Geometric Optimal Transport for Unsupervised Domain Adaptation
Authors: Gal Maman
Ronen Talmon
Venue: Accepted by TMLR ()

Abstract: Optimal Transport (OT) is a widely used and powerful approach in domain adaptation.
While effective, most existing methods rely on the pairwise squared Euclidean distances for the transportation cost, implicitly assuming a Euclidean space.
In this paper, we challenge this assumption by introducing Geometric Optimal Transport (GOT), a new transport cost designed for domain adaptation under the manifold assumption.
By utilizing concepts and tools from the field of manifold learning, specifically diffusion geometry, we derive an operator that accounts for the intra-domain geometries, extending beyond the conventional inter-domain distances.
This operator, which quantifies the probability of transporting between source and target samples, forms the basis for our cost. 
We demonstrate how the proposed cost, defined by an anisotropic diffusion process, naturally aligns with the desired properties for domain adaptation.
To further enhance performance, we integrate source labels into the operator, thereby guiding the anisotropic diffusion according to the classes.
We showcase the effectiveness of GOT through comprehensive experiments, demonstrating its superior performance compared to recent methods across various benchmarks and datasets.

Published in Transactions on Machine Learning Research (10/2025)
Geometric Optimal Transport for Unsupervised Domain Adaptation
Gal Maman
Viterbi Faculty of Electrical and Computer Engineering
Technion
Ronen Talmon
Viterbi Pacully of Electrical and Computer Engineering Technion
Reviewed on OpenReview: https: // openreview. net/ met/ forum? id= 44UUU
Abstract
Optimal Transport (OT) a widely used and powerful approach in domain adaptation. While effective, most existing methods rely on the pairwise squared Euclidean distances for the transportation cost, implicitly assuming a Euclidean space. In this paper, we challenge this assumption by introducing Geometrie Optimal Transport (GOT), new transport cost designed for domain adaptation under the manifold assumption. By utilizing concepts and tools from the field of manifold learning, specifically diffusion geometry, derive an op- erator that accounts for the intra-domain geometrics, extending beyond the conventional inter-domain distances. This operator, which quantifies the probability of transporting be- tween source and target samples, forms the basis for our cost. We demonstrate how the proposed cost, defined by an anisotropic diffusion process, naturally aligns with the desired properties for domain adaptation. To further enhance performance, integrate source la- bels into the operator, thereby guiding the anisotropic diffusion according to the classes. We showcase the effectiveness of GOT through comprehensive experiments, demonstrating its superior performance compared to recent methods across various benchmarks and datascts.
1 Introduction
Unsupervised Domain Adaptation (UDA) is an extensively studied field within machine learning. In UDA, given labeled data from domain denoted as the source domain, we aim train model that will generalize well to unlabeled data from different domain denoted as the target domain. Optimal Transport (OT) stands out key method for addressing the UDA problem. The essence of OT lies in finding the most efficient way to transport from one distribution to another while minimizing a transportation cost. It offers a robust approach to measuring the similarity between probability density functions and facilitating an optimal mapping between them. This capability makes it well-suited for UDA tasks, where there a need for distribution alignment.
An established approach utilizing OT for UDA is the Optimal Transport for Domain Adaptation (OTDA) framework (Courty et al. 2016), which directly employs the transport plan to map source samples to the target domain via barycentric mapping. While OTDA has proven effective in addressing the UDA task, the OT solution may not always yicld the optimal mapping for maximizing target accuracy. To improve accuracy, a common practice is lo add a regularization term into the OT problem formulation (Courty et al.) 2014 2016 Flamary et al. 2014). Alternatively, in recent years, the OT problem has been incorporated into deep learning models. One notable method, termed DeepJDOT (Damodaran et al.] (0188), uses the OT problem address the discrepancy between source and target distributions, subsequently leveraging the obtained transportation plan to train feature extractor implemented through deep model. Following DeepJDOT,

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numerous studies have proposed OT-based deep models, aimed at improving target accuracy in the UDA task. Some have suggested different OT formulations to be used within the DeepJDOT framework (Fatras ltt al. 2021 Nguyen et al. 2022ah, while others have introduced entirely new architectures and objective functions (Chen et al. 2018 Lee et al.) 2019 Balaji et al.) 2020). While research on utilizing OT for DA tasks is exlensive and constantly evolving, one aspect that remains relatively unexplored is the selection of the transportation cost function. The transportation cost can take the form of any dissimilarity measure between samples in the source and target domains. Nevertheless, the predominant choice tends to be a distance function, with the most commón choice being the squared Euclidean distance, computed between source and target samples. Only small number of papers have suggested using a cost function diverging from the typical Euclidean distance measure (Tai et al. 2021; Nguyen et al.] 2022b Duque et al.) 2023). Notably, this exploration of the Iransportation cost function is even rarer in the DA context, with only a few studies proposing alternative approaches, such as employing metric learning with a weighted Euclidean distance (Li ct al. 2020 Xu al., 2020) or Mahalanobis distance (Kerdoncuff et al. 2020 Jawanpuria et al. 2025) as the transport cost.
In this paper, we depart from reliance on distance functions as the transportation cost, and instead, focus on the probability of transportation from source samples to target samples. To achieve this, we introduce Geometric Optimal Transport (GOT), new transportation cost based on an operator composed of three diffusion operators. The construction of the operator uses concepts and tools from the field of manifold learning, specifically, diffusion geometry (Coifiman & Lafon 2006). Initially, we construct a graph solely from the source samples, where each node represents a sample in the source data. From this graph, we build the first diffussin operator, which aims at preserving the local geometry of the source domain. When source labels are available, they are utilized to further guide the diffusion process, ensuring that cach source sample can only diffuse (transport) to other samples within the same label category. Subsequently, we build a bipartite graph incorporating both source and target samples to capture the inter-domain relationships. We construct the second operator from this graph, which facilitates cross-domain diffusion. For the third diffusion operator, we construct a graph exclusively from target samples, aimed at capturing intra-domain similarities within the larget domain. Taking the product of the three aforementioned operators forms the composite operator underlying GOT, We analyze the asymptotic behavior of this operator, demonstrating that the cost function is defined by an anisotropic diffusion procoss between the source and target domains. Our analysis underscores that this process is primarily influenced by the intrinsic geometry of the domains and their discrepancy.
Our main contribution. We introduce a new transportation cost, termed GOT, which is based on a composite diffusion operator consisting of three diffusion steps: (i) in the source domain, (ii) across domains, and (iii) in the target domain. This framework enables the learning of the geometries and relationships both between and within the two domsins, distinguishing it from conventional approaches by considering both inter-domain distances and intra-domain structures. By incorporating source label information into our cost, we eliminate the need for regularization terms. Furthermore, the proposed cost is straightforward to compute and is derived directly from the data, in contrast to competing cost functions that necessitate learning the cost and are thus limited to specific frameworks. These features make GOT applicable to any OT problem formulation and ensure compatibility with most O'T-based methods, with the experiments in the paper illusrrating only a fraction of its broader potential. Our experiments demonstrate GOT's effectiveness compared to competing approaches. Specifically, when integrated into a deep domain adaplation framework, GOT achieves superior performance on benchmark datasets over the baseline and other OT-based methods. Additionally, we evaluate our cost on Motor Imagery (MI) benchmarks, showcasing its applicability to non- Euclidean data.
2 Related work
UDA is a subfield of machine learning that addresses the challenges of domain shifts and generalization, where a model trained on data from a source domain performs poorly when applicd to a target domain with
a different data distribution. Various methods and metrics have been proposed to align the distributions
of the domains, including Maximum Mean Discrepancy (MMD) (Gretton et al. 2012 Pan et al. 2010).
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Correlation Alignment (CORAL) (Sun et al. 2017 and Domain Adversarial Neural Networks (DANN) (Ganin et al. 2016), to mention just a few.
The Optimal Transport for Domain Adaptation (OTDA) framework was introduced by Courty et al. (016)) In addition to the general framework, the authors propose the integration of several regularization terms to improve performance, including class-based regularization term (Courty et al., (2014) that utilizes source label information. By employing specific parameter configurations, the optimization problem can be effec- tively addressed using the Sinkhorn algorithm (Cuturi, 2013). The authors also proposed applying graph regularization on the transported samples (Flamary et al. 2014). While this regularization, termed Laplace regularization, considers neighborhood information, it is limited to Euclidean geometry. Furthermore, their optimization problem is quadratic, rendering it unsolvable by efficient algorithms and resulting in high com- putational complexity. A notable work is the JDOT framework introduced in CCurty et al. (2017), which presents an optimization problem designed to simultancously optimize both the transportation plan and the classifier. Building upon this concept, the DeepJDOT framework (Damodaran et al.) 2018) further extends the idea by leveraging deep learning algorithms. This approach addresses two significant drawbacks in the OTDA framework. Firstly, handling large datasets becomes infeasible with OTDA due to the quadratic in- crease in complexity with the number of samples N, and the fact that it involves computations with N x N matrices. Conversely, DeepJDOT can train deep models using mini- batches, making it suitable for large datasets. Secondly, OTDA predominantly employs the squared Fuclidean distance as the transportation cost, and, as a result, it often performs poorly on non-Euclidean datasets. In contrast, although the DeepJ- DOT framework also uses the squared Euclidean distance as the cost, it operates in a latent space learned by a deep model, potentially enabling superior performance the authors proposed enhancing the DoepJDOT framework by employing Unbalanced Optimal Transport (UOT), yielding improved results. A related work by Nguyen et al. (2022a) suggested solving it using Partial Optimal Transport (POT) and presented a two-stage implementation for the DeepJDOOT framework, further enhancing target accuracy, particularly when coupled with the POT formulation. The OT problem was later incorporated into various deep learning and adversarial training frameworks (Chen et al., 2018 Lee et al., 2019 Xie et al. 2019: Balaji et al. 2020), all relying on the squared Euclidean distance as the ground cost.
non-Faclidean data. In Fatras et al. (2021)
Few papers in the literature have proposed alternative cost. functions diverging from the standard distance metric. For instance, Tai et al. 2022 introduced an innovative approach termed SLA, which utilizes the OT formulation a novel label assignment. method. In this approach, each element (i. k-) in the transportation cost matrix is associated with the probability that sample i belongs to class k. Gu et al. (2022) introduced a semi-supervised cost for Hetcrogeneous DA, which assumes access to a set of source-target pairs referred to as keypoints. Using these keypoints, the authors propose to compute a relation score for all source and target samples, which subsequently used to construct the cost matrix. A related semi-supervised approach is presented in Duque et al. (2023) for manifold alignment. This method similarly relies on prior correspondence knowledge between some source and target samples to align the domains. It constructs intra-domain diffusion operators for each domain and combines them by multiplying the operators. The known correspondences are used to select the relevant rows, resulting in a joint diffusion operator. The final cost matrix is computed using the cosine distance between rows of the joint and intra-domain diffusion operalors. While diffusion plays a central role in shaping the alignment, the approach is cemi-supervised and does not incorporate inter-domain distances. Furthermore, the resulting cost lacks a probabilistic interpretation. [Aadulacy et al. (2022) proposed a neural beawork-based framework for learning lask-specific semi-supervised cost functions. Although the have explored approaches that integrate both intra- and inter-domain information, albeit not in the context of transportation cost or specifically for DA. These methods are worth noting, as they share conceptual similarities with our approach in incorporating structural information from both domains. For example, Yan et al. (2018) introduced a semi-supervised DA method that adds a regularization term to the optimization problem. This term incorporates cross-domain distances, computed in the target domain after transportation, and relies on target labels. Alternatively, Xu et al. (2019) presented embedding space for the nodes, enabling the computation of inter-domain relationships, which are included in the optimization via an additional term.
distance typically leverages only intra-domain relationships, several works
method that learns an
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In the context of UDA, Kerdoncuff et al. (2020) proposed using Mahalanobis metric learning for the cost matrix. Specifically, they learn the metric and the transport plan simultaneously through alternating op- timization, where source labels are incorporated via a regularization term to guide the metric learning, Similarly, Jawanpuria et al. (2025) also employed Mahalanobis distance with metric learning. Hore, a regu- larization lerm introduced, which not only prevents trivial solutions but also enables a closed-form solution for the metric given the transport plan. In both methods, the Mahalanobis distance accounts only for inter- domain relationships (similar to the Euclidean cost), and intra-domain structure is not incorporated. Dhouib et al. (2220) derived bound on the target margin violation rate and proposed DA algorithm based on this bound. By using the dual representation of the infinity norm in their discrete bound, they parameterize the cost with a learned vector (that acts as a weights vector non-negative elements and sums to 1), and solve a min-max problem to optimize the plan and the learned vector. This method considers only inter-domain relationships and does not incorporate source labels.
In the context of deep UDA, Li ct al. (2000) introduced ETD, which utilizes a weighted Euclidean distance for the cost function. They proposed leveraging an attention mechanism to learn the correlation ween source and target samples, subsequently using this correlation as the weights in their proposed transporta- tion cost. The OT distance is then employed in the training objective, serving as a quantification of the domain discrepancy. The work in Xu et al. (2020) also proposed using weighted Euclidean distance as the transportation cost. In that method, termed RWOT, the element (i, k) in the weight matrix quantifies the probability that the i-th sample belongs to class k. This probability is dynamically computed based on both spatial prototypical information and the classifcassi probability of target samples. Both methods have fundamental drawbacks compared to our proposed approach. First, ETD and RWOT rely learned weights, limiting their applicability to specific deep learning frameworks, while GOT is derived directly from the data. Second, the costs in ETD and RWOT, including the learned weights, consider only the inter-domain relationships and source label information, whereas our cost also incorporates intra-domain relationships for both source and target domains.
By integrating principles from diffusion maps (Coifman & Lafon 2006), our approach incorporates intra- domain relationships and aims to reflect the underlying geometric structure of the data in the proposed cost. Diffusion maps, originally a dimensionality reduction tochnique, provide method to analyze and visualize high-dimensional data. The core concept involves constructing a transition kernel, where transition probabilities are defined by the local similarities between data samples. Diffusion geometry has been widely applied in various fields, including data analysis (Coifimn & Lafon 2006), computer vision (Bronstein et al., 2010 Liu et al. 2012), and medical imaging (Haghverdi et al. 2015 Zheludev et al. 2015), among others. For further background on diffusion geometry, see Appendix A
3 Optimal Transport for Domain Adaptation
We formulate the problem we consider and briefly describe the OTDA framework proposed in Courty et al. (2016).
Problem formulation. Let X' be an input space and y be label space. We define a domain as a distribution D over the input space X. Consider N, samples {xf e NY,, drawn iidd from a source domain D., associated with labels {uf E V and N, unlabeled samples {a² X3 drawn i.i.d from a target domain D,. In UDA, the goal is to enable a predictive model S, trained on labeled source samples, to generalize well to target samples. There are two dominant approaches for UDA. In the first approach, a mapping function g: X' + X' is learned to align the source domain to the target domain, e.g., by minimizing the divergence between them. A prediclive model ʃ is then trained using the mapped source samples, denoted by 2 = g(z)). In the second approach, we learn an embedding function g X + Z that maps both domains to a latent space Z of domain-invariant features. Thus, a predictive model f Z + V, trained using the source embeddings {2] Na with labels { i will generalize well on the target embeddings {2}= -
OTDA framework. We start by outlining the discrete OT formulation, which is consistently employed throughout this paper. For any source sample of and target sample a/, let c(xt,a²) denote the cost to
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move a probability mass from x; to z/, which is typically chosen to be the squared Euclidean distance. Let C N.N. be the cost matrix, with elements Ci5 = c(x3. x²)). The discrete OT formulation is given by:
Y = arg min (7, C)F, Yer
(1)
where I'= = {o E 1 =a,7" = b) denotes the set of transport plans that satisfy conservation of mass, a E R and b E R are the empirical distributions of the source and target samples, respectively, and (, denotes the standard Frobenius inner product.
This optimization problem has been studied for many years and addressed through various algorithms. To date, one of the most commonly employed algorithms is the efficient Sinkhorn algorithm (Cuturi 2013
which incorporates entropy regularization into the objective function. This leads to the following modified problem that can be solved itcratively using lincar projections, offering reduced complexity:
Y = arg min((, C)F + Ml2(1);
(2)
Tr
where 1=())== Ei, (1. (ig is the negative entropy of 7, and À is hyperparameter.
The OT problem can be utilized for UDA in both of the aforementioned approaches. Recent deep learning- based methods, such as Damodaran et al. (2018); Chen et al. (2018); Lee et al. (019)), incorporate OT into the loss function to learn domain-invariant representations, thereby following the second approach for UDA. In contrast, the OTDA framework follows the first approach, where, after solving the OT problem, source samples are transported to the target domain using the obtained transport plan. The transportation of any source sample zę is given by the barycentric mapping (Courty et al. 2016
Perrot et al. 2016), defined by:
2 arg min
4 Geometric Optimal Transport
In this work, we present GOT, new cost. function for the OT problem, designed specifically for the UDA setting described in Section g Our premise is the manifold assumption, a widely accepted principle in modorn high-dimensional data analysis that has given risc to an extensive body of work tormed manifold
learning (Roweis & Saul 2000 Tenenbaum et al., 2000: Belkin & Niyogi, 2003 Coifman & Lafon 2006). In the UDA setting, it implies that both domains are supported on the same hidden manifold M embedded in X. When the squared Euclidean distance is used as the transport cost, whether explicitly stated or not, it is assumed that the source and target domains are embedded in Rd. The manifold assumption extends this concept, allowing for the possibility that, the samples may not lie within Fuclidean space. Building on this foundation, we propose defining the OT cost matrix C based on diffusion operator, composed of three diffusion steps. Notably, the construction of this operator stemss from diffusion geometry (Coifinan & Lafon 2006), which is a prevalent manifold learning approach (see Appendix A for details).
Initially, we build an operator solely from the source samples, aimod at capturing the local geometry within the source domain. We construct a graph G, with N, nodes, where each node represents source sample. Under the manifold assumption, geodesic distances on M are locally Euclidean. We use the Euclidean distance to build the affinity matrix of the graph, K, E R.N.N., with a Gaussian kernel:
|| 1
Ki,i,) exp
(4)
where e, > 0 is a scale hyperparameter. Importantly, although the kernel based on Euclidean distances, this is fundamentally different from using the Euclidean distance directly as the OT cost: by employing this local kernel, which quantifies pairwise local similarities between data samples, the graph effectively captures the local neighborhood relationships.
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M
M
(a) Step #1
(b) Step #2
(c) Step #3
Figure 1: ralsttation of the diffusion process for computing Súj, the transition probability from the source i-th sample to the target j-th sample. Source samples are represented by circles, target samples by triangles, and the source markers are color-coded by labels. (a) Transition from x to ncarby source samples of the same class. (b) Transition from those source samples the nearest neighbors of z².. (c) Transition from the target neighbors to zty- The final probability Siy aggregates all paths from of to z/y-
Subsoquently, we obtain the source diffusion operator, denoted by P. E RRN,N, by normalizing the affinity matrix K, to be row-stochastic, i.e., ensuring that the sum of each row equals Following this standard normalization (Coifman & Lafon 2006), the rows of P. represent the transition probabilities of diffusion process on M defined at the source samples (af)
In the second diffusion step, we apply a cross-domain diffusion, which captures the inter-domain relationships by incorporating both the source and target samples. To this end, we define a bipartite graph G. with a vertex set V. = (uf.....00 this graph, each node us has edges only to target nodes, and each node vn has edges only source nodes. Consequently, the graph adjacency matrix consists of two off-diagonal blocks. For our purpose, utilize node of is given by:
of...., v that represents the N, source samples and N, target samples.
only the N. N₂ top-right block of the matrix. The weight of the edge between source node of and target
|| 11 EC
Ko(i.)) exp
(5)
where 6, > 0 is a scale hyperparameter. We normalize this kernel matrix to row-stochastic, and denote the resulting cross-domain diffusion operator by Q N.N.
In the third and last diffusion step, designed to capture intra-domain similarities within the target domain, construct the graph G, exclusively from target samples. This graph comprises Ne nodes, representing the target samples, and an affinity matrix K, RR,, N,N,, constructed using a Gaussian kernel with scale similarly to Equation 4 After normalizing K; to be row-stochastic, we obtain the target diffusion operator, dcnoted by P. E N.
The final operator S, which serves as the basis for the transport cost, is defined by:
S PPQ.P.
(6)
By construction, S(i.j) the probability of transporting from source sample x target sample z/ through local neighborhoods. Specifically, S(i,j) encapsulates the probability of (i) diffusing from source sample of to its source neighbors, (ii) diffusing from these source neighbors to target samples that are neighbors of a/. and finally (iii) diffusing from these target neighbors to z/.- An illustration of this triple diffusion process is presented in Figure [ The intra-domain graphs use local neighborhoods, defined based on class labels for G, and nearest. neighbors for G, (details in Section 4.2
As final step, we apply a negative element-wise logarithm the diffusion probabilitics to translate the notion of similarity into notion of cost, aligning with the requirements of the OT problem. Thus, the proposed transportation cost matrix, referred to GOT, is defined by: C = log(S.)

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The computational complexity of GOT is O(n"); see Appendix B.1 for details, and Figure 10 for empirical run-time comparison with competing methods.
4.1 Theoretical results
While competing methods typically do not include theoretical insights, our approach is driven by the following theoretical basis. Specifically, prove that the proposed operator S is diffusion operator and analyze its asymplotic behavior, showing how it aligns with the desired properties. This analysis provides a deeper understanding of the motivation behind our method. Building on this, we now transition to examining the diffusion operator in a continuous setting.
Consider infinite source and targel datasets, sampled i.id from the probability distributions μ and v, respec- tively, defined over the continuous manifold M. Consider Gaussian kernel (x, y) with scale parameter In analogy to the row-stochastic normalization in the discrete case, we define the continuous source diffusion operator Po.c by:
(e((x,)) f(x))(x)) dac(x)
Pa,ef(u)= - Pr.c(E, ()
where Ps,c(x,)) v) the normalized kernel, and (x) = (i, y)b(y)d
Similarly, define the continuous target diffusion operator Pr,er which operates the target domain, by employing Equation with respect to u and with the normalized kernel y) = (x(a.)) where de, f (x.
d..(z))
The cross-domain diffusion operator is defined by:
Qef(v) = / ke (2 (x)
y) f(z)
(8)
Note that in this analysis, without loss of generality,
the same kernel ke(2, for all three operators.
In the discrete case, applying the diffusion operator defined in Equation 6 to the source samples necessitates using the transpose of S. Thus, in the continuous case, the diffusion operator Se, defined as the composition of the three operators Ps.c, Qe, and Pt,c, can be expressed as Sef(x) = -P..QeP...fx))
Proposition 4.1. Suppose f E (MM, and suppose μ, E (MM) denote the probability measures of the source and larget domains, respectively, where μ is dominated by v. For sufficiently small e, the usymaplotic exponsion of operator S. is given by:
(
S.f(x) = (xx)
m2
3Δʃ +2
+ + log (
(9)
- (
- Au )
(x) (z))
where A, V are the Laplace- Laplace-Beltrami operator and the covariant derivative on M, respectively, and mp, m2 two constants defined by the Gaussion kernel and by the manifold M.
The proof appears in Appendix D
We remark that if p(x)= vx)) for all x E M, indicating no domain shift, the asymplotic expansion given by:
S.f(x) f())
m2
3 - 3f #
(x) + +O()),
(10)
mo
coinciding with the asymptotic expansion a diffusion operator on M with distribution Coifman &
Lafon 2006), applied three times.
The expression derived in Proposition suggests that GOT inherently considers both the domain shift and the intra-domain relationships of the source and target domains. In particular, the result implies that

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the proposed diffusion operator leads to an anisotropic diffusion process the manifold M, inffuenced by the manifold geometry (conveyed by the ia operators A and V w.r.t. the manifold), the source and target domains (captured by the terms within the second set of round parentheses), and the discrepancy between them (captured by the term within the first set of round parentheses, involving both the first- and second-order derivalives of 5). We illustrate the induced anisotropic diffusion process with a 2D loy example outlined in Appendix B.2
In OT, the goal is to find transport plan that minimizes the total cost of moving mass between distributions. In our approach, this cost is obtained by applying negative logarithm to the diffusion operator S. Viewing the expression derived in Proposition 4. from this perspective highlights the advantages of using the operator as the basis for our transport cost, particularly in the context of domain adaptation.
When solving the transportation problem, samples that minimize the cost are favored that is, they are assigned more mass in the transport plan. More specifically, by maximizing the term (A (or assigning more weight lo samples for which this term is large), we focus on regions with non-uniform density, where the source and target densities change rapidly. These regions tend to be sparser (where j is small). More importantly, they often correspond to key geometric structures in the data (where Ap is large). In the context of transport, this focus provides an additional advantage: samples in such regions carry richer structural information and are less ambiguous, whereas samples uniform-density areas tend to lack distinctive local features, making them harder to match.
# + 2 2 (x)
A natural question arises: what happens at locations where Ae is high while Au is low, vice versa? u
In other words, how do we avoid assigning high weight to regions where, for example, the source has low
density while the target has high density? This is where the term A(#) す 2V log (5) becomes important. Minimizing this expression encourages transport between regions where the ratio E varies smoothly. In such cases, high-density regions in the source are aligned with high-density regions in the target, and likewise for low-density regions. This prevents density inversion, where a dense source region is mapped to a sparse target region- mismatch that would distort the local geometry. Figure 5 in Appendix [.3 provides a detailed illustration of this analysis.
4.2 Enhanced intra-domain kernels
In UDA, we assumee access to labeled source samples. To leverage this information, we incorporate source labels into the proposed cost, guiding the diffusion based on the class labels. Specifically, rather than using fully connected graphs for the intra-domain kernels, as in Equation 4 we adopt the widely used approach of integrating neighborhood information into the Gaussian kernel. The resulting intra-domain kernel:
exp (- -
if K otherwise
K(i,k) k)
(11)
where M3 is defined as the set of all indices {k} such that the sample Iy is within the neighborhood of x;,
For the source operator Ps, where labels are available, we define the neighborhood as N₁ = {1: y = }:}: en- suring that each source sample x diffuses only to other source samples within the same class. This approach stands in contrast to other methods that propose incorporating source labels by adding regularization term to the problem formulation. Such regularization terms often impose constraints on the OT solver the learned geometry, potentially restricting the method's adaptability and performance.
For the target operator Pt, or in scenarios where labels are unavailable in specific DA setup, we define the neighborhood Nú, as the K nearest neighbors of vy. This allows leveraging local structure elfectively, even in the absence of label information. In this work, we set K = 3 for the target neighbors across all toy and real experiments. Appendix C.1.2 provides a discussion of this choice and demonstrates its robustness.
The proposed method is summarized in Algorithm 1
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Algorithm GOT
Input:
(((r), wf) 3=1 6a, Et.
1: Compute the intra-domain kernel of the source, K using Equation and supervised neighborhood information.
Compute the intra-domain kernel of the target, K,, using Equation 11 with three nearest. neighbors neighborhood.
Compute the cross-domain kernel K. according to equation 5
4: Compute the row-stochastic probability matrices, P., Pt. and Q.
5: Compute the diffusion matrix S according to Equation 6: Apply element-wise negative logarithm = - log((S).
Return the cost matrix C.
5
Experiments
In this section, we provide a visual comparison of GOT with competing methods using a toy example (Section 5.1. Additionally, we present experiments on deep domain adaptation tasks (Section 5.2) and non-Euclidean dataset adaptation (Section 5.3), comparing the performance of GOT against the leading methods. For all experiments, we employed the label-enhanced kernel for the source diffusion operator (Section [.2)). For all three diffusion operators Ps, Q, Pt; before computing the Gaussian kernels, conducted median normaliza- tion on the distance matrix. After computing the kernels, we performed doubly-stochastic normalization the square matrices P. and P, using the Sinkhorn algorithm, suggested in Landa et al. (021) The madrix Q which may not necessarily be square, was normalized to be row-stochastic. Unless specified otherwise, we employed the Sinkhorn algorithm (Cuturi 2013) to solve the OT problem, utilizing the publicly available POT package (Flamary al.j 2021]. While largel labels were not used during training, we did rely on them a validation set for hyperparameter tuning, consistent with competing methods. Parameters were optimized for each dataset as whole, rather than for each individual source target pair. Full details of the GOT parameters
provided in Appendix C1ll and the parameters used in each experiment are reported in Appendix C In Section 5.3 we explore an unsupervised tuning strategy on multiclass BCT task. All
the experiments were conducted on Nvidia DGX A100. The source code is available here
5.1 Two Moons: Visual Illustration
We utilize the loy example presented in Courty et al. (2016), which involves two entangled moons as the source domain, each representing a different class, The target domain is independently sampled from a rotated version of the source distribution. As the rotation angle increases, the problem becomes more challenging. Further details are provided in Appendix C.2
We employ GOT with the retroiy-regularized OT problem (Cuturi 2013), as presented in Equation 2 to derive the transport plan. The bascline method, OT, uses this formulation with the standard cost. Additionally, due to the limited number of methods proposing alternative transport costs particularly those that integrate source label information and can be applied non-deep frameworks we include two regularization approaches: OT-reg (Courty al., 2016 2014), which incorporates source labels, and OT- Laplace (Courty et al. 2016 Flamary et al. 2014). which uses Laplace regularization term to preserve data structure. We emphasize that all three met hods use the squared Euclidean distance as the transportation cost. Table in Appendix C.2 summarizes the key differences between the methods.
In Figure 2 we visualize the obtained plans for a 40 rotation angle, and focus on one target sample, marked by a bold black triangle. For clarity, we plot lines from this sample to the ten source samples with the highest values in the corresponding column of the plan. In Figure 2a we observe that GOT successfully achieves the optimal transportation plan, with all source samples contributing mass the red-labeled larget sample associated with the red class. In OT (Figure 2b). since the cost is based solely on the pairwise Euclidean distances, the mass is coming from the closest source samples, most of which belong to the wrong class. In
https://github. com/ Caan//aaaaaaa/OTT

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Source Target
Class 1 Class 2
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Dub
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0
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4
-4
-2
-4
-2
X1
X1
(a) GOT
(b) OT
(c) OT-reg
(d) OT-Laplace
Figure 2: Two moons illustration for a 40 rotation angle. Source samples are marked with circles, target
samples with triangles, and the colors of the markers indicate the labels. Each line represents an entry 7(, in the transportation plan, where the j-th sample is marked by bold black triangle. For clarity, we plot lines from this sample to the ten source samples with the highest values in the j-th column of the plan. The line color corresponds to the label of the source sample 2;, and the line intensily reflects the values in the plan. (Best viewed in colors).
OT-reg (Figure 2ch, the regularization term encourages solution where each target sample receives mass only from source samples of the same class. In this case, it results in the most undesirable plan, where all source samples are from the opposite class. In OT-Laplace (Figure 2d), the regularization aims to keep nearby samples close after transportation. However, the encoded neighborhood information is insufficient to correct the Euclidean cost, and the plan mirrors the behavior of the standard OT, We note that the plan is sparser as the method does not include entropy regularization.
This scenario highlights GOT's effectiveness in handling challenging mappings. GOT leverages source labels not as strict constraints, as OT-reg, but to guide the diffusion process in the source domain, while still accounting for the local geometry of the data. Additionally, it shows that by taking into account. intra-domain relationships, rather than relying on pairwise Euclidean distances, the chosen sample can receive mass from source samples that are remote in Euclidean terms, as the diffusson distance effectively incorporated. Indeed, we see that GOT successfully connects the chosen target sample to the correct, more distant source samples, which other method considered. We provide further visual analysis and numerical comparison in Appendix C.2
5.2 Deep Domain Adaptation
In the following experiments, we incorporate GOT into the Deep.DDOT framework (Damodaran et al. 2018 which comprises two components: embedding function X Z, designed to learn leatures that are both discriminative and domain-invariant, and classification function f : 2 -- V. The DeepJDOT cost matrix for estimating the transport plan 7, given the fixed-parameter models f and I, is given by:
C₁ 77 l0(=;) (( + r2 oz f(o(xt)))
(12)
where the cross-entropy loss, and 1/, 1/2 are hyperparameters.
The cost combines (i) feature alignment via Euclidean distance in the latent space, and (ii) label alignment via cross-entropy between source labels and target pseudo-labels. In our work, we replace the Euclidean term in Cūj with the proposed GOT cost. For additional information about the framework and the optimization process, see Appendix C.3
We apply our method to three benchmark domain adaptation datasets: Digits, Office- Home, and VisDA. One of the key advantages of our method its adaptability any OT formulation. To demonstrate this flexibility, we employ different formulations for each dataset, as detailed below, resulting in a different baseline method across datasets. We compare our method to the DeepJDOT framework with the standard
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Table 1: Target domain accuracy. Results on (a) the Digits dataset, and (b) the VisDA dataset.
(b)
Method
SVHN-MNIST
USPS-MNIST MNIST-USPS Avg
Method
Accuracy
DANN
95.80 +: 0.29
94.71 +: 0.12
91.63 + 0.53
94.05
DANN
67.63 + 0.34
ALDA
98.81 = 0.08
98.29 = 0.07
95.29 = 0.16
97.46
ETD
97.9 = 0.4
96.3 0.1
96.4 : 0.3
96.9
ALDA
71.22 + 0.12
RWOT
98.8 = 0.1
97.5% 0.2
98.5 0.2
98.3
DeepJDOT
69.58 + 0.34
DeepJDOT
96.04 + 0.66
97.22 + 0.23
86.12 + 0.60
93.13
JUMBOT
72.97 + 0.26
JUMBOT
98.99 = 0.06
98.68 - 0.08
96.93 = 0.42
98.20
TS-POT
75.65 + 0.78
m-POT
98.98 = 0.08
98.63 = 0.13
96.04 = 0.02
97.88
Ours
99.19 + 0.04
98.87 + 0.03
97.81 - 0.30
98.62
Ours
78.56 = 0.15
Table 2:
Target domain accuracy on the Office- Home dataset (ResNet-50).
Method
A-C
A-P
A-R
C-A
C-P
C-R
P-A
P-C
P-R
R-A
R-C
R-P
Avg
ResNet-50
34.9
50.0
58.0
37.4
41.9
46.2
38.5
31.2
60.4
53.9
41.2
59.9
46.13
DANN
47.9
67.1
74.9
53.8
63.5
66.4
53.0
44.4
74.4
65.5
53.0
79.4
61.93
ALDA
54.0
74.9
77.1
61.4
70.6
72.8
60.3
51.0
76.7
67.9
55.9
81.9
67.04
ETD
51.3
71.9
85.7
57.6
69.2
73.7
57.8
51.2
79.3
70.2
57.5
82.1
67.29
RWOT
55.2
72.5
78.0
63.5
72.5
75.1
60.2
48.5
78.9
69.8
54.8
82.5
67.63
DeepJDOTT
52.0
70.9
76.1
60.5
66.6
69.2
58.4
48.7
75.3
68.9
54.9
79.9
65.12
JUMBOT
55.7
75.0
80.7
65.1
74.5
75.1
65.3
53.3
79.6
74.5
59.3
83.9
70.17
TS-POT
57.4
77.1
81.6
68.3
72.8
76.5
67.4
55.1
80.6
75.4
59.9
84.0
71.36
Ours
57.2
78.0
82.1
70.2
74.9
78.8
68.1
56.5
82.0
75.6
60.9
84.8
72.43
cost (DeepJDOT, JUMBÖT, and m-POT), O'--based methods with alternative costs (ETD and RWOT), and DA basclines (DANN and ALDA). Further details are provided in Appendix C.3
Results. Table Ta presents the results for the Digits datasets. Details regarding the model architecture and parameters for cach method available in Appendix C.3.1 For the digits datasct, follow Fatras et al. (2021) and use the UOT formulation to obtain the transportation plan, making JUMBOT the baseline method. The table displays the mean and standard deviation of the target test set over three runs. Results for each individual run can be found in Table in Appendix C.3.1 The highest accuracy is indicated in bold, while the second highest is underlined. Our method achieves the highest accuracy for SVHN and USPS- Additionally, it attains the second highest accuracy for MNIST-USSPS, followed by RWOT. We acknowledge that this dataset is considered an easy DA task, and appears that the generator learns features that are well separated by the classes, even when using the standard cost. Therefore, the obtained minor improvement. is expected.
Table 2 shows the results for the Office-Homme dataset. In this experiment, we again incorporate the proposed cost into the UOT problem formulation to derive the optimal plan. Here, DeepJDOT and JUMBOT were reproduced using the original DeepJDOT framework, while TS-POT was reproduced using two-stage (TS) implementation, as proposed the original paper (Nguyen et al. (022a).. The results are averaged over three runs, with full details, including the parameters used for each method, provided in Appendix C.3.2 Table 2 summarizes the target test accuracy for each scenario of the Office- Home dataset. Remarkably, the proposed method achieves the highest average accuracy across the 12 scenarios and outperforms the tested existing methods in 10 out of the 12 scenarios. Specifically, it surpasses the baseline JUMBOT, which was implemented using the UOT formulation with the standard cost, by more than 2%.
Table Dilceeepprreeett the mean and standard deviation over three runs for the VisDA datasct. In this experiment, we integrate GOT into the POT problem formulation, using the TS implementation proposed by Nguyen al. 2022a). Consequently, TS-POT serves the baseline. DeepJDOT and JUMBOT reproduced using
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Table 3: Binary classification accuracy on two BCI datasets. Results for (a) the cross-subject task, and (b)
the leave-one-uubecct-ort task.
(a)
(b)
Method
MI1
MI2
Avg
Method
MI1
M12
Avg
CSP-LDA
57.23
58.70
57.97
CSP-LDA
59.71
67.75
63.73
RA-MDRM
64.98
66.60
65.79
RA-MDRM
69.21
70.91
70.06
EA-CSP-LDA
66.85
65.00
65.93
EA-CSP-LDA
79.79
73.53
76.66
MEKT-R
70.99
68.73
69.86
MEKT-R
83.42
76.31
79.87
METL
71.81
69.06
70.44
METL
83.14
76.00
79.57
OT
63.83
64.59
64.21
OT-reg
67.17
67.26
67.22
WBT
85.21
74.31
79.76
GOT (ours)
72.24
70.41
71.33
WBT + GOT (ours)
87.29
77.24
82.26
the original Dee framework, while TS-POT was reproduced using the TS implementation. Additional details regarding the TS implementation and the parameters used for each method are available in Appendix C.3.3 We observe that employing the TS implementation with the POT formulation is highly effective for this dataset, TS-POT obtained significantly higher accuracy than the other competing methods. Notably, incorporating GOT into this framework further enhances performance, incrcasing accuracy by nearly 3% on average. In Figure 11 in Appendix C.3.3 we analyze the t-SNE representation of the VisDA target features obtained from the deep model trained with GOT, comparing them to those learned using the baseline method, TS-POT. In Table [[ of Appendix C.3.3 we present the accuracy achieved by integrating GOT into framework with a vision transformer (ViT) backbone, comparison to thaaa-of-tho-art (SOTA) methods utilizing similar architectures.
5.3 Domain Adaptation for non-Euclidean Data
features and instead directly utilize the optimal plan to transport source features into the target domaín. We evaluate the proposed method electroencephallgrram (EEG) data, addressing both binary and multi-class classification tasks. Specifically, we apply our approach to two Motor Imagery (MI) benchmarks from the BCI Competition IV. The first dataset, dataset 1 from Blankertz et al. (2007), is referred to as MII, and the second dataset, 1a from Tangormann et al. (2012) is referred to as MI2. We follow the mothodologics of |Barachant et al. (2011: 2013); Zanini et al. (2017: Rodrigues et al. (2018); Yair et al. (2019). I and solve the DA problem on the (non- Euclidean) Riemannian manifold of Symmetric Positive Definite (SPD) matrices. For all experiments in this soction, utilize the entropy-regularian
In this section, we depart from employing OT learn
problem formulation (Cuturi 2013), referring to methods GOT. The baseline method, which uses this formulation with the standard transport cost, is labeled OT in the tables. Additionally, we include comparison with OT-reg (Courty et al. 2016 2014), a regularization approach that incorporates source labels into the problem formulation, which we consider as a secondary baseline. For detailed information on the framework, datasets, pre-processing, and parameters, please refer Appendix C.4.
Results. To facilitate meaningful comparison with leading methods, which are limited two-class sce- binary classification task. For the evaluation,
narios, initially evaluate our proposed method utilize both BCI datasets; specifically, for the M12 dataset, we select the left hand (class 1) and the right hand (class 2) as the two classes of interest. Our method is benchmarked against bascline algorithms commonly employed in BCI classification, namely CSP-LDA (Grosse- Wentrup & Buss 2008), RA-MDRM (Zanini et al., 2017). and EA-CSP-LD: (He & Wu 2019). Additionally, we compare our approach to two leading methods: MEKT (Zhang Wu 2020) and METL (Cai al.] (022). Results for CSP-LDA, RA- MDRM, EA-CSP--DA, and METL are taken from the tables in Cai (2022), while results for MEKT
sourced from the original paper (Zhang & Wu 2020
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Table 4: Multi-class classification accuracy on dataset M12. (a) Results for the cross-session task. (b) Results for the cross-subject task.
(a)
(b)
Method
1
3
7
8
9
Avg
Method
3
7
8
Avg
OT
81.08
84.03
78.12
77
68.06
77.71
OT
55.38
62.74
57.53
56.25
52.28
56.84
OT-reg
80.9
85.59
78.99
83.33
75.17
80.8
OT-reg
54.36
62.48
57.25
59.09
54.62
57.56
MADAOT
68.75
71.53
70.14
72.22
65.8
69.69
MADAOT
54.41
59.79
54.97
57.29
53.49
55.99
MLOT
84.9
84.72
83.33
83.85
77.26
82.81
RMLOT
83.33
78.3
67.36
81.08
68.23
75.66
MLOT
61.48
70.01
64.04
66.43
59.7
64.33
GOT
83.51
86.98
74.31
88.19
80.73
82.74
RMLOT
43.4
54.51
43.12
46.2
47.35
46.92
MLOT GOT
86.63
86.28
84.72
83,85
75.52
83.4
GOT
63.24
75.13
67.3
71.53
64.45
68.33
Table 3a presents the results for the cross-subject task. In this experiment, each subject, in turn, serves as the target domain, with the remaining subjects alternately acting as the source domain. Accuracy is computed by training a linear SVM classifier on the transformed source mappings and evaluating it on the target mappings. The results show the average target accuracy across source subjects, averaged over all subjects. The proposed method achieved the highest accuracy for both datasets, improving the baselines OT and OT-rog by 7% and 4%, respectively. Detailed subject-wise results appear in Table 12 and Table 13 in Appendix C.4.1. along with the experimental parameters.
In Table 3b we present the results for the leave-ore-subeect-ort task, where each subject alternately acts as the target domain while all others are used as source domains. We employ the multi-source OT-basod DA framework from Montesuma & Mboula (2021), referred to as Wasserstein Barycenter Transport (WBT). The table reports the average target accuracy, computed by training a linear SVM all the transported source representations and evaluating on target samples. Interestingly, the baseline method (WBT with the standard cost) outperformed all competing methods on dataset MI1. Notably, incorporating GOT resulted in further accuracy improvements, achieving the highest performance across both datasets. For details on the framework and the experimental parameters, see Appendix C.4.1
Finally, we assess the performance of GOT in multi-class classification task on dataset MI2. In addition to the baseline methods OT and OT-reg, we compare GOT to the competing methods MADAOT (Dhouib) ct al. 2020 MLOT (Kerdoncuff al. 2020), and RMLOT (Jawanpuria ct al., 2025 In Kerdoneuff et al. (2020) the authors propose lo rely solely source labels for parameter luning. We follow similar approach, and implement a more robust selection procedure applied consistently across all competing methods. Further details on the tuning procedure are provided in Appendix C.4.2.
Table 4a shows the results for the cross-session task using source-label tuning. In this experiment, for each subject, the first session (day) serves as the source domain, while the second session serves as the target domain, and vice versa. The reported target accuracy represents the average accuracy across both sessions, obtained by training linear SVM on the mapped source samples. Following Zanini et al. 2011)) Yair et al.) (2019), we present results for five subjects out of the available nine, as the remaining subjects ex exhibited poor performance and were considered invalid. We observe that GOT achieves the highest accuracy for three out of the five subjects, improving upon the baseline OT by 5%. Its overall average is comparable to MLOT, with a difference of less than 0.1%. Notably, when tuning with target labels, GOT achieves the best performance overall (Table 19b in the appendices). A practical advantage of GOT is that it can be integrated into almost any OT-based method. Exploiting this property, we also present results for MLOT combined with GOT (denoted by 'MLOT+GOT") for the cross-session task, which further improves performance.
Table 4b presents the cross-subject results, analogous to the binary classification results in Table 3a Same as in the cross-session lask, only the five valid subjects are considered. Here, GOT consistently achieves the highest accuracy across all subjects and yields an average accuracy that surpasses OT and OT-reg by more than 10%. It also outperforms all competing methods, with MIOT -the sccond-best method lower by 4%. Comprehensive results for all nine subjects and all methods, under both tuning strategies and both tasks, are provided in Appendix C.4.2 For completeness, Appendix C.4.2 also includes results for the multi-class
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Table 5: Multi-class classification accuracy (%) on the M12 BCI dataset, averaged over all source target experiments. Results are reported for two parameter tuning strategies: tuning based on source labels and tuning based on target labels.
Tuning stralegy
OT
OT-reg
MADAOT
MLOT
RMLOT
GOT (ours)
Source-label
40.99
41.97
39.07
44.73
35.90
47.18
Target-label
44.20
49.47
43.33
52.85
50.63
55.00
classification task obtained using single parameter configuration tuned on target labels across the entire dataset. Interestingly, these results are closer to those obtained with unsupervised tuning than to those from larget-tuning per source- target pair.
In Table 5, we report average results for multi-class classification on the MI2 dataset, which includes a total of 306 ccoss-ssssion and cross-subject experiments. The results show that GOT achieves the highest accuracy under both parameter luning strategies: source-label tuning (unsupervised) and larget-label luning. Across all methods, we observe clear gap between the two tuning strategies, indicating that selecting parameters solely based on source accuracy is suboptimal. We note that for methods with limited hyperparameters such as OT and RMLOT, the selected parameters often coincide with those that maximize source accuracy (as originally done in Kerdoncuff et al. (2020)) This effect is particularly evident for RMLOT, which exhibits a gap of nearly 15% between source- and target-based tuning. In Appendix C.4.2 we provide detailed results for both tuning stralegies and all methods.
6 Conclusions
In this work, we introduce a novel transportation cost for OT in DA that leverages diffusion geometry capture both intra-domain structures and inter-domain relationships. By analyzing the asymptotic behavior of the proposed diffusion operator S, we identified the key factors inftuencing the anisotropic diffusion process and demonstrated how it aligns with the goals of DA. Motivated by the manifold hypothesis - that high-dimensional data often lie on diverddimesional manifolds our approach exploits local neighborhood information to define a geometry-aware cost. Intuitively, such costs are better suited to capture meaningful relationships between samples than purely Euclidean distances, particularly in real-world domains. While there is no theoretical guarantee that GOT will always outperform traditional OT costs, our experimental results show that GOT consistently achieves superior performance across various UDA tasks, datasets, and frameworks. That said, GOT introduces three parameters related to the Gaussian kernel scale, and poor choices can lead to degraded performance. We therefore view GOT as a strong, robust, and broadly applicable choice for OTDA when accompanied by careful parameter tuning.
Limitations and future work. We acknowledge a fundamental limitation our method: since it incor- porates inter-domain relationships, operates under the assumption that both source and target domains reside in the same space. While this assumption is common to all DA methods, we believe that GOT has the potential to be extended to other applications, which view as directions for future work. In addition, the GOT cost is not distance metric, as it lacks symmetry and does not satisfy the triangle inequality, Consoquently, the OT solution does not define a valid distance (unlike the Wasserstein distance computed using the Euclidean cost). While this limitation did not affect the DA tasks we considered where only the transport plan is needed - adapting the cost to satisfy metric properties could greatly expand its util- ity. Finally, we note that recent. SOTA methods for DA, which we did not include in our comparisons, are basod on transformer architectures. While OT-basod methods, including ours, may not naturally fit into the transformer framework, we believe they remain valuable partieularly in scenarios where deep learning is impractical, or when explicit modeling of geometric relationships is important.
Acknowledgments
We thank the anonymous reviewers for their insightful feedback. This work was supported by the European Union's 's Horizon 2020 research and innovation programme under grant agreement No. 802735-FRC-DIFFOP..
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Supplementary Materials
A
Background on Diffusion Geometry
Let { be given set of data samples on a hidden manifold M with a metric Constructing a graph is a common strategy for approximating the manifold's structure (Roweis & Saul 2000 Tenenbaum et al.) 2000
Belkin & Niyogi 2003 Coifman Lafon 2006). By leveraging a local kernel, which quantifies pairwise local similarities between data samples, this graph captures local neighborhood relationships, facilitating effective representation of the underlying manifold geometry. Consider a weighted graph G =V, E) with vertex set V {.i,..., h where each node e V corresponds to the sample 2;. Let K E W be an affinity matrix, whose (it element encodes the weight of the edge between nodes and By and is given by:
( (1(611)5)) (1B(0102)))
(i,,j)=
(13)
where :)(:;)) a distance function induced by the metric 9, 6 > a scale hyperparameter, and is a positive function with exponential decay.
Next, the affinity matrix K is normalized to be row-stochastic:
K(i,))
P(i,j)
(14)
d(i)
where d(i) Σ ji. The matrix P is often viewed diffusion operator (Coifman & Tafon 2006
Pii,)) defines a transition probability from node to node j a Markov chain on the graph. Analogously, P can be viewed as the transition probabilities of diffusion process defined at the data samples {xa) on the manifold M (Coifman & Lafon 2006).
B Geometric Optimal Transport Additional Information
B.1
Computational Complexity of GOT
The most commonly used transport cost, the pairwise squared Euclidean distances, has a computational complexity of Onn"), assuming balanced datasets. In contrast, our proposed cost involves the following steps:
Computing pairwise distances for three matrices: O(n²).
Applying the exponential function to the matrices: Onn²).
Normalizing the matrices to be row-stochastic: O(n²).
Multiplying the three probability matrices: (n)3).
Applying the logarithm to the final diffusion operator: O(n²).
As noted in Section 5 often doubly-stochastic normalization instead of row-stochastic normalization, employing the Sinkhorn algorithm. Since the complexity of Sinkhorn is also O this does not increase the overall complexity. In total, the computational complexity of the proposed cost is Onr²). While higher than the traditional cost, our method avoids the need for a regularization term, which can often add to the optimization complexity.
Additionally, we note that up the fourth step, the computation involves three n matrices rather than one, resulting in higher memory requirements.
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Complexity comparison. While the standard Euclidean cost, which involves computing pairwise distances between source and target samples, has complexity O(n²), obtaining the transport plan depends on the specific optimal transport formulation. Solving the original OT problem requires a computational cost of O(n"), while using the Sinkhorn algorithm reduces the complexity to O OT-reg, used in Scctions 5.1 and 5.3 relies on Sinkhorn iterations for the optimization and, therefore, has complexity O(n²), whereas OT- OT-Laplace requires ((n)). In Figure 10a we report the empirical run-time for a toy example, comparing GOT with OT, OT-reg, and OT-Laplace. The results illustrate the distinction between complexity arising from computing the cost matrix and complexity arising from the optimization process (for example, due to added regularization terms). Notably, although both GOT and OT-Laplace have theoretical complexity of O OT-Laplace considerably slower in practice. GOT, which leverages the Sinkhorn algorithm, not only outperforms OT-Laplace in speed but also converges in far fewer ilerations than Sinkhorn with the standard Euclidean cost. As a result, the total run run-time of GOT can be faster than OT-reg, despite OT-reg having a lower thcoretical complexity of (nn)).
The competing methods ETD and RWOT introduce weighted Euclidean distance the transport cost and differ in how they learn the weights. Both methods are lea--leannnn--aseee and even when disregarding the complexity introduced by the neural networks, their computational cost remains at least O(n²) duc to the matrix multiplications between the learned weight matrix and the pairwise distance matrix. Our method, GOT, also has complexity ((n²) for the same reason matrix multiplication. However, unlike ETD and RWOT, GOT is not restricted to deep-learning frameworks, making more memory-eficcint. Moreover, as shown in Section 5.2. GOT achieves superior performance while maintaining comparable computational cost.
B.2 The Diffusion Process - Simulation
T
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04
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10
(a)
(b)
(c)
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Figure 3: (a) The PDF of the target domain, colored by PDF values. (b) Visualization of the vector field induced by the proposed diffusion operator. (c) and (d) show the PDF of the target domain projected to
2D, colored by the proposed diffusion process initiated at high-density and low-density regions, respectively. (Best viewed in colors).
In this section, we illustrate the behavior of the proposed diffusson operator, and more specifically, the expression outlined in Proposition 4.1. through a toy example. Consider the 2D space [o, 1]3, where the source domain / represented by uniform distribution over the entire space, and the target. domain is concentrated along 1D contour. Specifically, 20% of the target data. are sampled from a uniform distribution over [0.1]². For the remaining 80% of the target samples, the coordinates follows. The first dimension Fi is computed the sum of two variables sampled from uniform distributions: E U (7002, 0.98) and U2 E U[00.02 0.02), giving un + U2. The second dimension 22 is generated with 0.4 sin (2Tu,) ) + 0.5, where μn is the same variable used for x1. Figure 3a presents the target domain, istimeted using Kernel Density Estimation (KDE).
generated
To illusrrate the behavior of the proposed operator, we compute the diffusion operator S, which contains the probabilities of transporting from the uniform source samples to the target samples concentrated on the contour, presented in Equation 6 Figure 3b visualizes the resulting vector field the unit square [0, 1]2
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68-
16-
01
04.
161
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02-
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.71
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27g
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022
00 01
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1-1 26
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101
65
8.1 66 10
101
06
1.
X1]
.01
(b)
(c)
(d)
Figure 4: The 2D space colored by the diffusion induced by (a) the source operator Psy (b) the composed operator QP,, and (c) the final operator P.QP,. (d) shows the diffusion induced by applying only the cross operator Q, analogous to using the Euclidean cost. In the first row, diffussio starts from a source sample located in a high-density target region, and in the second row, it starts from a low-density target rogion.
induced by S. As expected, we observe that for all source samples, the average direction induced by the diffusion operator is toward the 1D contour. Figures 3c and 3d present two specific cases that illustrate how the anisotropic diffusion balances inter-domain and intra-domain relationships. In Figure 3cl the diffusion process is initiated at source sample located on the contour, a region of high target probability density, marked by the black star. As the source distribution uniform, by definition, the value of / (x) at this initial sample small, indicating a high probability that this source sample belongs to the target domain. Here, the diffusion assigns more weight the geometry of the target domain, spreading rapidly along the contour where the target density is highest. In contrast, Figure 3d shows the diffusson process starting from source sample located off the contour in region with sparser target distribution. Similar to the previous case, we observe an anisotropic diffusion directed toward regions with high probability density. However, due to the location of the starting point, the diffusion first draws it toward the denser region, utilizing inter-domain relations, and resulting in a noticeable asymmeln in the diffusion process.
For further intuition, Figure - illustrates the space after each diffusion step applied to the two points. Figure 4d shows the diffusion obtained by applying only the cross-domain operator Q, which equivalent to using the traditional cost. that. considers only inter-domain Euclidean distances. Since the source domain is uniform and source labels case. Importantly, it is only after applying P. that the diffusion sprcads into the dense regions of the target, thereby accounting for the geometry of the target domain.
not incorporated into the operator, the effect of Ps is not apparent in this
B.3 Illustration of Proposition 4.1
In Figure 5 we illustrate the analysis of Proposition 4.1 using a simple example. The source distribution Gaussian centered (2, 2), while the target mixture of two Gaussians centered at (-2, -1) and (2, -1). Figures 5a 5d present the source μ and target v, along with their normalized Laplacians, Ap/p and Av/v. respectively.
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0.225
5
0
-2
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(b)
(d) Av
(1) B Au
2 Au
(h) - log(S)
Figure 5: Illustration of Proposition 4.1 The source is Gaussian and the target v is a mixture of two Gaussians, shown in (a) and (b). Panels (c) and (d) display the normalized Laplacians, (e) and (f) show the
terms A and as defined in Equation 15 (g) shows the combined effect 2A B, and (h) shows the final diffusion cost. In (e), (g), and (h), the quantities enter the transport objective with a positive sign and are therefore minimized (blue regions are favored for transport), while in (f), the term enters with negative sign and is effectively maximized (red regions are favored).
For simplicity, we consider the constant function f = 1 and assume m2/mo- resulting in the simplified asymptotic expression:
S.1 = 5 (x) 1- E22 v
B)) (2A-c(22BB)))) +0()),
(15)
where A = A(//v) a/v
and B
Au + 2
Figure shows A, revealing a clear split along the y-axis: values are high on the left and low on the right. This asymmetry occurs because the source distribution concentrated on the right, while the target is dense both sides due to the symmetric Gaussians. Since A favors matching dense-to-dense and pearse-to-sparse regions, this implics that souree samples from the right side, where is dense, are transported toward the left, where v is also dense. Note that the source and target geometry is not visible when considering A alone, Figure 51 shows B, highlighting two Gaussian components: values are low at the centers and increase toward the boundaries. The right Gaussian has slightly lower values than the left, reflecting the overlap between the source and target distributions in that region. At the boundaries, where densities are sparse and vary rapidly, B reaches its maximum. Due the negative sign of B in the cost, samples with higher values mostly outside the Gaussians receive greater weight in the transport. Figure shows 2A - B, illustrating the combined effect. The two Gaussians reappear with higher overall values, with the left Gaussian being larger and stronger. Moving outward from the Gaussians, the values decrease, indicating that the transport maps source samples from the surrounding regions into the two target clusters, with a stronger emphasis on the left Gaussian. Finally, Figure 5h shows the final diffusion cost C = - log(S) for in Equation 15 with
1. Minimizing this cost. transports samples primarily from the top-right region the dense source area to the bottom region containing the two target Gaussians, focusing mostly on the left cluster.
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C Experimental Details
C.1 Hyperparameter selection GOT
C.1.1 The Gaussian scale
The Gaussian kernel is known to be highly sensitive to its scale parameter. When the scale is too large, the kernel fails to distinguish between samples, leading to loss of important information. Conversely, when the scale is too small, the kernel becomes overly sensitive to noise, which may result in overfitting. Moreover, the choice of scale is inherently dependent selecting the Gaussian kernel parameter. Moreover, determining the appropriate kernel scale is an open problem that remains an active area of research across various fields, beyond OT and domain adaptation. A common practice is to set it as the median of the pairwise distances in the dataset. We followed this approach consistently in all our experiments, as stated at the beginning of Section 5 In the deep experiments, we fixed the Gaussian scale to 6, = e, = 1, = 1 and applied only the median normalization. This choice proved robust in practice, though further tuning could potentially yield even better performance. For the remaining experiments, in addition to the median normalization, we tuned the parameters 6xy Ce.
the data scale. There is single optimal methodology for
To further evaluate the effect of the Gaussian scale, we conducted sensitivity analysis on the MI2 multi-class classification task. We fixed À = 0.01 and varied one of {Ea, Ec. ) at time while keeping the others fixed to 1, after applying median normalization. Figure T reports results for both cross-subject and cross-session adaptation, for all subjects and for the five valid subjects. The analysis confirms that c, and e are indeod sensitive to the choice of scale: both very small and very large values lead to performance degradation. By contrast, ét yields relatively stable results, which is expected given that the neighborhood structure is already constrained by the choice of three neighbors. Importantly, sctting all scales to 1 (the choice we adopted in the deep experiments) provides robust middle ground, consistently yielding reasonable accuracy even if not always optimal.
CS
CS
CS
CSe
C8e
80
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Accuracy (%)
CS
700
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(5)
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Aopuraay
Acqurany
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60
800
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Rc
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(c)
Figure 6: Ablation study the scale parameters (a) c, (b) Ee, and (c) e (log scale) in the BCI multi-class classification task. Results are shown for Cross-subject (CS), Cross-session (CSe), Cross-subject with 5 valid subjects (CS-5), and Cross-session with 5 valid subjects (CSe-5).
C.1.2 The number of neighbors
The neighborhood of the kernel controlled by the scale parameter For the source and target operators, we further refine the neighborhoods by either incorporating label information (when available) selecting fixed K = 3, and found this choice to be empirically
the K nearest neighbors. In all our experiments, robust. While tuning K could potentially improve performance, its influence was less pronounced compared to other parameters, so we opted to keep it fixed.
Figure 7 illustrates the impact of K on target accuracy in a set of BCI multi-class experiments. We fixed the scale parameters (c, 3, 0.1, = 0.1) and varied only the number of target neighbors. With joint parameter tuning, different values of K could potentially yield better results. Dataset details appear in Appendix C.4 and the multi-class classification setup described in Appendix C.4.2
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90
1E-7E
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Number of target neighbors
Figure 7: Effect of the number nearest neighbors on target accuracy for multi-class experiments on the MT2 BCT dataset.
C.1.3 Hyperparameter tuning
For the reproduced results of the competing methods, we followed the hyperparameter settings reported in their respective papers, with one exception DeePJDOT. The reported parameters for DeepJDOT led to poor results in our setting, so we re-tuned them ourselves.
Our method, GOT, requires tuning four parameters: the diffusion operator parameters 6s Ce: Ca and the entropy regularization weight (due to the use of the Sinkhorn algorithm).. While target labels were not used during training, we did rely on them as a validation set for hyperparameter tuning, consistent with competing methods. Importantly, we optimized parameters for each dataset as a whole, rather than for each individual source-target pair. For instance, the same set of parameters was used across all multi-class experiments on the BCI dataset (MI2), as presented in Appendix C.4.2 For reproducibility, the parameters used for each dataset and each experiment are detailed in this appendix in the respective sections. Additionally, in Section 5.3 and Appendix C.4.2 we present results obtained using unsupervised tuning approach. Notably, these results are very similar to those obtained using single set of parameters for the entire dalaset, tuned with target labels.
C.2 Two Moons: Visual Illustration
In this experiment, we analyze the resulting transportation plan and evaluate the performance of GOT compared to competing methods using the toy example from Courty et al. (2016). For the source domain, we generate two entangled moons, each containing 50 samples, with Gaussian noise added (standard deviation 0.05) to represent two distinct classes. The target domain is independently sampled from a rotated version of this distribution. the rotation angle increases, the adaptation problem becomes more challenging.
We compare GOT with the entropy-regularized OT problem (Courty al. 2016 Cuturi 2013), referred to OT, which serves the bascline method. Additionally, due to the limited number of methods proposing alternative transport costs particularly those that integrate source label informat ion and can be applied in
non-deep frameworks we include two regularization approaches: O'T-reg (Courty et al. 2016 2014p. which incorporates source label information, and OT-Laplace (Courty et al.) 2016 Flamary et al. 2014). which uses a Laplace regularization term to preserve data structure. Notably, all three methods use the squared Euclidean distance as the transportation cost. Table summarizes the key differences between GOT and these three competing methods. While the first two characteristies in the table are straightforward, the last two may require further clarification. OT-Laplace's optimization relies on simplifying the regularization term under the assumption of Euclidean geometry. For non-Euclidean geometries, the intra-domain distances, inter-domain distances, and baryeentric mapping become too complex to optimize, limiting it to Euclidean settings. While the squared Euclidean distance is commonly used as the transport cost in OT and OT- reg, both methods can handle non- Mon-Euclidean geometries, as demonstrated in Section 5.3 The specific optimization processes proposed for OT-reg and OT-Laplace restrict their applicability to other formulations or solvers. In contrast, our GOT, which is new transport cost rather than a regularization term, versatile and applicable any problem formulation solver, shown in Section 5.2
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Table Summary of key differences between GOT and competing methods.
OT OT-reg OT-Laplace
GOT (ours)
Intra-domain relationship integration
Label information utilization
Supports non-Euclice -nnEuuclidean geometry
Versatile OT compatibility
Source Target
Class 1
Class 2
x 0
X1
X1
X1
(a) GOT
(b) OT
(c) OTr-reg
(d) OTT-aaplace
Figure 8: Two moons illustration for 40 rotation angle. Source samples marked with circles, target samples with triangles, and the colors of the markers indicate the labels. Each line represents an entry 7(, in the transportation plan, where the j-th sample is marked by a bold black triangle. For clarity, we plot lines from this sample to the ten source samples with the highest values the j-th column of the plan. The line color corresponds to the label of the source sample 2;, and the line intensity reflects the values the plan. (Best viewed in colors).
To analyze the differences between the methods, we visualize the obtained plans for a 40 rotation angle, focusing two arbitrary target samples that demonstrate the typical advantages of GOT over the other methods. We obtain the optimal plan for GOT and the other three methods follows. For GOT, we first compute the proposed cost and then use the entropy-regularizeed OT problem (Cuturi 2013) to derive the transportation plan For the other methods, we calculate the standard transportation cost the squared Euclidean pairwise distances and then solve the OT problem using entropy rogularization, label regularization, and Laplace regularization to obtain the transportation plans for OT, OT-reg, and OT- Laplace, respectively.
Figure 8 presents the plan values for target sample located in the middle of the blue moon, marked by bold black triangle. For clarity, we plot lines from this target sample only to the ten source samples with the highest values in the corresponding column of the plan. In GOT (Figure Sap, we observe the desired optimal plan, where all lines connected to the chosen target sample originate from the same blue class, even though a red-labeled source sample closer. This outcome is due to incorporating label information into the diffusion process. OT (Figure Sb),, the transportation cost based solely on the pairwise Euclidean distances between source samples (circles) and target samples (squares). As a result, the blue-labeled target sample receives mass from the closest source samples, including those associated with the other red class. In OT-reg (Figure Scb. the regularization term promotes group sparsity, encouraging solution where each target sample receives mass only from source samples within the same class. As result, this method also provides the desired plan in this case. In OT-Laplace (Figure [d],, the core idea is that samples with small Euclidean distances before transportation should remain close after transportation, and vice versa. As a result, the bluc-labeled target sample receives mass not only from the nearby red-labeled source sample, but also from a more distant blue-labeled souree sample. We note that there are fewer than ten lines because this method does not include entropy regularization, resulting in sparser transportation plan.
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Cost analysis. To illustrate the connection between the diffusion process and the resulting transportation plan, we present in Figure 9a. the average direction of transport from each target point to the source domain, as induced by the diffusion operator S, for a 70 rotation angle. Although the standard OT cost lacks a probability notion, we present a comparable visualization in Figure 9b by applying the exponential function to the negative cost matrix and normalizing the resulting matrix be columm-stochastti. The diffusion induced by the cost aligns with the resulting plan values depicted Figures and For example, all target points within the blue dashed ellipse in Figure 9a are directed toward the correct red moon of the source. Conversely, in Figure 9b most target points are directed toward the closer but incorrect blue moon of the source. Similarly, all target points within the red dashed ellipse in Figure Dal including the chosen point shown in Figure []] are directed toward the correct blue moon of the source, even though closer source samples, labeled red, are present. It is worth noting that, while the presented directions correspond to the target samples, the GOT cost relies solely on source label information. In contrast, in Figure [9b] the same target points are directed incorrectly, with many directed toward the wrong red source samples. For clarity, only these areas are highlighted in the figures; however, additional regions, such as the bottom-left edge of the blue target moon and the center of the red target moon, could also be analyzed similarly, This example demonstrates how the proposed method effectively controls the resulting transportation plan by guiding the diffusion process toward the desired directions.
Source
Class
Target
Class
2
2
0
4
-2
-2
2
X1
X1
(a) GOT
(b) OT
Figure 9: Visualization of the vector field induced by the GOT cost and the standard OT cost, for 70 rotation angle. (Best viewed in colors).
Classification results. In addition to the visual illustration, we evaluate the performance of the methods across various rotation angles. Table presents the target classification error rates for these different angles. For the evaluation, follow the OTDA framework (Courty al. 2016). This involves computing the optimal plan using GOT or one of the competing methods, transporting the source samples to the target domain via barycentric mapping, and then training an SVM classifier with a Gaussian kernel on the source data. The classifier is subsequently tested on the target samples. The classification results are consistent with the previous analysis. We observe that GOT the only method that achieves 100% accuracy up to a 60 rotation angle. Furthermore, it evident that GOT outperforms all other methods with significantly higher accuracy across all tested angles.
Computational time comparison. We compare the run-time of the methods, measuring both the time to compute the cost matrix C (Euclidean or GOT) and the time to obtain the transport plan under cach optimization approach. Figure 10a reports the average results over 100 runs for varying number of samples. As described Section B.1 computing the GOT cost has complexity O(z3), whereas computing the standard Euclidean cost has complexity O(??). Regarding the optimization process, OT and GOT are solved using the efficient Sinkhorn algorithm with complexity O(?)). OT-reg solved via an iterative procedure that calls Sinkhorn, with complexity O(kn?), where l is the number of outer iterations. OT-Laplace is solved
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Table 7: Two moons data. Target error rates for different rotation angles, obtained by training an SVM classifier with a Gaussian kernel on the transported source data.
rotation angle
GOT (ours)
OT
OT-reg
OT-Laplace
10a
0
0
0
20
0
0
0
0
30
0
0
0
0.01
40
0
0.06
0.09
0.14
50
0
0.14
0.14
0.16
60
0
0.18
0.2
0.22
70
0.07
0.23
0.3
0.31
80
0.23
0.34
0.34
0.34
90"
0.33
0.37
0.39
0.42
using gradient-based algorithm, with complexity O(n²). As expected, OT is the fastest method in practice. OT-Laplace, due to its expensive optimization, is substantially slower than the other methods. Interestingly, OT-reg is slightly slower than GOT, despite using the Euclidean cost. This can be explained by the number of fterations: as shown in Figure Tob OT-reg always runs the maximum nunber of iterations (200 inner Sinkhorn iterations and 10 outer iterations the default in the POT package (Flamary et al. 2021)), whereas Sinkhorn with the GOT cost converges after small number of iterations (far fewer iterations than OT which runs Sinkhorn with the Euclidean cost). OT-Laplace also typically reaches its default maximum of 101 iterations. These results suggest that although computing the GOT cost is more expensive than the Euclidean cost, its compatibility with a wide range of O'T solvers and its ability to accelerate convergence could make it empirically faster in practice, even compared to methods with lower theoretical complexity.
2000
OT
Number of iterations
1750
4000
OT-reg
OT-reg
Run-time (S)
OT-Laplace
1000
OT-Laplace
0000
GOT
GOT
750
4000
N
(b)
Figure 10: Comparison of (a) mu-time and (b) iteration count over 100 runs on the two moons dataset with a 40 rotation, evaluated over varying numbers of samples.
C.3 Deep Domain Adaptation
In the following section, outline the framework utilized in the experiments presented in Section 5.2 For these experiments, we adopt the framework introduced by Damodaran et al. (0118). This framework, termed DeepJDOT, comprises two components: an embedding function, denoted by g: X + 2, which aims to learn features optimizing both classification accuracy and domain invariance, and a classification function, denoted by f: + }/, which predicts the labels based on the learned features.
The Dee objective, which operates on minibatch of size m, is given by:
min
f + (mill(==)) 2(2))11 me
(16)
where L is the cross-entropy loss, 7n and 7/2 hyperparameters, and the optimal plan.
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The training process follows an alternating optimization scheme. Initially, the parameters of both the feature extractor g and the classifier are fixed. The fixed-paramete models are denoted as f and g. Subsequently, the OT problem is solved with the associated cost matrix Ci. = m llā(=²)) 0 1 + + f((x))) Finally, the optimization procoeds by fixing the obtained plan, Y, and optimizing the model on the mini- batch using Stochastic Gradient Descent (SGD). The objective function for this training phase defined by Equation 16
In our experiments, propose to derive the plan using the GOT cost. We solve the OT problem with the associated cost Ci. -mn log( log(S..) m2f(y), f(o(z))), where:
S = E E P.(CA)))((
(17)
uu LEN)
The probability matrices P., Q and Pt, are obtained by applying row-stochastic normalization to the kernels defined in Equations 5 and 11] However, in this case, the kernels are applied in the latent space, using the source and target representations {0(x;));- and {0(z)) where m denotes the batch size. For the source and larget kernels, we compute the kernels by employing the Cosine distance, while for the cross-domain we use the Euclidean distance. .N denotes the neighborhood of zJ,, determined by the three nearest neighbors in the features space. Cx, Ce and C are scale hyper- pappe-parameters.. In all deep experiments, we set Eg - Ec = E, 1.
We note that for simplicity in handling derivatives, we omit the use of GOT in the second phase of training, and utilize Equation 16 as is, although integrating the GOT cost remains a viable option.
Once the plan is obtained, we fix y and train f and g using Equation 16
Baselines and competing methods. In addition to DeepJDOT (Damodaran et al. 2018 which utilizes the framework with the standard transport cost, we present comparisons to more recent methods that leverage the Dee framework. However, unlike DeepJDOT, which employs the network simplex flow algorithm, these approaches utilize different OT problem formulations: Unbalanced OT (UOT), referred to as JUMBOT Fatras et al. 2021 and Partial OT (POT), termed as m-POT (Nguyen et al., 2022a),, to derive the optimal plan. Notably, all three methods use the squared Euclidean distance as the transportation cost and differ solely in their approach to obtaining 7. We also compare our approach to two other OT- based deep models, which propose to employ a different transportation cost function from the conventional squared Euclidean distance, namely ETD (Li el. al. 2020) and RWOT (Xu et al.] 2020). In addition to OT--based methods, we benchmark our method against established DA baselines, including DANN (Ganin et al. 2016) and ALDA (Chen et al. 2020). For DeepJDOT (Damodaran et al., 2018)) JUMBOT (Fatras al., 2022a),, all employing the Deep.DDOT framework with different OT problem formulations, we reproduce the results ourselves using the POT (Python Optimal Transport) package (Flamary et al. 2021), 1 an open-source Python library. Results for DANN (Ganin et al., 2016) and
et al. 2021,, and m-POT (Nguyen
ALDA (Chen et al. 2020) are sourced from Nguyen et al. (2022a). For the remaining methods, we rely on the reported results from their respective papers.
Datasets. The Digits dataset comprises three widely-used digit datasets, serving as the different domains: (i) MNIST (28x28 grayscale images of handwritten digits), (ii) USPS (16x16 grayscale images of digits derived from scanned handwritten letters), and (iii) SVHN (32x32 colored images of digits with diverse backgrounds and fonts collected from Google Street View). All datascts consist of 10 classes, corresponding to the digits O through 9. The Office-Home dataset is composed of images from four distinct domains, each representing a specific visual style and content distribution. The four domains are Art (A), Clipart (C), Product (P), and Real- World (R), all with 65 classes. The VisDA-2017 dataset comprises synthetic and real-world images across 12 diverse classes and often serves as a benchmark for domain adaptation in computer vision. In our experimental setup, we designate the training set containing synthetic images as the source domain, while the validation set, consisting of real images, serves as the target domain.
C.3.1 Digits
For the digits experiments, we adopt the architecture proposed by Damodaran et al. (0118). The generator 9 is a CNN model trained from scratch, with convolutional layers containing 32, 32, 64, 64, 128, and 128
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filters, followed by fully connected (FC) layer of 128 hidden units and Sigmoid activation function. Layer normalization is applied after each convolution layer. The classifier f is FC layer with 10 units, corresponding to the number of classes. We maintain a batch size of m = 500 for both the source and target datasots. We construct balanced batches for the source training set by utilizing the labels, ensuring an equal number of samples for each class. The models are optimized using the Adam optimizer. Initially, the training is performed on the source data using cross-entropy loss for 10 epochs. Subsequently, the models are trained for 100 epochs using both the source and target data, with the objective as described above.
For the proposed method, we the objective function, = 1 for all the Gaussian kernels constructing our diffusion operator, and À = 0.02, T= 1 for the unbalanced Sinkhorn problem. For the reproduced methods we use the parameters reported in the original papers. For all reproduced methods, we use a learning rate Ir = 0.0002. For Deep.IDOT, we use ?/1 = 0.001, 1/2 0.0001. For JUMBOT, we use 71 = 0.1, 122 = 0.1 for the objective function, and A 0.1, т 1 for the unbalanced Sinkhorn problem. We note that since we struggled to reproduce the m-POT results for this specific dataset, Table Ta presents the m-POT results reported in the original paper [ggyyen et al. 2022a We ran each method 3 times, Table 8 presents the results for each run.
learning rate Ir 0.001. We set the hyperparameters vn 1, 72 for
Table 8: Accuracy for Digits dataset over three seeds for (a) DeepJDOT, (b) JUMBOT, and (c) JUMBOT + GOT (ours).
(a)
(b)
(c)
Run
S-M
U-M
M-U
Run
S-M
U-M
M-U
Run
S-M
U-M
M-U
1
95.81
96.96
86.05
99.05
98.64
96.66
1
99.15
98.90
97.81
2
95.53
97.34
86.75
98.99
98.62
97.41
2
99.23
98.87
98.11
3
96.78
97.36
85.55
98.93
98.77
96.71
3
99.19
98.84
97.51
C.3.2
Office-Home
For this dataset, the generator g is pre-trained ResNet50 with the final FC layer removed, and the classifier f consists of an FC layer. We construct balanced batches for the source training set by utilizing the labels, ensuring equal number of samples for each class. The models are optimized using the SGD optimizer with a learning rate of 0.001. Given that our cost incorporates source neighborhood information based on source labels, it is required for each mini-batch to contain at least two samples from every class. Notably, this dataset comprises 65 classes. Thus, we employ a batch size of m = 195, and train the model for 4000 iterations. For all reproduced methods, we follow the original papers by using a batch size of m = 65 and training the model for 10000 iterations.
For the proposed method, we set the hyperparameters 7) 0.01, 7)2 2. for the objective function, c for all the Gaussian kernels constructing our diffusion operator, and À 0.02, 0.5 for the unbalanced Sinkhorn problem. For the reproduced methods, we use the parameters reported in the original papers. For DeepJDOT, we set 77 = 0.01 and = 0.05 for the objective function. For JUMBOT and TS-POT, we use ! = 0.01, 72 = 0.5 for the objective function. For JUMBOT, we use X = 0.01, T = 0.5 for the unbalanced Sinkhorn problem. For TS-POT, we set the fraction of mass to 0.6, and the number of mini-batches to 2.
C.3.3 VisDA
Similar to the approach taken with the Office-Home dataset, we utilize pre-trained ResNet50 the generator g, with the classifier f implemented an FC layer. For all methods, we use a batch size m 72. We construct balanced batches for the source training set by utilizing the labels, ensuring an equal number of samples for each class. The models are optimized using the SGD optimizer with a learning rate of 0.0005, and trained for 10000 iterations. We employ the two-stage (TS) implementation proposed by Nguyen et al. (2022a) for our method and for TS-POT. In this implementation, the OT problem initially solved using a batch size of & m on the CPU, leveraging its ability handle larger matrices. After obtaining the optimal plan, the gradient step is executed the GPU with batch size of m. To itilize the large plan in loss
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Table 9: Accuracy for SfficeHHomt dataset over three seeds for (a) DeepJDOT (b) JUMBOT (c) TS-POT
and (d) JUMBOT + GOT (ours).
Run
A-C
A-P
A-R
C-A
C-P
C-R
P-A
P-C
P-R
R-A
R-C
R-P
Avg
51.8
70.7
76.4
59.7
66.9
69.1
57.6
49.1
75.3
69.4
54.6
80.1
65.07
2
52.2
71.1
76.2
60.8
67.2
69.5
58.8
48.4
75.4
68.7
54.8
79.8
65.22
3
51.9
70.9
75.8
60.9
65.9
68.9
58.8
48.7
75.2
68.6
55.3
79.8
65.06
(b)
Run
A-C
A-P
A-R
C-A
C-P
C-R
P-A
P-C
P-R
R-A
R-C
R-P
Avg
55.7
75.0
80.7
65.3
74.3
75.1
65.3
53.1
79.5
74.6
59.3
83.8
70.16
2
56.0
74.6
80.7
64.4
74.3
74.9
65.5
53.4
79.6
74.5
59.4
83.9
70.11
3
55.5
75.4
80.6
65.5
74.9
75.1
65.2
53.3
79.7
74.4
59.2
84.0
70.23
(c)
Run
A-C
A-P
A-R
C-A
C-P
C-R
P-A
P-C
P-R
R-A
R-C
R-P
Avg
57.7
76.3
81.7
68.4
73.7
76.9
67.3
55.2
80.6
75.4
60.2
83.7
71.43
2
58.4
77.4
814
67.9
72.5
76.2
67.4
54.8
80.7
75.4
59.1
84.2
71.29
3
56.3
77.5
81.7
68.7
72.2
76.5
67.7
55.3
80.4
75.5
60.4
84.1
71.36
(d)
Run
A-C
A-P
A-R
C-A
C-P
C-R
P-A
P-C
P-R
R-A
R-C
R-P
Avg
1
57.0
77.2
82.4
69.8
74.8
79.4
68.0
56.7
82.2
75.3
60.5
84.6
72.33
2
57.0
78.3
82.0
70.3
74.9
78.4
68.2
56.4
82.0
75.7
61.2
84.5
72.40
3
57.6
78.4
81.8
70.6
74.9
78.7
68.2
56.3
82.0
75.8
61.0
85.2
72.55
functions computed on smaller batch sizes, an adaptation the original loss function proposed. For more details, see Nguyen et al. (2022a)
for For the proposed method, we set the hyperparameters m = 0.01, 7/2 1 for the objective function, all the Gaussian kernels constructing our diffusion operator. For the TS implementation, we set s = 0.5 as the fraction of mass for the partial OT problem, and l = 2 as the number mini-batches. For the reproducod methods, we use the parameters reported the original papers. For DeepJDOT, we use m = 0.005 and 1/2 0.1 for the objective function. For JUMBOT and TS-POT, we use 7 0.005 and 7)2 = for the objective function. For JUMBOT, we use À 0.01, TS-POT, set the fraction of mass to 0.75, and & 1 for the TS implementation. In Tables 10 we present the results per class for GOT and the competing methods.
0.3 for the unbalanced Sinkhorn problem. For
Features Analysis. Additionally, we analyze the t-SNE representation of the VisDA target features ob- tained from the deep model trained with GOT, compared to those learned using TS-POT (Nguyen et al. 2022a), the baseline for this dataset. For better visualization, we randomly sampled 10% of the samples from cach class before applying t-SNE.
While the deep model is designed to learn representations that are both domain-invarian and optimized for source classification accuracy, resulting in well-separated classes for both methods (as shown in the figure), the visualization still highlights differences between the methods. These differences help explain why GOT achieves better larget accuracy compared to the standard transportation cost. For example, in Figure Ia which shows the TS-POT features, the 'knife' class (colored magenta and circled by a red dashed line) completely overlaps with the 'skatcboard' class (colored black). In contrast. Figure 11b which presents features obtained using GOT, shows that these classes are well-separated. Additionally, the TS-POT model appears to have learned representation with more than the expected 12 clusters. For instance, the cluster
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Table 10: Accuracy for VisDA dataset per class over three seeds for (a) DeepJDOT; (b) JUMBOT, (c)
TS-POT, and (d) TS-POT + GOT (ours).
Run Avg
plane bicycle bus car horse knife mcycle person plant sktbrd train truck Avgelass
69.66
88.6
59.2
70.1 67.2 87.6
4.0
90.4
68.3
93.5
49.6
88.4
31.2
66.5
69.88
91.3
57.8
69.0
72.5
84.5
1.9
89.9
61.1
91.9
64.2
84.9
30.5
66.6
69.21
87.8
57.1
70.9
69.0
87.2
3.2
91.7
70.2
92.3
46.8
87.5
25.2
65.7
(b)
Run Avg
plane bicycle
bus
car
horse knife meyyll person plant sktbrd train truck Avgelass
72.84
93.0
53.4
76.9
76.6
89.8
2.1
94.5
76.2
94.4
71.2
89.7
18.4
69.7
72.81
93.7
56.9
75.5
71.8
90.5
3.5
94.2
77.6
96.1
60.6
89.0
27.4
69.7
73.27
89.7
57.0
74.8
77.3
90.6
1.5
93.9
74.6
94.4
69.9
90.3
24.6
69.9
(c)
Run
Avg
plane bicycle
bus
car
horse knife meycle person plant sktbrd train truck Avgelase
75.08
94.4
66.6
78.8 73.2
93.4
1.9
95.2
73.9
94.8
79.8
88.4
31.6
72.7
2
76.54
95.1
61.4
82.1
75.2
91.9
67.1
94.8
79.2
96.2
80.0
89.8
13.7
77.2
3
75.32
93.1
68.8
81.6
74.7
93.8
2.1
94.4
75.1
95.9
76.7
90.8
26.4
72.8
(d)
Run
Avg
plane bicycle
bus car horse knife meycle person plant sktbrd train truck Avgelass
78.48
95.0
58.6
82.3 76.9
95.5
67.6
95.1
79.2
95.3
84.2
89.7
26.9
78.9
2
78.74
74
95.0
64.3
85.2
77.2
95.5
68.7
94.0
78.6
94.7
82.8
88.9
25.7
79.2
3
78.47
94.8
61.7
85.5
75.9
95.7
70.8
94.1
79.3
95.2
82.8
87.8
25.9
79.1
circled by a gold dashed line in Figure Ia does does not achieve perfect separation between classes (as expected, given the approximately 78.5% target accuracy), the label associated with each cluster is easily identifiable.
does not correspond to any specific class. In contrast, while GOT
plane
plane
100
bicycle
75
bicycle
bus
50
bus
50
horse
horse
knife
knife
mcycle
-25
mcycle
-25
person
person
plant
plant
-75
sktbrd
-75
sktbrd
train
-100
train
-100
-100
-30
60
100
truck
-100
-30
50
100
truck
X1
X1
(a)
(b)
Figure 11: t-SNE visualization of VisDA features, learned from the TS-POT model using (a) the Fuclidean cost and (b) GOT (ours).
ViT-Based Domain Adaptation. Vision Transformers (VITs) have gained significant attention in recent years due to their ability to process visual data in manner similar to natural language processing tasks. Unlike traditional convolutional neural networks (CNNs), which use localized convolutions to capture spatial
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hierarchies, ViTs leverage self-attention mechanisms to capture long-range dependencies between pixels across the entire image. This characteristic enables ViTs to effectively model global features, making them highly suitable for tasks such as domain adaptation.
In this experiment, we integrate the GOT cost into the state-of-thh-art ViT-based framework, FFTAT Yu et al.] 2024), to enhance the model's domain adaptation capabilities. The GOT cost is incorporated by adding it as an additional term the FFTAT loss function, described by the second term in Equation I6 The cost is applied to the global representations learned by the ViT backbone, which are then used for classification.
Following FFTAT, the backbone is pre-trained on ImageNet. For the reproduced FFTAT results, we use the parameters reported in the original paper. In our experiment (FFTAT + GOT), we use balanced source batches with a batch size of m = 36 and a learning rate of 0.07. The parameters for the OT-based loss are set to 171 = 0.01 and 1/2 = 1, while the FFTAT loss uses the default parameters specified in the paper. We apply the UOT formulation to derive the transportation plan, with X 0.005 and T 0.5. As in the DeepJDOT framework, set
for all Gaussian kernels used to construct diffusion operator. Additional details
about the FFTAT framework can be found in the original paper YYu et al. 2024).
For comparison, we include the latest ViT-based methods: TVT Yang et al. 2023 PMTrans (Zhu et al. 2023), CDTrans (Xu et al. 2021), and MTC (Hoyer et al. 2023). Table 11 presents the average target accuracy per class for the VisDA dataset. We observe that incorporating the GOT cost leads to a performance boost, increasing the accuracy by more than 1.% compared to FFTAT. also achieves a 0.36% increase over the baseline method, FFTAT UOT.
Table 11: Per-class accuracy comparison of ViT-based methods on the VisDA dataset.
Method
plane
bicycle
bus
car
horse
knife
mcycle person
plant
sktbrd
train
truck
Avg
TVT
97.1
88.8
86.4
64.4
96.4
97.4
90.6
64.1
92.0
90.3
93.7
59.6
85.1 1
PMTrrnn
99.4
88.3
88.1
78.9
98.8
98.3
95.8
70.3
94.6
98.3
96.3
48.5
88.0
CDTrans
97.1
90.5
82.4
77.5
96.6
96.1
93.6
88.6
97.9
86.9
90.3
62.8
88.4
MIC
99.0
93.3
86.5
87.6
98.9
99.0
97.2
89.8
98.9
98.9
96.5
68.0
92.8
FFTAT
99.78
97.41
93.73
80.4
99.59
99.13
98.05
87.68
98.07
99.69
99.32
75.74
94.05
FFTAT UOT 99.95
98.99
93.69
87.29
100.0
99.52
99.17
89.4
98.42
99.82
99.6
71.45
94.77
FFTAT + GOT 100.0
99.14
94.78
87.67
100.0
99.66
99.15
89.35
98.41
99.87
99.67
73.88
95.13
C.4 Domain Adaptation for non-Euclidean Data
In the following section, outline the framework utilized in the experiments presented in Section 5.3
Previous studies (Barachant et al. 2011 2013 Zanini et al. 2017 Rodrigues et al. 2018 have demonstrated the efficacy of using the empirical covariance matrices the EEG recordings as an informative feature representation for this type of data. Following this established approach, we adhere to the framework introduced by Yair et al. (2019), which solves the DA problem using OT on the (hon-uucli Riemannian manifold of Symmetric Positive Definite (SPD) matrices.
Consider the source data samples and labels as ((X))))) and the target data samples ( N. Each data sample X; is a covariance matrix, which is an SPD matrix that lies on the Riemannian manifold of SPD matrices M C Rddxd where d is the number of EEG channels. Applying Algorithm with the Log- Euclldlid metric (Arsigny et al. 2006), we derive the GOT cost. Next, we solve the OT problem using the Sinkhorn algorithm (Cuturi 2013). as described
Section 3 Subsequently, leveraging the obtained optimal plan 7: we map the source features into the target domain. This mapping can be computed using the barycentric mapping (Courty al. 2016 Perrot et al. 2016), as described in Section
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When utilizing the Log- Euclidean metric to compute the cost function, the barycenter can be represented as the weighted Riemannian mean, expressed by the following closed-form:
N Σ
Yi log (x:) (x:)).
z = exp
(18)
where exp(-) and log(-) are the matrix exponential and logarithm. The framework is summarized in Alg.
Algorithm 2 GOT on the Riemannian manifold of SPD matrices
Input: (()) v [X و=رر f fo,
Compute the GOT cost C by applying Algorithm | to the source and target matrices.
Obtain the optimal plan y by applying the Sinkhorn algorithm.
Compute the barycentric mappings of the source samples using equation 18
4: Return the source mappings.
Since the representations, (2 No and (X,) 5= |= are covariance matrices that lie on the Riemannian manifold of SPD matrices M, prior to conducting any linear computation, such as training a linear classifier, we project both the source and target covariance matrices onto the tangent space TMM, using the logarithmic map:
LogM (Z.) log log(M),
(19)
where M represents the Riemannian mean of all the covariance matrices.
Datasets. The MII dataset (Blankertz et al. 2007] contains EEG recordings from 7 subjects utilizing 59 electrodes. Subjects were instructed to perform two out of three MI tasks: imagining the movement of the left hand, the right hand, or the foot. Our evaluation focuses exclusively on the calibration data, comprising 100 trials for each subject. EEG signals were recorded at 1000 Hz, bandpass- filtered between 0.05 Hz and 200 Hz, and down-sampled from 9 subjects using 22 electrodes. The experiments include four MI tasks: the imagination of moving the left hand (class 1), right hand (class 2), both feet (class 3), and longue (class 4). The recordings in MI2 of each subject contain two sessions, recorded on separate days, with 72 trials conducted for each MI task in each session, resulting in 288 trials per session. EEG signals were sampled at 250 Hz and bandpass-filtered between 0.5 Hz and 100 Hz. In both datasets, subjects were presented with a cue at each trial, instructing them to imagine the corresponding MT task.
100 Hz. The MI2 dataset (Tangermann et al. 2012) consists of EEG recordings
Pre- Pre-processing. We apply the same pre-processing pipeline proposed in Zhang & Wu (020)) to both datasets. Initially, we extract 3-second time window, spanning from t = 0.5s to t 3.5s following the cue. Next, bandpass (BP) filter ranging from 8 Hz to 30 Hz is applied to the data. Following this, we compute the empirical covariance matrix of each trial. Finally, centroid alignment (CA) ŞZanini et al. 2017 Zhang & Wu 2020 is applied lo each set of covariance matrices, representing a domain, according to:
x (k) = (M (()) (M(*)) -
(20)
where x(()) denotes the empirical covariance of trial i of the k-th subject, and ((k) represents the Riemannian mean of the collection of covariance matrices of the k-th subject. The pre-processod set of cach subject is denoted by (x(*) ( Ji= where N is the total number of trials, possibly with the set of corresponding class
labels 1.
C.4.1 Binary classification
Single-source domain adaptation. In the cross-subject experiment, we apply Algorithm 2 where each subject, in turn, serves as the target domain, with the remaining subjects alternately acting as the source domain. Accuracy is computed by initially mapping the obtained source and target representations onto a fangent space, following Equation 19 and subsequently training a linear SVM classifier on the transformed
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source mappings and evaluating it the target mappings. The presented results showcase the average target accuracy across source subjects.
For the cross-subject task, we employed the Sinkhorn algorithm with entropy regularization À = 0.02. For the first dataset, MII, we used the following scale parameters for the probability kernels to produce the results shown in Table 3a 6 0.4, fc = 1, and et 1. For the second dataset, MT2, the scale parameters for the probability kernels were set as follows: e, 0.9, e. 2, and E,
In Tables 12 and Zhaang& & Wu 2020
present the accuracy per subject on this task, including comparison with MEKT utilizing their provided source code for our analysis.
Table 12: Binary classification accuracy for the cross-subject task on MIl, presented per subject.
1
2
3
4
5
6
7
Avg
STD
MEKT-R
71.58
65.58
64.25
65.58
83.33
64.67
81.92
70.99
8.33
GOT 70.92
63.5
63.25
69.75
86.33
72.67
79.25
72.24
8.3
Difference
-0.66
-2.08
-1.0
4.17
3.0
8.0
-2.67
1.25
3.92
Table 13: Binary classification accuracy for the cross-subject task on MT2, presented per subject.
2
3
4
5
6
7
9
Avg
STD
MEKT-R
74.57
50.69
84.55
64.06
53.39
64.06
59.38
86.55
74.31
67.95
12.84
GOT
75.52
53.39
83.94
67.27
57.55
65.02
63.8
91.06
76.13
70.41
12.28
Difference 0.95
2.7
-0.61
3.21
4.16
0.96
4.42
4.51
1.82
2.46
1.8
Multi-source domain adaptation. For the leave-one-out task, we utilize the Wasserstein Barycenter Transport (WBT) framework, proposed in Montesuma & Mboula (2021). This method involves transporting all source samples to an intermediate domain, termed Wasserstein Barycenter Transport (WBT), before subsequently transferring the source samples from WBT to the target domaín using standard OTDA. For the transportation of the source samples to the WBT only, utilized the label-enhanced kernel for both the source domain and the WBT, leveraging the available labels. For simplicity, we initially map both the source and target covariance matrices onto a tangent space, using Equation [9] and then apply the WBT algorithm with the Euclidean metric. Notably, we utilize GOT instead of the standard cost for all OT applications within the WBT framework, including transportation from each source domain to WBT and from the Wasserstein barycenter of sources to the target domain. The results for these experiments appear in Table 3D
For both datasets, we set X = 0.02 for transporting all the source domains to the WBT and X 0.1 for transporting samples from the WBT to the target domain. In the latter case, we apply max normalization to the cost matrix before using the Sinkhorn algorithm. For the first dataset, MI1, the scale parameters for the probability kernels used to transport all source domains to the WBT are èi = 0.05, e = 0.3, and ei = 0.05. For the transportation from the WBT to the target domain, the parameters are 6, 0.05, 6 = 0.02, and e, 0.03. For the second dataset, M12, the scale parameters for the probability kernels used to transport all source domains to the WBT are ex 0.01, ei 0.3, and ei 0.01. For the transportation from the WBT to the target domain, the parameters e, 0.3, e, 0.02, and E, 0.02.
In Tables 14 and 15 we present the accuracy per subject on this task, including a comparison with MEKT (Zhang & Wu 2020
C.4.2 Multi-class classification
In this section, we assess the performance of GOT in a multi-class classification task using the MI2 dataset, taking into account all four classes available in the dataset. Tn addition to the baseline methods, OT and
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Table 14: Binary classification accuracy for the leave-one-out task on MIl, presented per subject.
2
3
4
5
6
7
Avg
STD
MEKT-R
86.5
69.5
73.5
89.0
94.0
89.5
91.5
84.79
9.43
GOT
90.0
78.0
75.0
92.0
96.0
87.0
93.0
87.29
7.91
Difference
3.5
8.5
1.5
3.0
2.0
-2.5
1.5
2.5
3.28
Table 15: Binary classification accuracy for the leave-one-out task on M12, presented per subject.
2
3
4
7
Avg
STD
MEKT-R
93.06
49.31
95.83
73.61
56.25
70.83
70.14
94.44
83.33
76.31
16.76
GOT
91.67
51.39
97.92
75.0
61.81
70.14
68.06
96.53
82.64
77.24
16.14
Difference
-1.39
2.08
2.09
1.39
5.56
-0.69
-2.08
2.09
-0.69
0.93
2.38
OT-reg (Courty al. 2016
2014
Yair et al. 2019 we compare GOT to standard DA methods, including
CORAL (Snn et al. 2017p, SA (Fernando et al. 2013 and TCA (Pan et al. 2010).
For all the multi-class classification experiments we employ the Sinkhorn algorithm with entropy regulariza- tion parameter À = 0.1 and apply Algorithm D with C, = c, = 0.1, < = 0.1. For the reproduced methods, OT and OT-reg, we apply median normalization to the cost matrix before applying the Sinkhorn algorithm. We set A 0.005 for the entropy parameter. For OT-reg, we use r 10 for the label regularization, as suggested in the original paper (Yair et al. 2019) For CORAL, SA, and TCA, we utilize the ADAPT package de Mathelin et al. 2021) with the default parameters.
Table 16a shows the results for the cross-session task. In this experiment, for each subject, the first session (day) serves as the source domain, while the second session serves as the target domain, and vice versa. The reported target accuracy represents the average accuracy across both sessions, obtained as follows: First, we apply Algorithm 2 to obtain the transported source samples. Next, we map both the target and the transported source covariance matrices to tangent plane, as described in Equation [9] We then train a linear SVM on the mapped source samples and test it on the target samples. Following Zanini al. (2017)
Yair et al. 2019),, present results for five subjects out of the available nine, as the remaining subjects exhibited poor results and were considered invalid in those works. We observe that the proposed method achieves the highest accuracy for three out of the five subjects. For subject 1, however, it shows lower accuracy compared to the baseline OT-reg. This is the only instance across all experiments where we do not achieve an improvement over the baseline. Notably, GOT yields the highest average accuracy, surpassing OT and OT-reg by over 6% and 2%, respectively. Detailed results for all subjects appear in Table 17a.
Table 16: Multi-class classification accuracy on dataset MI2 (five valid subjects). (a) Results for the cross- session task. (b) Results for the cross-subject task.
(a)
(b)
Method
3
7
9
Avg
Method
1
8
Avg
CORAL
82.3
87.7
80.9
83.0
72.9
81.35
CORAL
53.1
58.2
52.0
52.5
53.3
53.82
SA
83.7
87.8
81.4
83.9
74.8
82.33
SA
58.6
63.2
53.8
59.3
58.2
58.63
TCA
71.0
78.1
64.6
77.8
70.1
72.33
TCA
59.1
61.2
49.4
60.7
52.7
56.64
OT
84.0
85.9
81.1
76.4
68.2
79.13
OT
57.2
65.4
59.7
57.5
52.7
58.51
OT-reg
85.2
84.5
81.1
85.4
78.1
82.88
OT-reg
63.2
69.7
63.8
66.1
62.1
64.97
Ours
84.7
86.5
84.5
88.9
81.4
85.21
Ours
66.7
75.1
69.4
73.4
65.5
70.03
In Table 16b
we present the results for the cross-subject task, analogous to the results shown in Table for the binary classification case. Same in the cross-session task, only the five valid subjects are considered.
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Our method achieves the highest aceuracy across all subjects and demonstrates an average performance that significantly exceeds both baseline methods. Detailed results for all nine subjects can be found in Table 17b
Table 17: Comparison of (a) cross-scssion and (b) cross-subject multi-class classification on the MI2 datasct.
(a)
(b)
Subject CORAL
SA
TCA
OT
OT-reg GOT
Subject CORAL
SA TCA OT
OT-reg
GOT
82.29
83.68
71.01 84.03
85.24
84.72
1
43.66
49.57 51.91 49.1
54.24
57.99
50.35
51.91
44.27 56.08
57.99
58.85
2
27.19
27.82 27.58 27.79
28.23
27.44
87.67
87.85
78.13
85.94
84.55
86.46
3
47.34
53.34
50.59 52.89
57.7
62.08
57.47
59.9
37.5
58.51
57.81
60.24
4
35.31
34.71
32.82 37.7
38.85
39.29
5
41.32
41.67
28.47
46.7
45.14
44.27
5
30.96
30.57
29.42 32.79
33.26
33.51
6
47.4
50.87
36.81
52.95
50.69
48.61
6
33.18
32.54 32.08 34.13
34.09
34.42
7
80.9
81.42
64.58
81.08
81.08
84.55
7
39.69
41.41 40.82 44.95
48.11
52.21
82.99
83.85
77.78
76.39
85.42
88.89
8
44.67
52.07
51.81 47.73
55.53
62.04
72.92
74.83
70.14
68.23
78.13
81.42
9
42.76
48.67
45.5
42.45
51.68
54.94
Avg
67.03
68.44
56.52 67.77
69.56
70.89
Avg
38.31
41.19 40.28 41.06
44.63
47.1
Parameter tuning strategies. For the multi-class classification task on the MI2 datasel, we additionally explored different hyperparameter tuning strategies. Following Kerdoncuff et al. (2020), we applied an unsupervised tuning approach based solely on source labels. To make this procodure more robust, we extend the selection rule originally used in MLOT, which simply chooses the parameter set yielding the highest source accuracy. Specifically, we first collect the top 5% of parameter configurations with the highest source accuracy, alternatively the configurations within 5 points of the maximum source accuracy (whichever results in a larger set). For each paramcter, we then select the most frequent value in this subset. In case of lies, we resolve by choosing the value with the highest source accuracy. For the parameters 6,z fe, and et: which are naturally more sensitive to the data, we instead take the average value across the subset. This procedure is applied consistently across all competing methods.
We compare GOT to the baselines OT and OT-reg in the Riemannian framework, where the cost function is defined as the Log-Euelidean distance between covariance matrices rather than the standard Euclidean cost. In addition, we evaluate against three competing methods: MLOT (Kerdoncuff et al. 2020) and RM- LOT (Jawanpuria et al.) 2025), which incorporate Mahalanobis metric learning into the cost function, and MADAOT (Douib et al. 2020), which augments the cost with a weight vector and solves a minmax prob- lem to jointly obtain the transport plan and the classifier. For MLOT and MADAOT we used the authors' publicly available code, while RMLOT had no released implementation and was therefore re-implerennted by us. Since MADAOT assumes Euclidean geometry and both MLOT and RMLOT are limited to Maha- lanobis distances, projected the covariance matrices from the SPD manifold to a tangent plane (Eq. 19 before applying these methods. To ensure fairness, trained linear SVM top of the representations obtained by MADAOT, instead of using its built-in predictions, after verifying that this yields higher ac- curacy. Furthermore, as suggested in MLOT, we applied PCA preprocessing to the data and treated the number of components as a tunable parameter (with the option of disabling it). This preprocessing step was also included for RMLOT and MADAOT, so that the parameter search was performed consistently across mothods. We did not apply PCA preprocessing for OT, OT-reg, or GOT, as these methods dircctly operate on covariance matrices within the SPD manifold.
We report results for both cross-session and cross-subject adaptation. Tables 18a and 18b present the cross-session results, obtained using parameters tuned by source labels (with the procedure described above) and by larget labels, per source target pair. Tables 19a and 19b report the cross-session results restricted to the five valid subjects. In this setting, GOT consistently improves upon the baseline OT with Log Eog-Euclidean cost under both tuning strategics, for both the valid five subjects and the entire dataset. When compared to MLOT, GOT achieves similar performance with optimally tuned parameters (Table 18b), but lower accuracy under source-tuned parameters (Table 18ap. Interestingly, for the five valid subjects,
Results.
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GOT is comparable MLOT in both tuning strategics. A key advantage of GOT is its fiexibility: it be seamlessly integrated into other OT-based methods. We incorporated the GOT cost. into the MLOT framework, reported under the "MLOT+GO"" column. Unlike GOT, which requires tuning only four parameters (e.,,, MLOT+GOT involves seven parameters in total. This allowed us to adopt a more robust selection stralegy for the three GOT-specific parameters, which are interdependent and directly affect the cost: instead of averaging (as GOT), we selected the most frequent combination of (cs, ce; Q)) values within the top-performing subset. We observe that MLOT+GOT improves the cross-session performance of MLOT by more than 1% with optimally tuned parameters. Under unsupervised tuning, the results are comparable when considering all subjects, and improved when focusing on the five valid subjects.
It is worth noting that the cross-session task includes only 18 experiments out of the total 306 source target pairs in the dataset. To provide a more comprehensive comparison, we now turn to the cross-subject task, presented Tables and 20b for source- and target-tuned results. In this setting, GOT alone outperforms all competing wothods--surpasiin and by more than 5% under unsupervised tuning. The advantage even more evident for the five valid subjects (Tables 21a and 21b). In both tuning strategies, and for both the valid five and all subjects, GOT consistently outperforms all competing methods, with MLOT-the sccond-best-lowe
the baseline OT by more than 10% with optimally tuned paramcters
by at least 2%.
Table 18: Cross-scssion Multi-class classification accuracy dataset MI2. (a) Tuning by source. (b) Tuning by target.
(a)
(b)
Subject
OT
OT-reg MADAOT MLOT RMLOT GOT
-OT
Subject OT
OT-reg MADAOT MLOT RMLOT GOT MLOT+GOT
81.08
80.9
68.75
84.9
83.33
83.51
$6.63
83.16
86.46
74.48
88.19
84.9
86.98
89.58
2
55.38
52.26
43.4
54.17
48.09
55 55.21
55.56
2
57.29
39.2
51.56
61.46
61.11
62.67
63.54
3
84.03
85.59
71.53
84.72
78.3
86.98
86.28
3
86.63
86.98
72.05
89.06
85.02
88.72
89.76
59.2
39.58
61.11
46.35
57.81
64.24
64.93
47.22
45.83
36.46
50.69
35.24
42.88
45.83
48.09
37.15
51.22
44.44
50.35
6
40.1
48.09
38.89
53.12
31.6
42.36
51.74
55.21
55.03
39.24
56.94
52.6
56.25
57.29
7
78.12
78.99
70.14
$3.33
67.36 16
71.31
81.72
7
81.77
83.51
76.56
87.67
86.46
87.15
89.06
8
77.26
83.33
72.22
83.85
81.08
88.19
83.85
8
78.3
85.91
71.48
89.41
86.11
90.45
90.28
9 68.06
75.17
65.8
77.
68. 23
80.73
75.52
9
68.06
81 81.25
69.97
81. $1.08
79.51
86.16
83.51
Avg
65.61
67.03
56.31
70.35
59.95
67.
69.77
Avg
68.81
72.55
59.
71.36
72.01
71.71
75.58
Table 19: Cross-scssion Multi-class classification accuracy dataset MT2 (five valid subjects). (a) Tunnnn by source. (b) Tuning by target.
(a)
(b)
Method
Avg
Method
Avg
OT
81.08
84.03
78.12
77.26
68.06
77.71
OT
83.16
86.63
81.77
78.3
68.06
79.58
OT-reg
80.9
85.59
78.99
83.33
75.17
80.8
OT-reg
86.46
86.98
83.51
85.94
81.25
84.83
MADAOT
68.75
71.53
70.14
72.22
65.8
69.69
MADAOT
74.48
72.05
76.56
74.48
69.97
73.51
MLOT
84.9
84.72
83.33
83.85
77.26
82.81
MLOT
88.19
89.06
87.67
89.41
81.08
87.08
RMLOT
83.33
78.3
67.36
81.08
68.
68.23
75.66
RMLOT
84.9
88.02
86.46
86.11
79.51
85.0
GOT
83.51
86.98
74.31
88.19
80.73
82.74
GOT
86.98
88.72
87.15
90.45
83.16
87.29
MLOT+GOT
86.63
86.28
84.72
83.85
75.52
83.4
MLOT+GOT
89.58
89.76
89.06
90.28
83.51
88.44
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Table 20: Subeett Multi-class classification accuracy on dataset M12. (a) Tuning by source. (b) Tuning
by target.
(a)
(b)
Subject
OT OT-reg MADAOT MLOT RMLOT GOT
Subject
OT
OT-reg MADAOT MLOT RMLOT GOT
1
46.56
46.07
43.84
50.75
37.39
55.53
50.22
57.86
48.73
62.67
60.77
64.32
2
26.18
26.92
24.78
26.54
25.82
28.55
29.63
30.96
29.67
32.14
31.24
34.44
3
49.98
51.35
46.27
55.99
43.53
61.65
54.47
61.79
50.95
65.61
63.28
69.37
4
36.06
36.78
35.35
38.35
31.1
38.99
38.66
41.79
38.39
44.22
43.13
45.94
5
32.18
31.85
31.76
32.47
29.21
32.62
34.35
35.56
33.76
36.02
34.51
37.87
6
32.88
33.33
30.97
34.22
29.86
32.71
35.33
36.7
33.47
37.86
37.14
39.72
7
12.84
43.87
41.1
17.17
34.56
49.89
16.47
53.48
45.71
59.6
54.46
60.25
8
45.89
48.99
45.64
5415
39.12
60.36
49.88
59.19
50.18
65.99
62.84
69.52
9 12.55
11.43
12.2
18.17
39.02
52.65
44.9
54.96
19.76
59.44
56.27
62.47
Avg
39.46
40.4
37.99
43.13
34.4
45.88
Avg
42.66
48.03
42.29
51.51
49.29
53.77
Table 21:
Cross-subject Multi-class classification accuracy dataset MI2 (five valid subjects). (a) Tuning by source. (b) Tuning by target.
(a)
(b)
Method
3
Avg.
Method
avg
OT
55.38
62.74
57.53
56.25
52.28
56.84
OT
58.53
66.67
60.61
58.59
54.73
59.83
OT-reg
54.36
62.48
57.25
59.09
54.62
57.56
OTT-reg
65.6
72.16
66.08
68.75
63.91
67.3
MADAOT
54.41
59.79
54.97
57.29
53.49
55.99
MADAOT
57.86
63.24
57.2
61.02
58.25
59.51
MLOT
61.48
70.01
64.04
66.43
59.7
64.33
MLOT
69.16
76.39
72.98
74.98
68.36
72.37
RMLOT
43.4
54.51
43.12
46.2
47.35
46.92
RMLOT
68.47
74.37
69.25
72.01
65.23
69.87
GOT
63.24
75.13
67.3
71.53
64.45
68.33
GOT
71.18
79.82
73.42
78.17
72.18
74.95
D Proofs
Lemma D.1. The asymptotic eaponsion of single-domain diffusion operator, defined in Equation 7 is expressed follows:
m2 Af AH (x) (x)+
(21)
mo
According Lemma from Coifman & Lafon (2066) given an isotropic kernel kg h |==-| with an exponential decay, and an operator T. defined by Tg((z) SM (x.)))())dd)
Proof of Lemma D.1
the asymptotic expansion of T, for all g (3(M and for all z E M is given by:
T.g(z) = mog(z) - m22 (Ag(x) - w())g(=)) 0(2),
(22)
where A is the Laplace Beltrami operator on M, mp, ma two constants depending on the kernel h, and w(x) is potential function.
For the diffusion operator P. defined in Equation 7 we use g(x) f(z)u(z) and from Equation 22 we get: de(x)
[(x)p(x)
m2f (A [(x))(())
w(x) [(c))(()) + 0(2),
Pvf(x) = -m (x)
(23)
dc(x)
dx
where (x)) K, (1)
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Published in Transactions on Machine Learning Research (10/2025)
Similarly, we use g(x) px()) in Equation 22 and obtain that the asymptotic expansion of de is given by:
d, (x) (1) m22 (Au(x) w(x)u(x)) O(e2)
moz(x) (1- (1- mo
m2
(
Au
(z) + +(E2)
(24)
By applying the Taylor expansion of 1 1-w and neglocting terms of order O(c2) and higher, we obtain:
(x) = (moz(())) (1 (1+
m2 AH -w) (x) + 0(E2)
(25)
m700
Multiplying Equation 25 by fu and subsequently applying the Laplace Beltrami operator yields;
fu (z)
f
(1+
m2
( A w (x) + +(2))
(26)
de
m70
mo
fP de
A
m2 + (1 (1( )) (o) (z) ) mo
w
(x)
(27)
mp
Finally, by substituting Equations
26 and 27 into Equation
23 and neglecting terms of order O(e²), we
derive the asymptotic expansion of the operator Pe:
Pzf(x)= = f(x)
m2
- f Ap
+ wof + +AS -ws)
(x) + 00))
ma
f(x)=
m2
Af - Au
(x) 0(67)
Lemma D.2. The asymptotic expansion of the cross-domain difffusion operator, defined by:
kc( f(z))(z)dz
Qef(v) dz)(
(28)
is expressed as follows:
Q.1(x) (())
m2
( (1()))
Av (x) +0
(29)
mn
Proof of Lemma D.2. For the cross-domain diffusion operator defined in Equation fa
28 with c. as the scale
hyperparameter of the cross-domain Gaussian kernel, we substitute g(x) obtain:
into Equation 22 and
de.ec
fu (x) m2ed (A Qr.(xx) mo de,cc
(a(dit)
((H) deree
fu (0) + +(E))
(x) w(x) dr.so
(30)
where dr.,, (I) ke. (x, (y)dd.
Similarly, we substitute g(x) = v(x) to Equation 22 and obtain the asymptotic expansion of de...
dr. (x) mor(x) - m22, (Av(x) (x)))(())) O(e)).
(31)
We utilize the Taylor expansion as in Equation 25 leading to the derivation:
(d..) (d.)) = (m)u(x))
m2 cc
Av
w (x)) + +((2)
))(())
(32)
39

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Published in Transactions on Machine Learning Research (10/2025)
Multiplying Equation 32 by f and subsequently applying the Laplace-Beltrami Beltrami operator yields:
deo
(x)
fμ mgy
1
m2 mg
cc
Av
w (x)
+ -(e))
(33)
V
A
fP de.
(x)
1
a
(5#)
m2 m
S 11 Av
/ ( w) )) (I))
(34)
mm
Finally, by substituting Equations 33 and 34 into Equation 30 and neglecting terms of order O(e?), we derive the asymptotic expansion of the operator Q.
(e)(x) -
m2
- ش + - A (1t) - (x) +
# Av
Ec
mg
H (z) f 1/
m2 Ec
A (s - Av
)) (x) O(E))
m0
Proposition 4.1. Suppose f E (MM), and suppose p.v E (+M) denote the probability measures of the source and target domains, respectively, where is dominated by v. For sufficiently small the asymptotic expansion of operalor S. is given by:
Szf(x) 1 (x) f
m2 mo
+2 log (-)
3 +2
(9)
(2"
+2 -w
)
f
(x) O(E?)
where A, V are the Laplace-Beltrami Beltrami operator and the covariant derivative on M, respectively, and mo, m2 are two constants defined by the Gaussian kernel and by the manifold M.
Proof of Proposition 4.1 The proposed diffusion operator is defined by the composition S.f(x)
Pt.e, Ps.,s.., where = Ees
We start by defining the function g(x) (()...(x). Substituting the asymptotic expansion of the source diffusion operator, as derived in Equation 21 into Equation 29 we obtain:
Pe,e
11 (
m2
Af - f )
(z)
m2
EE
(5
fth f4 Av
(x) + 0(2)
mi
mo
v
= 1 1
m1
6 P1
H (1
Av
v )) + O(6))
(r)
6,
+ cc
(35)
where all terms with order O(e²) were neglected.
Remark that assuming E E yields:
= 8 (x) m2 E
LAS +
( 1
Ay
1 2) (x) (x)+
(36)
40

---

Published in Transactions on Machine Learning Research (10/2025)
Next, we define the function g(x) Pee and substitute Equation 36 into the asymptotic expansion of the target diffusion operator, as defined in Equation 21
Per Qe, (()) = (x)
m2
P AS + + (1t) -
4 v ( + + (c
Au
= )
mg
m2 ct
A (s" v - -/
(x) + 0(2)
mo
v v
f H (z)
m2
c _ Af + (Ec + a) (1
m0
- v
Auu + (Ec Ec) 1/ (x) + 0(e))
)
Eo
(37)
where all terms with order O(e²) were neglected.
Assuming Eg = Ec = =ewe we get:
Pea Qe, Pac. f(x) 5
m2
("AS + +2 (11)
-s 1
(An
Ap
+2 v
)) (x) +0(2).
(38)
mo
v
Finally, by utilizing:
A ( =
1as+ (5) +2V (5)
(39)
we yield the expression:
(xf(x)= (()))
m2 mo
c
1 3f 2f (5) v
1 Aلl ("' +2 ))) AV/V (:)) (x) + +(()).
(40)
1/
v
p
Lastly, for a more comprehensive analysis, we utilize the property 7/ = Vlo((f) to derive the expression
presented in Proposition 4.1
41


Reviews:
[{"review_id": "BnXxX9272W", "reviewer": "Reviewer_UqCV", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The authors propose a new method for computing a cost matrix using the manifold assumption that makes the transport between domains more robust.\n- The new cost matrix went with theoretical insights about why this cost matrix is useful for domain adaptation, which is not the case in OT DA papers cited in the paper.\n- This new cost is tested first on a toy example showing its efficiency, then GOT is incorporated into different baselines to show its superiority over competitors for image classification and motor imagery.\n\nWeaknesses:\n- GOT is used inside the framework of DeepJDOT, but DeepJDOT is not properly introduced in the main paper. The cost matrix used in DeepJDOT is both the Euclidean distance + a term of cross-entropy loss between the target pseudo-label and the true source label. You explain everything in the appendix, but I think it would be nice to be more transparent directly in the main paper.\n- I think having a fair comparison for BCI experiments would bring more impact to your results. Right now, the authors report results from two different papers. With different software like moabb or pyriemmann, I think implementing the benchmark is doable and would bring more trust in the results.\n- The scale hyperparameters are fixed to 1 for deep learning adaptation. I'm wondering if the method is sensitive to these parameters? Maybe adding a sensitive study could help practitioners apply the method easily.\n- How did you choose the number of 3 nearest neighbors for the target? Is it a robust choice for different modalities?\n- The majority of the methods used in the paper need hyperparameters. How are these hyperparameters tuned in practice? A paragraph should be added for transparency and reproducibility.\n- The authors gave a computational cost comparison in the appendix; it would be nice to have a comparison in practice for computing the different OT distances.\n- Do you think GOT should automatically be used as a cost matrix in OT-based DA? Or is it a choice that practitioners should make? Maybe adding a discussion about that could be helpful.\n\nSmall weaknesses:\n- Maybe adding a new blue dot for Figure 1 could be clearer. Right now, it is propagated to three source dots and three target dots. But in your method, you can have more for the source, but only 3 for the target. Adding one blue dot can help to understand the difference.\n-The title of Figure 2 should be clearer that you are plotting only the column $\\gamma(., j)$ and the 10 highest values, even if it is said in the paragraph.\n- There is no name for the WBT acronym (Wasserstein Barycenter Transfer ?)", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "If Optimal Transport has been widely used for Domain Adaptation, the cost matrix used to compute the transport is the pairwise Euclidean distance. The authors introduce a new cost matrix called Geometric Optimal Transport (GOT), leveraging diffusion geometry and manifold learning, that allows for better transport between the two domains. The GOT cost matrix comes with theoretical insight and experiments over two modalities, image classification and  BCI Motor Imagery.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please go through the weaknesses to know what needs to change. This is a brief summary:\n\n- Add a better introduction of the DeepJDOT joint loss to understand that there are two terms.\n- Implement competitors for BCI experiments.\n- Add a paragraph about hyperparameter selection with a sensitivity study of scale hyperparameters of GOT.\n- Add computational cost comparison between different cost matrices in practice.\n- Modify Figures 1 and clarify the title of Figure 2.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "usKdsk5XQF", "reviewer": "Reviewer_svjb", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n\n1. Novel idea for constructing the cost matrix for the OTDA problem without relying on min max approach or learning it from data\n2. Theoretical justification\n3. Extensive experimental results with many baselines\n\n**Weaknesses**\n\n1. Missing references\n2. More recent non-OT baselines could have been included\n3. It is not clear how to interpret the theoretical results", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper studies the problem of applying optimal transport to the domain adaptation problem (OTDA). The main underlying contribution is to use the diffusion geometry for this task. In particular, the authors construct the cost matrix based on a diffusion operator between the domains. This cost matrix is a product of three similarity matrices that capture intra- and inter-domain diffusion operators. The authors provide a theoretical justification for their approach, showing that the diffusion operator they propose makes sense for the DA problem. Extensive experimental results suggest that the proposed approach works well in practice.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. The authors missed a bunch of references on learning the cost matrix in OT. I think those should be discussed in this manuscript, and a visualization of the different learned ground matrices may be provided to illustrate the differences between them. \n\nHere are some references:\n\n1. Cuturi et David, \"Ground Metric Learning\", JMLR, 2014.\n2. Dhouib et al. \"Margin-aware Adversarial Domain Adaptation with Optimal Transport\", ICML'20\n3. Huizing et al. \"Unsupervised Ground Metric Learning Using Wasserstein Singular Vectors\", ICML'22\n4. Jawanpuria et al. \"A Riemannian Approach to Ground Metric Learning for Optimal Transport\", arxiv'24.\n5. Kerdoncuff et al. \"Metric Learning in Optimal Transport for Domain Adaptation\", IJCAI'20.\n\nThe authors can insist on defining the ground metric without learning it to better position their approach but still they need to cite the relevant work and compare whenever possible. \n\n2. The most recent non-OT DA baseline dates back to 2020. It would be nice to include at least 1 or 2 recent baselines \n3. I think that the analysis of Proposition 4.1 is not very intuitive. It would be nice to illustrate its different parts to make it clearer for the audience not familiar withg diffusion geometry.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "eiJg3lBEcU", "reviewer": "Reviewer_eA7v", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Audience.**\nBased on these key facts, this paper addresses a long-term, meaningful problem. Its objective is to propose a better cost metric to more accurately reflect data structure. From this perspective, the data satisfies the first criterion of TMLR, meaning at least certain audiences will be interested in the paper's merits.\n\n**Evidence**\n\nUnfortunately, I do not believe this paper provides convincing evidence to demonstrate its technical merits. This conclusion is based on the following observations:\n\n1. The key selling point of Geometric Optimal Transport (GOT) lies in its consideration of geometric space. However, apart from toy data, this paper does not clearly demonstrate this key advantage. I remain unclear on how geometric space is beneficial. For instance, the geometric space relies on the kernel, which is also related to Euclidean distance. This makes it much harder to understand why your choice specifically better fits real-world data. There is no support for this, which is arguably the most crucial part of the paper.\n2. The theory seems somewhat unclear. Proposition 4.1 appears to lack insights into why the proposed metric is beneficial in UDA, or which data structures would particularly benefit from this approach. Based on the results, I didn't grasp this point.\n3. Regarding the experimental results, I noticed that most results indicate a performance improvement. However, based on the hypothesis, certain data structures should better fit the manifold assumptions. My question is, how can real-world data, such as images, verify this?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper explores geometric optimal transport (OT) for unsupervised domain adaptation (UDA).\n\nThe core idea is as follows:\n\n- Traditional OT in UDA has focused on Euclidean space, which presents potential limitations.\n- This paper,focuses on UDA under the manifold assumption.\n\nThe proposed method involves:\n- Introducing source, target, and cross-domain diffusion operators.\n- Defining the final operator as the product of these three diffusion operators.\n- Defining the transport cost as the negative log-likelihood of this operator.\n\nTheoretical and practical support includes:\n- Theoretical analysis presented in Section 4.1.\n- Experiments conducted on typical UDA benchmarks.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I recommend that the authors make major revisions to the current version, particularly concerning points (2) and (3). The paper could be improved by:\n\n- Adding clear theoretical or mathematical analysis to illustrate why and when manifold assumptions can help UDA (within the OT context).\n\n- Differentiating experiments, algorithms, and methods in deep and non-deep OT UDA, as both require quite different assumptions and scenarios.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)