitations": "The authors adequately addressed the limitations of their study, see also Weaknesses section of this review.", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The method is interesting, simple, uses few parameters and as the authors demonstrated efficient.\n2. The method is well described with all necessary details to comprehend it and implement.\n3. The dataset is large and comprehensive and extensive comparison with other relevant data alignment techniques is presented.", "weaknesses": "1. No performance analysis on a simulated data\n2. No interpretation of the obtained decision rule.\n3. Table 1 - not all the difference in performance of GOPSA with the other competing techniques seem significant.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer N9Jm,\n\nThank you for your positive assessment of our submission. In the following, we detailed answers to the weaknesses and questions you raised.\n\n**Addressing Weaknesses:**\n1. **Evaluation of the methods on simulated scenarios:**\\\nWe agree with the reviewer and added numerical experiments on simulated data to illustrate the advantage of “GOPSA” compared to other methods. All methods are evaluated in 3 simulation scenarios: shift in X only, shift in y only, and joint shift in X and y. The results are presented in Figure 1 of the attached PDF file. We demonstrate that “GOPSA” effectively compensates for the joint shift in both X and y, and is robust to shifts in only X or only y.\n2. **Interpretability of Riemannian models:**\\\nThe interpretation of Riemannian-based models, such as the one we refer to as “No DA,” involves transforming the model’s parameters into interpretable patterns, as demonstrated in prior work [c]. Applying similar techniques to “GOPSA” could yield valuable insights into its decision-making process, revealing the patterns and features that contribute most significantly to its predictions. This analysis is left for future research, and could enhance the transparency and interpretability of our method and guide further refinements and applications.\n3. **Statistical significance of the results:**\nThe reviewer is right that the difference in performance between “GOPSA” and other techniques is not always significant. The previously mentioned analysis done on simulated data showed that “GOPSA” clearly outperforms other techniques in the presence of a joint shift in X and y. Thus, with real data, we expect “GOPSA” to perform better when there is a joint shift in X and y, and to perform similarly to other methods when it is not the case. In Figure 2, which we updated and included in the attached file, we report the p-values of a t-test on the difference “GOPSA” minus ”DO Intercept” for three metrics on all site combinations. We chose to perform this test with “DO Intercept” as it is the competing method with the best performances. “GOPSA” significantly improved Spearman’s rank in 4 out of 5 site combinations. The results for R2 score and MAE are more nuanced, but on the majority of splits both scores tend to be improved with “GOPSA”.\n\n**Addressing Questions:**\n1. **Interpretation of the age prediction model:**\\\nAs explained previously, we agree with the reviewer that an interpretation analysis of the model would be interesting. In this work we mainly focused on the development of a novel approach to the specific setting of joint shift in both X and y. In our opinion, the specific work required to perform a clean and thorough interpretation represents a contribution in itself, see e.g. [c]. Indeed, the interpretation is not straightforward and would require taking into account many factors: age and frequency are linked, and confounding factors, like the sexe and the mental health of the participants. \n2. **Statistical test results:**\\\nA statistical t-test was conducted to compare the performance of “GOPSA” with that of the best-performing baseline method, “DO Intercept”. The p-values resulting from this test are presented in Figure 2 and have been updated in the rebuttal PDF. The statistical analysis from Figure 2 ensures a rigorous evaluation of the significance of the performance differences between “GOPSA” and the baselines. “GOPSA” significantly improved Spearman’s rank in 4 out of 5 site combinations. The results for R2 score and MAE are more nuanced, but on the majority of splits both scores tend to be improved with “GOPSA”. Regarding Table 1, we agree that including p-values would be beneficial. However, due to space constraints, we will add them in a subsequent version of the manuscript.\n3. **Comparison with DL model:**\\\nWe added in the benchmark a deep learning (DL) approach, called “GREEN”, that has recently been developed for EEG applications like age prediction [b]. We utilized the “g2” variant, which is an SPD network [d], because the data are SPD matrices. Even though the architecture does not include any adaptation layer, the results on simulated (Figure 1 of the rebuttal PDF) and real data (Figure 2) showed this model to be relatively robust to shifts. On real data “GREEN” performed better than “No DA”, the classical Riemannian-based regression pipeline but  it still falls short of the proposed “GOPSA” method which is tailored for the studied problem. However, as mentioned in the conclusion of the paper, since “GOPSA” is implemented using PyTorch, a future perspective would be to integrate “GOPSA” into Riemannian DL models like “GREEN”.\n4. **Application of the transport to the EEG time series:**\\\nIn this work, the dataset we used did not provide the raw EEG time series, so we focused on covariance matrices as features. However, the reviewer’s remark is correct, and the transportation operators learned from the covariance representation could be applied to the EEG time series. We think that it would be interesting to investigate this approach with datasets that provide the raw EEG signals.\n\n[b] Paillard, J., Hipp, J. F., & Engemann, D. A. (2024). GREEN: a lightweight architecture using learnable wavelets and Riemannian geometry for biomarker exploration. bioRxiv, 2024-05.\n\n[c] Kobler, R. J., Hirayama, J. I., Hehenberger, L., Lopes-Dias, C., Müller-Putz, G. R., & Kawanabe, M. (2021, November). On the interpretation of linear Riemannian tangent space model parameters in M/EEG. In 2021 43rd Annual International Conference of the IEEE Engineering in Medicine & Biology Society (EMBC) (pp. 5909-5913). IEEE.\n\n[d] Huang, Z., & Van Gool, L. (2017, February). A Riemannian network for SPD matrix learning. In Proceedings of the AAAI conference on artificial intelligence (Vol. 31, No. 1).", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "0FMwk2eAy1", "reviewer": "Reviewer_ucmS", "summary": "This paper presents a method for tackling domain adaptation challenges in EEG data analysis, specifically addressing shifts in both the feature space (represented by SPD matrices) and outcome variables $y$. The proposed method is designed on top of Riemannian mixed-effects model and is tailored for regression problems. Also, a key highlight of the proposed method is its capability to generalize from the source domain to any target domain without the need for retraining.", "questions": "Please refer to the weakness section.", "limitations": "I understand this is a professional paper on EEG study. However, I am currently unclear about its overall significance, particularly regarding the motivation or research question for investigating DA methods on the SPD manifold. However, my final decision is open to change pending the authors' rebuttal and further discussions.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "The paper is well-written at the beginning, with a clear introduction to concepts such as EEG data variability and its intuitive effect visualization, as in Figure 1. The paper provides clear theoretical foundations for establishing Riemannian geometry. The proposed method shows good performance improvement over baselines.", "weaknesses": "* Motivation Clarity. \nMy main concern is that the necessity of investigating domain adaptation (DA) methods on the SPD manifold is not convincingly presented. While SPD manifold representations are prevalent in EEG analysis, I did not see what specific issues were caused by domain shift. Or what unique issues does DA on the SPD manifold present that the proposed method wants to investigate and resolve? A clear problem setup is helpful for understanding the position of the research.\n\n* The proposed method is on top of a Riemannian mixed-effects model (lines 142-143), which has been used to tackle data shifts in $X$ and $y$ as introduced in Related work (lines 68-74). While the authors mention that there is an opposing setting in previous studies (lines 75-78), it is unclear which specific limitation this paper focuses on and what the challenge is. \n\n* The main characteristics/advantages of the proposed method are less explained. The paper overall explains very well about the derivation of the algorithm, but it is hard to understand the technical significance compared to existing methods and in what aspects it is better. While the work offers a thoughtful generalization of Riemannian mixed-effects model, it seems like it does not include any concepts that are wholly new. \n\n* It would be helpful to elaborate on why domain shifts occur in both the data and biomedical variables. A minor question is: Is this an EEG-specific issue or common in biomedical data?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer ucmS,\n\nWe thank you for your detailed review of our paper. We considered your feedback and addressed the concerns and suggestions you raised to improve the clarity and impact of our research.\n\n**Clarification of Motivation and Problem Setup:**\n1. **Motivation for domain adaptation on the SPD manifold:**\\\nThe necessity for domain adaptation (DA) methods is illustrated by the “No DA” approach, as detailed in the paper. This baseline corresponds to a SOTA covariance-based regression pipeline [e, f] without any adaptation between domains. As shown in Figure 2 from the PDF rebuttal, the “No DA” approach performs worse than a dummy model returning the average y per domain in terms of R² and MAE, demonstrating the critical need for adaptation techniques.\\\nClassical domain adaptation methods on the SPD manifold are the “Re-center” and “Re-scale” methods [a, f] which correct shifts in X between domains. However, they exhibit very poor performance when both distributions in X and y vary between domains, as shown in Figure 2.\\\nThis joint shift issue on EEG data recently emerged with the ‘HarMNqEEG’ dataset [h], introduced in 2022, which we use in our experiments. Thus, this submission aims at presenting this problem and providing a new method called “GOPSA” to solve it.\n2. **Riemannian mixed-effects models:**\\\nIn the existing literature (see e.g. [g]), Riemannian mixed-effects models typically assume that X belongs to a Euclidean space while y is in a Riemannian manifold. Our paper considers the opposite scenario, with X on a Riemannian manifold and y as a real-valued response. To the best of our knowledge, we are the first to address this problem, highlighting the novelty and significance of our approach. Thus, the proposed method is not made “on top of a Riemannian mixed-effects model” but the first to address mixed effects with X on a Riemannian manifold and y real-valued.\n3. **Advantages of  GOPSA:**\\\nWe acknowledge the reviewer’s concern regarding the need for a clearer explanation of the main advantages of our method, “GOPSA”. To address this, we have added numerical experiments on simulated data that demonstrate how “GOPSA” effectively handles joint shifts in both X and y. The results are presented in Figure 1 of the rebuttal PDF. First, we leverage the classical instantaneous mixing model:\n$$x_i(t) = A\\eta_i(t) $$\nwhere $x_i$ is the observed time-series, $i$ is the underlying signal of the neural generators and $A$ is the mixing matrix whose columns are the observed patterns of the neural generators. Furthermore, we use the log-linear model proposed in [e, f]\n$$y_i = \\beta_0  + \\sum_{\\ell=1}^d \\beta_{\\ell} \\log(p_{\\ell i})$$\nwhere $p_{\\ell i}$ is the variance of the $\\ell$-th element of the underlying signal $\\eta_i$. From this, we generate domains by applying two shifts. One on X that changes the mixing matrix per domain,\n$$x_i \\mapsto B_k^{\\xi} x_i$$\nwith $B_k$ a domain-specific SPD matrix, $k$ the domain number and $\\xi$ the variable that controls the intensity of the shift. A second shift is applied on y by shifting the variances per domain,\n$$p_{\\ell i} \\mapsto p_{\\ell i}^{(1+k\\xi)}.$$\nIt should be noted that $\\beta$ is kept the same across the domains.\\\nThe results are presented in Figure 1 of the rebuttal PDF. First, it shows that, if there is no shift in X, then \"No DA\" perfectly estimates the y because the log-linear model is respected across domains even when the y distribution changes (Figure 1B). However, the clear advantage of “GOPSA” is to estimate this log-linear model with shifts in (X, y) per domain (Figure 1C) which other methods can not do.\\\nThese experimental results on simulated results are consistent with the results on real data: “GOPSA” is the best method to estimate a shift on X while domains have different distributions in y.\n4. **Domain shifts in biomedical data:**\\\nDomain shifts in X often occur in biomedical data due to variations in data collection methods, such as differences in equipment, electrode placement, and environmental conditions. These factors can lead to inconsistent signal characteristics across different study sites. Shifts in y, the outcome variable, can result from demographic and clinical variations in the populations being studied, leading to differences in response distributions. While these issues are prominent in EEG data, they are common across various biomedical fields, necessitating robust adaptation techniques like our “GOPSA” method to ensure accurate cross-domain predictions.\n\n[a] Rodrigues, P. L. C., Jutten, C., & Congedo, M. (2018). Riemannian Procrustes analysis: transfer learning for brain-computer interfaces. IEEE Transactions on Biomedical Engineering, 66(8), 2390-2401.\n\n[e] Sabbagh, D., Ablin, P., Varoquaux, G., Gramfort, A., & Engemann, D. A. (2019). Manifold-regression to predict from MEG/EEG brain signals without source modeling. Advances in Neural Information Processing Systems, 32.\n\n[f] Mellot, A., Collas, A., Rodrigues, P. L., Engemann, D., & Gramfort, A. (2023). Harmonizing and aligning M/EEG datasets with covariance-based techniques to enhance predictive regression modeling. Imaging Neuroscience, 1, 1-23.\n\n[g] Schiratti, J. B., Allassonniere, S., Colliot, O., & Durrleman, S. (2015). Learning spatiotemporal trajectories from manifold-valued longitudinal data. Advances in neural information processing systems, 28.\n\n[h] Li, M., et al (2022). Harmonized-multinational qEEG norms (HarMNqEEG). NeuroImage, 256, 119190.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "XdfufWcrv8", "reviewer": "Reviewer_r2hf", "summary": "This paper proposes a novel method, Geodesic Optimization for Predictive Shift Adaptation (GOPSA), for predictive regression modeling with multi-source domain adaptation. The proposed method employs a domain-specific re-centering operator and a regression model using EEG data for age prediction. The method is innovative.", "questions": "The performance differences between the proposed GOPSA and the baseline, domain-aware (DO) intercept appear to be not large enough in several source-target site combinations. The T-test seems correct (100 repetitions per site combination). However, did you check the assumptions of the T-test: (1) whether the differences follow a normal distribution and (2) whether the variances are equal?", "limitations": "Only one dataset was used in this study. Experiments on different datasets should be conducted to validate the generalizability of the method.", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "1. This paper proposes a novel riemannian-based solution with multi-source domain adaptation for a regression task.\n\n2. Extensive experiments have been conducted, although only one dataset has been used.\n\n3. I carefully checked the code and lemmas; the method, experimental design, stratified split (ensuring independence), and significance tests appear to be correct.", "weaknesses": "1. This method assumes knowledge of the average value of the target label $\\bar y_T$, which is problematic due to the potential information leakage from the test set. As stated in lines 146-149, the method relies solely on $\\bar y_T$ for adjusting regression model predictions, making it difficult to evaluate the generalizability of the method on unseen data. \n\n2. The paper does not compare the proposed method with other alignment techniques such as re-scale and rotation correction [1].\n\n3. There is no comparison with other Riemannian-based methods or state-of-the-art deep learning techniques for age prediction using M/EEG.\n\n4. It may be inappropriate to claim 'achieved state-of-the-art performance' on a dataset without comparing it with other existing methods.\n\n\n\n[1] Mellot, Apolline, et al. \"Harmonizing and aligning M/EEG datasets with covariance-based techniques to enhance predictive regression modeling.\" Imaging Neuroscience 1 (2023): 1-23.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer r2hf,\n\nWe thank you for your detailed review. We took into account your comments and modified the experiments accordingly:\n\n**Addressing Weaknesses:**\n1. **Knowledge of the target label mean:**\\\nWe acknowledge your concern regarding potential information leakage due to the assumption of knowing the target label mean value. We now estimate target $\\bar{y}$ on target splits (50% of the data) that do not overlap with the evaluation target splits (50% of the data), rather than assuming target $\\bar{y}$ to be known, ensuring no leakage from the test set. The revised Figure 2, included in the rebuttal PDF, shows that the performance remains comparable to when target $\\bar{y}$ is assumed to be known.\\\nWe emphasize that target $\\bar{y}$ is the only quantity computed from the target y. In many scenarios, it is acceptable to assume that target $\\bar{y}$ is known, such as when a hospital has prior knowledge about its patient population.\n2. **Comparison with the state-of-the-art:**\\\nThe Riemannian method “No DA” we use is already a strong baseline from recent works on regression from SPD matrices [e, f]. In response to your review, we have added “GREEN” [b], a recently proposed deep-learning method for biomarker prediction, to our experiments. We utilized the “g2” variant, which is an SPD network [d], since the data are SPD matrices. The revised Figure 2 shows that “GREEN” performs better than “No DA,” but still falls short of “GOPSA”. Although “GOPSA” shows superior performance, it is worth noting that the learned parallel transport could be integrated into “GREEN,” which we plan to explore in future work.\\\nWe also included the “Re-scale” baseline [a, f] in our experiments on real data. This method corrects second-order statistics on the SPD manifold. However, as shown in the revised Figure 2, this additional baseline suffers from similar issues as “Re-center” and does not effectively resolve the problem of joint shifts in (X, y). Additionally, [f] studied rotation corrections but found that a meaningful method could only be derived where data must be paired between domains (i.e., the same patients are present in several domains), which is not feasible in our experimental setup.\\\nOverall, the revised Figure 2 clearly indicates that GOPSA remains the best-performing method compared to all tested methods.\n\n**Addressing Questions:**\n1. **The performance difference between “GOPSA” and “DO Intercept”:**\\\nFrom the revised Figure 2, we highlight that “GOPSA” significantly outperforms the “DO Intercept” which is the best performing baseline. Indeed, “GOPSA” significantly improved Spearman’s rank in 4 out of 5 site combinations (t-test p-values below 0.001). The results for the R² score and MAE are more nuanced, but on the majority of splits, both scores tend to be improved with “GOPSA” compared to “DO Intercept.”\n2. **Statistical assumptions of the t-test:**\\\nWe thank you for your remark regarding the assumptions of the t-test. Firstly, we recall that we revised the computation of the target $\\bar{y}$, leading to some changes in the p-values from the initial submission. To address your concerns, we examined the normality assumption of the score differences presented in the revised Figure 2B. Although space constraints prevented us from including Q-Q plots, our analysis confirms that the score differences generally follow normal distributions. Exceptions were noted for the MAE and R² scores in the site combination of the first row (Ba, Cho, G, S) and all scores in the site combination of the fourth row (Cu03, M, R, S). We also computed variances of scores of “GOPSA” and “DO Intercept” and obtained that they are equal except in the aforementioned cases. In these cases, the reported p-values are above 0.05, indicating that the score differences might not be statistically significant from zero and thus we do not mislead the reader. However, with more careful testing (without normal assumptions), some of the p-values could potentially be lower. We appreciate your attention to this detail, as it helped us ensure the robustness and reliability of our findings.\n\n**Addressing Limitation:**\n1. **Dataset concerns:**\\\nWe use the multi-centric dataset HarMNqEEG (2022) [h], which effectively serves as a group of datasets by combining measurements from various hospitals, with data from over 1,500 subjects. This dataset encompasses varying conditions of measurements, including data from 9 countries, 14 studies, and 12 different EEG devices. This combination of heterogeneous datasets explains the presence of a (X, y) shift between sites. Consequently, testing across different combinations of sites allows us to simulate experiments across various measurement conditions effectively. This approach enables us to treat these site combinations as distinct datasets, thus providing a robust assessment of the generalizability of our method.\\\nFor future work, we welcome any recommendations for additional datasets that could further validate the generalizability and robustness of our method.\n\n[a] Rodrigues, P. L. C., et al. (2018). Riemannian Procrustes analysis: transfer learning for brain-computer interfaces. IEEE Transactions on Biomedical Engineering\n\n[b] Paillard, J., et al. (2024). GREEN: a lightweight architecture using learnable wavelets and Riemannian geometry for biomarker exploration. bioRxiv\n\n[d] Huang, Z., & Van Gool, L. (2017, February). A Riemannian network for spd matrix learning. In Proceedings of the AAAI conference on artificial intelligence (Vol. 31, No. 1).\n\n[e] Sabbagh, D., et al. (2019). Manifold-regression to predict from MEG/EEG brain signals without source modeling. Advances in Neural Information Processing Systems, 32.\n\n[f] Mellot, A., et al. (2023). Harmonizing and aligning M/EEG datasets with covariance-based techniques to enhance predictive regression modeling. Imaging Neuroscience, 1, 1-23.\n\n[h] Li, M., et al (2022). Harmonized-multinational qEEG norms (HarMNqEEG). NeuroImage, 119190.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We thank the reviewers for the thorough and insightful reviews of our submission. 

We appreciate the acknowledgment of our method’s novelty, simplicity, and effectiveness in addressing domain adaptation challenges in EEG analysis. We are also glad that the presentation and technical soundness were noted positively across the reviews.

To address the weaknesses of the paper, we submitted a PDF file in this rebuttal with two figures:
- Figure 1 is new and contains simulated experiments where shifts are applied on either X, y, or both (X, y). These experiments clearly demonstrate the efficiency of the proposed method, “GOPSA”, in estimating shifts in X between domains, even in the presence of a shift in y, contrary to the baselines. This additional evaluation of domain-adaptation methods via ground-truth simulation of the data-shift generating process should help the reader to build intuition about the studied problem and the added value of the proposed solution “GOPSA”. A new section with these simulated data will be added to a subsequent version of the paper.
- Figure 2 is a revision of the Figure 2 of the submission. First, we estimate target $\bar{y}$ on target splits that do not overlap with the evaluation target splits, rather than assuming target $\bar{y}$ to be known. Second, we included two additional baselines in the experiments on real data: “Re-scale” [a], which corrects second-order statistics on the SPD manifold, and “GREEN” [b], a deep-learning architecture tailored for EEG data. “Re-scale” performs similarly to “Re-center”, i.e., is worse than all other methods. “GREEN” performs better than “No DA” but is still far from “GOPSA,” which is tailored for the studied problem. Overall, “GOPSA” still outperforms the strongest baseline, “DO Intercept.” Indeed, “GOPSA” significantly improved Spearman’s rank in 4 out of 5 site combinations (t-test p-values below 0.001). The results for the R² score and MAE are more nuanced, but on the majority of splits, both scores tend to be improved with “GOPSA” compared to “DO Intercept.” These findings underscore GOPSA’s superior performance and robustness in addressing the challenges of joint shifts in X and y.

In addition, we would like to emphasize that “GOPSA” was specifically developed to handle joint shifts in the data distribution and the outcome distribution, as illustrated by the simulations in Figure 1 of the rebuttal file. We thus expect “GOPSA” to outperform the baseline methods (e.g. “DO Intercept”) whenever joint (X, y) shifts occur. In our experimental benchmark, “GOPSA” significantly outperformed the baseline methods in some site combinations, but not all. This allows us to assume that not all site combinations show joint shifts. We, therefore, argue that this signature of “GOPSA” outperforming “DO Intercept” can be used by researchers as a decision rule to infer the presence of joint shifts and, hence, can serve as a tool for data exploration and model interpretation.

Finally, point-by-point answers to each reviewer are provided with respect to the different reviews.

[a] Rodrigues, P. L. C., Jutten, C., & Congedo, M. (2018). Riemannian Procrustes analysis: transfer learning for brain-computer interfaces. IEEE Transactions on Biomedical Engineering, 66(8), 2390-2401.

[b] Paillard, J., Hipp, J. F., & Engemann, D. A. (2024). GREEN: a lightweight architecture using learnable wavelets and Riemannian geometry for biomarker exploration. bioRxiv, 2024-05.

Meta-review: This paper presents a method for tackling domain adaptation challenges in EEG data analysis, specifically addressing shifts in both the feature space (represented by SPD matrices) and outcome variables. The approach is novel and well justified. The experimental evaluation seems not sufficient in the initial reviews, and the authors' rebuttal with additional experiments has successfully addressed this issue. The motivation of the study should be more clearly presented in the final version.
Decision: Accept (spotlight)