Elastic Optimal Transport: Theory, Application, and Empirical Evaluation
Authors: Pei Yang
Yuhang Zhuang
Qi Tan
Venue: ICLR 2026 Poster (2026)
Keywords: optimal transport
domain adaptation
transfer learning
Primary area: transfer learning, meta learning, and lifelong learning
TL;DR: A novel optimal transport method with solid theory, broad applicability, and promising empirical evaluation.

Abstract: The classical optimal transport such as Kantorovich's optimal transport and partial optimal transport could be too restrictive in applications due to the full-mass or fixed-mass preservation constraints. To remedy this limitation, we propose elastic optimal transport (ELOT) which is distinctive from the classical optimal transport in its ability of adaptive-mass preserving. It aims to answer the problem of how to transport the probability mass adaptively between probability distributions, which is a fundamental topic in various areas of artificial intelligence. The strength of elastic optimal transport is its capability to transport adaptive-mass in the light of the geometry structure of the problem itself. As an application example in machine learning, we apply elastic optimal transport to both unsupervised domain adaptation and partial domain adaptation tasks. It adaptively transports masses from source domain to target domain by taking domain shift into consideration and respecting the ubiquity of noises or outliers in the data, in order to improve the generalization performance. The experiment results on the benchmarks show that ELOT significantly outperforms the state-of-the-art methods. As a powerful distribution matching tool, elastic optimal transport might be of interests to the broad areas such as artificial intelligence, healthcare, physics, operations research, urban science, etc. The source code is available in the supplementary material.

Published as a conference paper at ICLR 2026
ELASTIC OPTIMAL TRANSPORT: THEORY, APPLICATION, AND EMPIRICAL EVALUATION
Pei Yang, Yuhang Zhuang Department of Computer Science
Qi Tan
Department of Computer Science South China Normal University Guangzhou, China
South China University of Technology Guangzhou, China
yangpei @scut. edu. cn
tanqi @scru. edu. cn
20242104468 49000046640000 001900mmaall..... scut. edu. cn
ABSTRACT
The classical optimal transport such as Kantorovich's optimal transport and partial optimal transport could be too restrictive in applications due to the full-mass or fixed-mass preservation constraints. To remedy this limitation, we propose elastic optimal transport (ELOT) which distinctive from the classical optimal transport in its ability of adaptive-mass preserving. It aims answer the problem of how transport the probability mass adaptively between probability distributions, which is a fundamental topic various areas of artificial intelligence. The strength of elastic optimal transport is its capability to transport adaptive-mass in the light of the geometry structure of the problem itself. As an application example machine learning, we apply elastic optimal transport to both unsupervised domain adaptation and partial domain adaptation tasks. adaptively transports masses from source domain to target domain by taking domain shift into consideration and respecting the ubiquity of noises or outliers in the data, in order to improve the generalization performance. The experiment results the benchmarks show that ELOT significantly outperforms the state-of-the-at methods. As a powerful distribution matching tool, elastic optimal transport might be of interests the broad areas such as artificial intelligence, healthcare, physics, operations research, urban science, etc. The source code is available in the supplementary material.
INTRODUCTION
The optimal transport (OT) problem first came up in Monge's seminal work (Monge. 1781). It aim- s to find an optimal way to move a pile of sand into hole, assuming the pile and the hole must have the same volume. OT is formulated the mathematical problem of comparing two probability distributions, which is a fundamental problem in many areas. Many researchers in different areas found that optimal transport was strongly linked their subjects, and helped expand the optimal transport foundations (Villani) 2003). Recent years have witnessed another revolution in the spread of OT, thanks to the emergence of approximate algorithms that can solve large-scale problems (Cu- turi 2013). As a consequence, OT is being increasingly used to unlock various problems in artificial inte ligenigeece, statistics, bioinformatics, economics, logistics, physics, etc.
However, the classical optimal transport has its limitations, which motivates us to develop the novel formulation of OT. Kantorovitch (1942) moved away from the idea that mass transportation should be deterministic to consider instead a probabilistic transport, which allows mass splitting from source toward several targets. But major bottleneck of OT in its traditional formulation is that requires the two distributions to have the same total probability mass and all probability mass has to be transported. This is too restrictive in many applications because full mass preservation is likely to fit noise and outliers, or undesired pairs, and prevent any form of partial matching. There have been two main directions, i.e., partial optimal transport and unbalanced optimal transport. to attempt to remove the constraints of full-mas. preservation. Caffarelli & McCann (2010) and Figalli (2010) proposed partial optimal transport to preserve the fixed amount of mass instead the full mass. Liero et al. (2017) developed unbalanced optimal transport which replaced the 'hard' marginal constraints

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of OT by 'soft' penalties using some divergence. Both partial optimal transport and unbalanced optimal transport provide great flexibility to model partial matching. However, they lack the ability of adaptive-mass transport. For partial optimal transport, it is challenging to determine the fixed budget of mass to transport. For unbalanced optimal transport, it is usually unknown to what extent the 'soft' penalties should be imposed on the marginal constraints.
Therefore, a natural question is: could it adaptively transport probability masses among probability distributions? This motivates us develop the elastic optimal transport, which is able to adaptively determine the transported masses in the light of the intrinsic structure of the problem itself. The distinctive advantage of elastic optimal transport against its classical counterparts lies in the ability of adaptive-mas, preserving. The elastic optimal transport is of great interests to the broad areas such as machine learning. biomedical, computational physics, operations research, urban science, etc. For example, optimal transport plays a crucial role in wide variety of machine learning applications. such as generative adversarial networks (Arjovsky
al. 2017). computer vision (Solomon et al,) 2016), natural language processing (Xu et al.) 2021 1). clustering (Ho et al. (2017), graph matching (Zang & Lauw [2024] Sun et al. 2024), semi-supervised learning (Chapel et al. (020), and domain adaptation (Courty al. 2017b). The essential problem in these applications is how to compare two probability distributions (e.g., align the Al-generated images with the natural images). Due to the ubiquity of noises, outliers, and divergences existing in the real-world data, the optimal transport with adaptive-mass preserving is preferred in such applications to alleviate the distortion of OT mappings. The main contributions of the paper are highlighted as follows.
We propose novel formulation of optimal transport. The elastic optimal transport is distinc- tive from classical optimal transport in the aspects of adaptive-mass preserving and mixed-sign ground cost matrix. Its strength is that the transported masses are adaptively determined by the geometry structure of the problem, providing a powerful tool for adaptive distribution matching,
The theoretical analysis sheds light on the mass transport mechanism of ELOT. Furthermore, we derive an equivalent formulation for elastic optimal transport, which can be solved by the off-the-shelff OT algorithms.
We propose a novel model for domain adaptation based on elastic optimal transport. It transports the masses from the source domain to the target domain adaptively, taking full consideration of the domain shift. The experiment results on the benchmarks show that ELOT outperforms the state-of-the-att approaches in variety of unsupervised domain adaptation and partial domain adaptation tasks.
2
ELASTIC OPTIMAL TRANSPORT
We introduce the classical optimal transport, and then propose the elastic optimal transport approach.
2.1 PRELIMINARY
Without loss of generality, we focus on discrete optimal transport in this paper. Assume that we are given pile of sand, and hole that have to completely fill up with the sand. Let μu and be the probability measures of the pile and the hole respectively. The probability measure p with the probability masses {p:}2= the locations {a. formulated as μ Et-1:d p=òz; where dz, is the Dirac function located T,, and p; belongs to the probability simplex. i.c. Σi=1 Ps 1. Likewise, the probability measure v with the probability masses {1z }m= on the locations { is formulated as / = Σ== gz The transport plans are formulated by the probability measures y, where Ti measures the amount of mass transported from location x; to location zj. The bounded ground cost Ci₃ tells how much it costs to transport one unit of mass from &; to Zj. We denote 17*" and p_xm as the non-negative and mixed-sign matrix, respectively. The d-dimensional vector of ones is denoted by 1ap
Kantorovich's Optimal Transport. In the discrete setting, Kantorovich's optimal transport prob- lem (Kantorovitch] 1942 is formulated as:
min (EI()
< <>
(1)
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where C E R and
11 v) {r 14* 7 p,, 1/).
Partial Optimal Transport. Partial optimal transport (Caffarelli & McCann 2010: Figalli. 2010) relaxes the full-mass constraints in Kantorovich's 's problem and preserves the fixed amount of mass:
min vEll,(u.))
<
where C. E Raxm and
Ip(,, = by 24 1 S S,,7" S SV, =
(4)
where s(0 S min([ll[l; the fixed budget of mass, which is user-defined. 1 vector norm.
Unbalanced Optimal Transport. Unbalanced optimal transport (Liero et al. 2017 relaxes the 'hard' marginal constraints in Kantorovich's problem with the 'soft' penalties by using some diver- gence D
min 7ER
S7 (m) T2 (1""ll")
(5)
where C E R4 The trade-off coefficients T₁ and 72 are used to adjust the weights of the marginal divergences.
2.2 ELASTIC OPTIMAL TRANSPORT
We propose the elastic optimal transport in order to overcome the limitations of the classical optimal transport, which are discussed in Section [ In the discrete setting, elastic optimal transport is formulated as:
W(, min yEll.(u.))
<7,C>
(6)
where C e R4 Raxm and
(1 E IR 71 S H,,"" Sv).
(7)
The distinctive characteristics of elastic optimal transport are twofold: adaptive-mass preserving and mixed-sign ground cost matrix. First, the partial masses are adaptively transported. There are no full-mass fixed-mass preserving constraint done in Kantorovich's optimal transport and partial optimal transport, respectively. Second, the ground cost matrix C a mixed-sign matrix which con- tains positive and negative entries. The classical optimal transport usually assumes that the ground cost non-negative [Villani 2003). However, in many scenarios, it is natural to allow for negative costs. Consider the example related CO2 abatement from economics and operations research context (Levihn] 2016). Many investment options both increase productivity and reduce CO2 e- missions, leading to financial returns. Therefore, the costs for these options should be negative. In contrast, the costs for those investment options with financial expenditure should be positive.
In the seminal work of partial optimal transport, Caffarelli & McCann (2010) (equations 1.7 and 1.8) introduced the Lagrange multiplier À 2 O and replaced the cost c(z, with c(I, y) A, allowing for negative cost. Then the mass s can be attained by selecting the appropriate value of X. The drawback of Caffarelll & McCann (2010) is that it needs to specify the fixed budget of mass s. In contrast, ELOT able to automatically find the optimal mass to be transferred. We overcome drawback of Caffarelli & McCann (2010) and rely on the combination of mixed-sign cost and the marginal inequality constraints to achieve the goal of adaptive-man -mss preserving.
In summary, the advantage of elastic optimal transport lies in the self-adaptive preserving of mass. The elastic optimal transport is capable of preserving the suitable masses in accordance with the native geometry of the problem. It provides elegant solution for adaptive matching. Unlike partial optimal transport or unbalanced optimal transport, it does not need to specify the fixed budget of mass the softness of marginal constraints, which challenging in essence.
Using the technique to convert the inequality into an equation by adding the slack variable in linear programming (Matoušek & Gärtner 2007), solve the elastic optimal transport problem by adding
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slack variables and extending the cost matrix. Define the augmented cost matrix and the marginal vectors as
dlm 12 20 1
C
o
-
2
[|llaa]
(8)
where a 2 is fixed non- negative scalar. The elastic optimal transport with inequality constraints can be reformulated as the following optimal transport problem with equality constrains
W( = min 7El((..)
< <7,0>
(9)
where
(x+1)x(m+1) M( D) { E R
(m(1)) Pr P.. (n-1) =
(10)
Theorem D shows the equivalence between the clastic optimal transport problem defined in Equation and the reformulation defined in Equation [ which allows using the aff-the-shelf optimal transport algorithms to solve the elastic optimal transport problem.
(") and W(H, v) (or W(J. D) be the opti- mal transport plan and the optimal transport distance for the elastic optimal transport problem (or the reformulation). Assume %o is the matrix 5 with the last row and column removed.
Theorem (Optimal Transport Solution) Let -f" (or 5")
Optimal Transport Plan. The optimal trunsport plan Y is equivalent to %
%o
argmin <7,0=m (Elle(ax))
(11)
ti) Optimal Transport Distance. The difference of W(u, v) and W(p 7)) constant term which can be ignored in optimization (i.e., the optimization is not affected by the specific value of a):
W(F D) W((p))
+ (111)++
(12)
The proof Theorem in Appendix A.I.1
Also, Theorem I Iinicattss that the optimal transport plan of ELOT keeps unchanged no matter what is the value of a. So we just set a = 1 in the ex- periments. The algorithm complexity of ELOT is slightly different from classical optimal transport with the scale of problem-solving changing from n x m to (n + 1) x (m + 1), where the extra one corresponds to the augmented dimension.
Entropic Elastic Optimal Transport. The entropy-reguulrrized optimal transport CCtuuri 2013) has the advantages that it defines a strongly convex problem which can be solved efficiently. Like- wise, one may add the entropy-regularie transport defined in Equation 6 where c is the entropic coefficient. The mntropy-rrgularizer term en- courages the sparsity the transport plan, and allows using the Sinkhorn Sinkhorn-Knopp algorithm (Cuturi 2013) for efficient computation.
term eH(()) = -e Ei, "Y.j log Yi. to the clastic optimal
Note that either ELOT or classical optimal transport does not guarantee unique solution, which stems from the interplay between its linear objective function and the convex polyhedral feasible region. Therefore, both the original ELOT and the reformulated problem could have multiple op- tima. However, the uniqueness of solution to the entropy-regularize ELOT and its reformulated counterpart is guaranteed because the entropy- regulai-ee objective function is strictly convex.
2.3 MASS TRANSPORT MECHANISM
We cast the elastic optimal transport problem constrained worst transport problem, which pro- vides insights into the mass transport mechanism of the clastic optimal transport problem. Let's detach the cost matrix into the positive part C+ and the negative part C i.e., C = C+ C-, where
CJ c
Cijs if Cij > 0; otherwise.
Cūj - 0, otherwise.
{
CCss Cijs
if Cfj < 0;
We stipulate that the transport mass is zero (i.e., Pij = 0) when the cost is zero (i.e., Ci = 0) to avoid ambiguity. The elastic optimal transport problem defined in Equation i rewritten as
min ET
(13)
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where
Te(u, = -100 E, 76i S pi. E, uzz
(14)
where Cij could be positive negative and (2n(2 2 0) satisfies both left and right marginal inequality constraints for all admissible i. Then, we define the constrained worst transport problem
Cigris 785
(15)
N/ (T)(
8.
where
{ Tho 7i5 = 0, if Cij = 0; E, Paj=-e 14 or m, if Cis < <o.
{18)
Te(us v) =
(16)
where -Cty is non-negative and (2 2 0) satisfies the conditions specified in Equation 16 for all admissible i. We prove the equivalence between elastic optimal transport and its corresponding constrained worst transport problem.
Theorem 2 (Mass Transport Mechanism) The optimal transport plan of the elastic optimal trans- port problem Equation [3 equivalent to that of its corresponding constrained worst transport problem in Equation!
15 i.e.,
arg min
Egony
arg max (ETc(u.))
C-c Cig--s
(17)
(pT) 2.j
Theorem 2 (proof in Appendix A.1.2) gives insights into the mass transportation mechanism of the elastic optimal transport problem. The source and target areas can be divided into active and inactive regions according the sign of ground cost. There is no mass transportation between the source and the target areas when their cost are positive. The mass allocation between areas with negative cost acts like that of the constrained worst transport problem, where the optimal transport plan satisfies either left or right marginal equality constraint rather than both inequality constraints.
From pragmatic point of view, Theorem indicates that elastic optimal transport provides robust solution to handle noise, outliers, and distribution shifts among data. Suppose that the items zth row (or jth column) in the cost matrix all positive, then there is no mass transported from x; (or to 2,) according to Theorem 2 i.e., Σj 70y 2j) is likely to be outlicrs or noisy point. The strength of clastic optimal transport is that it is capable of filtering out outliers automatically by enabling adaptive mass allocation. According to the above discussion, is straightforward to attain the following corollary (the proof is omitted):
< μ4; (or >. 7 0 < vj). It suggests that 2; (or
Corollary (Optimal Transport Mass) For the elastic optimal transport problem, the total mass of the optimal transport plan satisfies:
Σn min
Em. min {llllla Iellla
(18)
ij
jEJ
where I (or J) is the set of row (or column) in the cost matrix who has least one negative item.
3 APPLICATION TO DOMAIN ADAPTATION
Optimal transport plays fundamental roles in many machine learning applications. As a crucial task of machine learning, domain adaptation aims to estimate a transferable model for target domain by exploiting source domain data in the presence of domain shift. We focus on both unsupervised and partial domain adaptation, where no label is available in the target domain.
Let X {a: / be set of data samples drawn from distribution µ the source domain, associated with a set of class labels = {y: )r and 2 = (m a set of data samples drawn from a distribution y on the target domain. The empirical distribution μ for source domain is formulated as > p=ó2; where 5R, is the Dirac function located at x,, and p. is probability mass associated
with z; and belongs to the probability simplex, ie. Σ P: = 1. The distribution v for target domain can be formulated similarly.
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We propose the generalized unsupervised domain adaptation model based on elastic optimal trans- port, called ELOT. Let g be the feature extractor that maps the data samples into latent feature space, and f the classifier which maps the latent feature space into the label space. Both the ground cost matrix C(g. f') and the empirical classification loss L(g. f') depend on the extractor ! and the clas- sifier f. The objective of ELOT is to minimize the elastic optimal transport distance between the source domain and target domain, as well as the empirical classification loss on the source domain, i.e.,
min (EIl.
< <, +C(g,f)
(19)
where C(g. Ryxm and
()( IT 711m S 1,,7",, S Sv).
(20)
The strength of using clastic optimal transport for domain adaptation is its capability of transporting partial mass adaptively from source domain to target domain. Due to the domain shift, full-mass conservation may lead to fitting dissimilar pairs (and noise or outliers) between source and target domains. Also, since it is unknown to what extent the two domains are related, fixed-mass con- servation is too restrictive. Therefore, the most suitable way is fit similar sample pairs between domains while neglecting dissimilar pairs and outliers via adaptive transportation of partial masses.
Specifically, we provide an instantiation of ELOT as follows. Nevertheless, there are a variety of choices of ground costs, depending on the problems themselves.
min Ens (alpotos) - s(2,) || - By: tanh (1(()))) Σn ye log (1(1(1)))
(21)
where E (e, v), and tanh is the hyperbolic Tangent function. a and B are non-negative coefficients to balance the impact of feature-wise cost and label-wise cost. Here we use cross- entropy loss as the empirical classification loss.
ELOT requires the costs to be mixed-sign to avoid the potential degeneracy. This could be limi- tation in some tasks, but ELOT is widely applicable to many scenarios such as domain adaptation where the cost could be induced in multiple feature spaces (e.g., original feature space and latent feature space, or marginal distribution and conditional distribution). Specifically, the underlying idea in constructing the ground cost matrix C is align the domains in the feature space and label space simultancously. Since the target labels are unknown, we use the surrogate version f((()) which is a common practice use the pseudo labels. The ground cost consists of two types of cost, i.e. feature-wist cost and label-wise cost. The intuition is that the more similar the pair is in both feature and label space, the more mass transported between them.
Note that ELOT is general and many kinds of ground cost functions could be used in ELOT. For example, one may define the alternative ground cost function as follows:
C₁j = o ((((((=)))
(22)
It is worth mentioning we use exactly the same number of hyper-parameters as JUMBOT (Fatras al., (02T) and m-POT (Nguyen et al. 2022) to construct the cost function. Since one may move the coefficient a from the cost term to the cross-entropy loss term as usually done to achieve the same effect, there actually only one hyper- paper-parameter 33 used the cost function. Besides, both JUMBOT and m-POT need extra trade-off parameters (s, TI, 72 ) to set either the marginal penal- ization coefficient or a priori amount of mass to be transferred, which remains a challenging issue. Therefore, the advantage of ELOT over partial OT or unbalanced OT is that it not only achieves adaptive-mase transport but also requires less trade-off coefficients.
It is unavoidable tune the of the cost structure, depending on the problem itself. As usually, it could be tuned via cross- validation. Hence, ELOT relies the cost structure for adaptive- mass transport, rather than playing the exchange game of hyper-parameter.
in these cases since this dyper-paramees is part
The detailed ELOT algorithm is presented in Appendix A.2

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4 EXPERIMENTS
We evaluate our method on both unsupervised domain adaptation and partial domain adaptation tasks in comparison with the state-of-the-ard algorithms. The implementation details are introduced in Appendix A.4.1
4.1 DATASETS
VisDA (Peng et al. 2017 is a large-scale dataset for unsupervised domain adaptation. It contains 152,397 synthetic images the source domain and 55,388 real-world images as the target domain.
The two domains share 12 object categories. Following the common setting (Fatras et al. 2021 Nguyen al. 2022), we evaluate all methods on VisDA validation set. Office-oome Venkateswara et a 2017) contains 15,500 images from four domains: Artistic images (A), ClipArt (C), Product images (P) and Real-World (R). For each domain, it consists of 65 object categories that are common in home and office scenarios. All methods are evaluated on 12 adaptation tasks. Office-31 (Saenko) ctt al.) 2010) consists of 4652 images from 31 categories, collected from three domains including Amazon (2817 images), Webcam (795 images) and DSLR (498 images), respectively. There are totally 6 adaptation tasks for evaluation. DomainNet (Peng et al. 2019) is a large-scale dataset with six distinct domains and approximately 0.6 million images distributed among 345 categories. We follow Gu et al. (2024) to adopt four domains including Clipart (C), Painting (P), Real (R), and Sketch (S)) with 126 classes for partial domain adaptation task. All above datasets are widely used for the evaluation of domain adaptation algorithms.
4.2 UNSUPERVISED DOMAIN ADAPTATION
We compare ELOT with a variety of unsupervised domain adaptation algorithms: 1) OT-based meth- ods such ROT (Balaji al. 2020 DeepJDOT (Damodaran al. 2018), JUMBOT (Fatras al., 2021) and m-POT (guuyen et al. 2022); and 2) Non-OOT-hased methods such as DANN (Gaiin
ct al. ) CDAN Long et al. 2017a). ALDA (Chen et al. 2020). and CaCo (Huang al. 2022)
DEEPJDOT (Damodaran et al. 2018) is the deep learning-based extension of JDOT (Courty et al., 2017a) which is a joint distribution optimal transport method. ROT (Balaji al., 2020) followed the unbalanced optimal transport formulation while keeping the f f-divergence relaxations of marginal distributions as inequality constraints. JUMBOT (Fatras al. (02T) adopted the unbalanced opti- mal transport to alleviate the issue of undesired matching during the mini-batch sampling. In con- trast, m-POT (Nguyen et al. 2022) used partial optimal transport to mitigate the misspecified map- pings by limiting the amount of masses. Domain Adversarial Neural Network (DANN) (Ganin et al.) 2016) adversarially learned a feature extractor and a domain discriminator, Conditional Domain Ad- versarial Network (CDAN) (Long et al., 2017a) applied conditional domain discriminator instead. ALDA CChnn et al. 2020) combined self-training and adversarial training for correccione do- main discrimination. The contrastive-learnig based method CaCo (Huang al.) (2022) adopted the category contrastive loss for adaptation. The more discussion of related work on domain adaptation and distinctive characteristic of ELOT are given in Appendix A.5
The main goal of our experiments is to verify the superiority of elastic optimal transport over the classical OT theory such as partial OT and unbalanced OT. We are not aiming to pursue SOTA performance on these datasets, which could be attained by using more advanced backbones. For fair comparison, the backbones of all the methods are based on the deep neural network ResNet- 50 (He et al. 2016) pretrained on ImageNet. We conduct each experiment three times and report the average Accuracy score (in %) and standard deviation. The accuracies of the comparison methods are reproduced, or quoted from JUMBOT (Fatras et al., (021). m-POT (Nguyen et al., (022] or their papers unless otherwise stated. The standard deviations for comparison methods are shown whenever available in their papers.
Tables I 2 show the proposed ELOT method significantly outperforms the comparison baselines. The bold and underlined accuracies represent the best and the second best performance. On the large-scale VisDA dataset which is one or two order of magnitude higher than the other two datasets, ELOT beats the rumner-up method P-PO by large margin of 2.73. On Office-31 dataset, ELOT performs consistently better than the comparison methods on all 6 adaptation tasks. For Office- Home dataset, ELOT also achieve higher accuracies in 12 out of 12 adaptation scenarios.
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We take closer look at the unsupervised domain adaptation methods based optimal transport. DeepJDOT (Damodaran et al. (0188) adopted Kantorovich' 's optimal transport to align the joint fea- tures/labels distribution. JUMBOT (Fatras et al. 202I) and m-POT (guyyen et al. 2022) followed DeepJDOT to align joint distributions, while using unbalanced optimal transport and partial optimal transport respectively instead of Kantorovich's s optimal transport. JUMBOT and m-POT leveraged the geometrically robust versions of OT to alleviate the influence of undesired coupling between samples, leading to better performance than DeepJDOT. ELOT consistently outperforms the above OT based methods the large-scale and medium-scale datasets including VisDA, Office-31, and Office-Home. It demonstrates the superiority of elastic optimal transport in domain alignment by throwing away the full-mass or fixed-mass constraints. The strengthen of ELOT is that it is adept at making choices on whether the pairs should be mapped or not, and how much masses should be transported, by taking the domain shift into full consideration. Moreover, ELOT relies on the geometry structure of the problem itself to transport suitable masses adaptively, providing a better choice than the fixed-mass strategy for domain adaptation.
Table I: Classification accuracy VisDA (ResNet-50) for unsupervised domain adaptation tasks, where (*) and (#) denote the results quoted from m- m-POT and JUMBOT.
Method
Accuracy
(
67.6311033
CDANGE)
70.10
ALDA (+)
71.22:10.12
ROT
66.30
DeepJDOT
68.00
JUMBOT(#)
72.50
m-POT
13.59++0.15
ELOT (ours)
76.22-+0.28
Table 2: Classification accuracy on Office-31 (ResNet-50) for unsupervised domain adaptation tasks. The results reproduced from m-POT are denoted with (o).
Method
A-W
D-W
W-+D
A-+D
D-*
W-+A
ResNet-50
60.7
DANN
82.0
96.9
99.1
79.7
68.2
67.4
82.2
92.9
ALDA
95.6
97.7
100.0
94.0
72.2
72.5
88.7
CaCo
89.7
98.4
100.0
91.7
73.1
72.8
87.6
DeepIDOT (o] 87.8=2022
97.9+0.3
99 8:10.1
88.7+.0.1
70.8=0.3
71.20.22
JUMBOT (o) 91.3=20.4 98 98.5200.2
100.0::0
89.410.3
68.820.3
70.2=0.2
86.4
99.1
ELOTT(uus)) 96.0.10.3
99.1
100.0.10
94.0.102
17.4.0.1
76.4.0.1 90.5
Table Classification accuracy on Office-Home (ResNet-50) for unsupervised domain adaptation
tasks, where (*) denotes the results quoted from m-POT.
Tas
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
S0s
34.90
50.00
53.00
37.40
41.50
46:20
38,50
31.20
60.40
$3.90
41.20
59.90
46.10
DANN(.)
$2.96
CDAN(+
59.70
77.50
70.50
57.90
ALDA(-)
54.04
74,89
77.14
61.37
70.62
72.75
60.32
51.03
76.66
67.90
35.94
$1.87
67.4
RCT(())
47.20
70.50
76.40 40
58.60
68.
70.20
56.50
45.00
75.80
69.40
$2.10
80.60
64.30
(=)
JUMBOT(+)
74,93
74.58
POTT
55.65
75.50
80.76
6A 34
74.88
76
64.46
$3.38
$0.60
74.55
59.71
83.81
34
KLOTouus)
56.79-00.2
77.4==011 82.18::0.2
70.26-
75.03:2003
0.3 78.03---0.22
66.64-20.1
53,68-0.3
81.12:00
75.39
60.26::-022
84.162:0.1
71.75
4.3 PARTIAL DOMAIN ADAPTATION
In order to further demonstrate its generality and robustness, evaluate ELOT on partial domain adaptation tasks, where the target labels are the subset of the source labels. Samples belonging to the missing classes become outliers which poses great challenge the robustness of the methods.
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Table 4: Classification aceuracy on Pffice-Homt (ResNet-50) for partial domain adaptation tasks.
Avg
ResNer-500
59.10
74.90
48.20
14.20
61,40
PADA(=)
18.70
52.20
33.80
15.80
73.30
77.30
ETN( +)
59.20
77.00
79.50
62.90
65.70
75.00
68.30 30
55.40
84.40
75.70
57.20
84,50
70:40
B+3US(-)
59.34
78.73
95.42
72.30
72.34
83.54
73.19
60.20
$5.92
79.13
63.00
85,90
75.20
77.47
JUMBIT(-)
75.60
72.89
79.79
61.95
50.78
67.38
75.96
m-X(I(+)
64.60
50.62
87.17
20.43
22.61
83.58
77.02
63.24
57.43
$1.42
08.50
87.38
77.95
ELOT(cu)))
64.5420011
86.66:00
0..4:20.2
77,87
784330
83.55:0.22
80,44
65.19.00
87.80:20.1
82.19::0.1
69.88
86, 0.1 79.19
Table 5: Classification accuracy on DomainNet (ResNet-50) for partial domain adaptation tasks.
Method
C-P
C-R
C-+
S-C S-R
S-+P
Avg
ResNet-50
41.2
60.0
42.1
45.4
39.3
49.8
46.3
DANN
27.8
36.6
29.9
25.8
29.5
32.7
30.4
CDAN
37.5
48.3
46.6
35.4
38.5
43.6
41.7
PADA
22.5
32.9
30.0
17.5
23.9
26.9
25.6
BA3US
42,9
54.7
53.8
50.4
42.7
49.7
49.0
STCPDA
69.6
60.7
67.8
ARPM
67.9
79.8
66.3
62.5
64.8
71.7
68.8
ELOT (Ours)
68.8
77.7
68.3
68.0
69.9
75.9
71.4
In order to verify that various cost functions could be used in ELOT, we use the alternative ground cost function (22) in partial domain adaptation task. Our empirical study shows both the cost func- tions in Equations 21 and 22 attain the similar performance. Therefore, we report the results using
the cost function 22 only.
We evaluate ELOT on Office-Home dataset, following BA3US (Liang et al. (0200 to select the first 25 out of 65 categories (in alphabetic order) in each domain as partial target domain. ELOT is compared with variety of non-OTTbased methods including PADA (Cao et al. 2018), ETN (Cao (O al. 2019), BA3US (Liang et al,] 2020), and OT-based methods including DeepJDOT (Damodaran
al., 2018), JUMBOT (Fatras et al. 2021). and m-POT (Nguyen et al. 2022). The results shown in 4
For fair comparison, we do not use ten crop technique testsets, following from JUMBOT (Fa- tras et al., 202T) and m-POT (Nguyen et al., 2022). The network architecture and training details similar the unsupervised domain adaptation setting the Office-Home dataset. The exper- iments are trained for 5000 iterations. The batch size is 65 which consistent with the OT-based methods. The trade-off coefficients
follows: a 0.01, 3 4.5, 1. The results of the comparison methods are quoted from m-POT (Nguyen et al. 2022) and JUMBOT (Fatras et al., 2021)
reinforess the superiority of ELOT over the comparison methods. ELOT achieves the highest accuracy on out of 12 partial domain adaptation tasks, and obtains the comparable results with the strongest competitors on the remaining tasks. Partial domain adaptation provides an ideal test bed for evaluating the robustness of the methods because the outliers belonging the missing labels could cause negative transfer. The existence of considerable number of outliers would amplify the risk of mapping undesired pairs such the samples with different classes. The experiment results suggest that ELOT provides more robust solution for domain alignment than the conventional OT methods. Furthermore, ELOT not only yields adaptive-mass transport but also uses less hyper- parameter. In contrast, both unbalanced OT (e.g., JUMBOT) and partial OT (e.g., m-POT) need extra trade-off parameters to set either the marginal penalization coefficient or a priori amount of mass to be transferred, which remains nontrivial issue.
Table reports the average classification accuracy for six partial domain adaptation tasks on the Net dataset. DomainNet is a large-scale challenging dataset with 345 classes. We follow ARPM (Gu et al.) 2024) and use the first 40 classes in alphabetical order to build the target domain in each task. ELOT significantly outperforms the comparison methods such the recent methods including STCPDA (He et al.) (024) and ARPM (Gu et al. 2024), demonstrating its superiority on the large-scale challenging dataset.

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Heatrao OT Flan (msee 0,3311, S050
0000
0.06
-10
00000
(a) c
(b) c= 0.1
Figure 1: Heatmap of optimal transport plan. The masses are allocated along diagonal blocks,
aligning labels between domains.
4.4 HEATMAP OF TRANSPORT PLAN
We go deep into the transport plan investigate how the masses are adaptively transported from source domain to target domain. For VisDA dataset, Figure plots the heatmaps of transport plans mini-batch with batch size 72. Similar to the previous work (Fatras et al. 2021 Nguyen et al. 2022), we adopt the stratified sampling to select mini-batch of source samples that each class has the same number of samples. For the target domain, a mini-batch samples are randomly selected since the target labels are unavailable in training.
Each row (or column) corresponds to one source (or target) sample. The source (or target) samples reordered into clusters by the order of labels (or pseudo labels). Therefore, the 72 x 72 transport plan matrix is partitioned into 12 x 12 blocks since there are 12 classes, except for the cases some target labels are absent due random sampling. As expected, the masses are almost allocated along the diagonal blocks, aligning labels between source and target domains. It suggests that the optimal transport in ELOT is class-aware. Also, since pseudo labels are used, it is likely that some predictions are wrong. To check which prediction is wrong, we use '+' located in the row cluster to indicate the true label (corresponding to the row cluster) of the target sample.
Figure I plots the heatmaps with O and c 0.1 respectively. It intuitively shows that the transport plan becomes sparser as the raguopy-regularized coefficient increases. shown on the top of the figures, the total masses transported from source domain to target domain are 0.61 11 and 0.7468 for e by our method with those of m-POT (Nguyen et al. (022). An interesting finding is that m-POT achieved the best accuracy when the fraction of masses is set to 0.75 (see Figure of (Nguyen et al.) method. However it is challenging to pre- define the optimal fraction of masses in m-POT. In contrast, our method can automatically learn the optimal fraction of masses be transported, leading to elegant solution for adaptive matching.
and e 0.1, respectively. We compare the optimal transport masses obtained
2.22), which is very close to 0.7468 obtained by
In summary, the heatmap intuitively verified that ELOT transports masses between domains in an adaptive and class-aware way.
The more experiment results such as discussed in Appendices and A.4.3
sensitivity analysis and T-SNE visualization
and
5 CONCLUSION
We propose a novel formulation of optimal transport along with the solid theoretical analysis to remedy the limitation of classical optimal transport. Elastic optimal transport leverages the geom- ctry structure of the problem itself to transport adaptive-mass rather than full or fixed masses. provides a flexible and powerful tool to model the problem of aligning the probability distributions adaptively by respecting the ubiquity of noises, outliers, and divergences existing in the data. Far beyond domain adaptation, we believe that elastic optimal transport opens the pathway to unlock the problems of adaptive distribution matching in a variety of areas such as artificial intelligence, data science, bioinformatics, economies, operations research, etc.
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ACKNOWLEDGMENTS
This work is partially supported by National Natural Science Foundation of China (No 62211530114) and Basic and Applied Basic Research Foundation of Guangdong (No 2023A1515010750).
REFERENCES
Martín Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein generative adversarial networks. In Proceedings of the 34th International Conference on Machine Learning (ICML), pp. 214-223,
2017.
Peizhen Bai, Filip Miljkovi'c, c, Bino John, and Haiping Lu. Interpretable bilinear attention network with domain adaptation improves drug-target prediction. Nature Machine Intelligence, 5:126 136, 2023.
Yogesh Balaji, Rama Chellappa, and Soheil Feizi. Robust optimal transport with applications in gen- erative modeling and domain adaptation. In Neural Information Processing Systems (NeurIPS), 2020.
Karsten M. Borgwardt, Arthur Gretton, Malte J. Rasch, Hans-Peter Kriegel, Bernhard Schölkopf, and Alexander J. Smola. Integrating structured biological data by kernel maximum mean discrep- ancy. In Proceedings 14th International Conference on Intelligent Systems for Molecular Biology, pp. 49-57, 2006.
Luis Caffarelli and Robert J. McCann. Free boundaries in optimal transport and Monge-Ampere obstacle problems. Annals of Mathematics, 171:673-723 730, 2010.
Zhangjie Cao, Lijia Ma, Mingsheng Long, and Jianmin Wang- Partial adversarial domain adaptation. In European Conference on Computer Vision (ECCV), volume 11212, pp. 139-155. Springer, 2018.
Zhangjie Cao, Kaichao You, Mingsheng Long, Jianmin Wang, and Qiang Yang. Learning to transfer examples for partial domain adaptation. In IEEE Conference Computer Vision and Pattern Recognition, CVPR, pp. 2985-2994. Computer Vision Foundation / IEEE, 2019.
Laetitia Chapel, Mokhtar Z. Alaya, and Gilles Gasso. Partial optimal tranport with applications on positive-unlabeled learning. In Neural Information Processing Systems (NeurIPS), 2020.
Minghao Chen, Shuai Zhao, Haifeng Liu, and Deng Cai. Adversarial-leareed loss for domain adap- tation. In Proceedings of the AAAI Conference Artificial Intelligence (AAAI), volume 34, pp- 3521-3528, 2020.
Nicolas Courty, Rémi Flamary, Amaury Habrard, and Alain Rakotomamonjy. Joint distribution optimal transportation for domain adaptation. Neural Information Processing Systems (NeurIPS), 2017a.
Nicolas Courty, Rémi Flamary, Devis Tuia, and Alain Rakotomamonjy. Optimal transport for do- main adaptation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 39(9):1853- 1865, 2017b.
Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. In Neural Infor- mation Processing Systems (NeurIPS), pp. 2290-2300. 2013.
Bharath Bhushan Damodaran, Benjamin Kellenberger, Rémi Flamary, Devis Tuia, and Nicolas Courty. Deepjdot: Deep joint distribution optimal transport for unsupervised domain adapta- tion. In Proceedings of the European Conference on Computer Vision (ECCV), pp. 447-463, 2018.
Kilian Fatras, Thibault Séjouméé, Rémi Flamary, and Nicolas Courty. Unbalanced minibatch optimal transport; applications to domain adaptation. International Conference on Machine Learning (ICML., pp. 3186-3197, 2021.
11

---

Published as a conference paper at ICLR 2026
Alessio Figalli. The optimal partial transport problem. Archive for Rational Mechanics and Analysis, 195:533-560, 2010.
Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, Mario Marchand, and Victor S. Lempitsky. Domain-adversare training of neural networks. Journal of machine learning research, 17:59:1-59:35, 2016.
Somdatta Goswami, Katiana Kontolati, Michael D. Shields, and George Em Karniadakis. Deep transfer operator learning for partial differential equations under conditional shift. Nature Ma- chine Intelligence, 4:1 141155-1164 2022.
Xiang Gu, Xi Yu, Yan Yang, Jian Sun, and Zongben Xu. Adversarial reweighting with a-power maximization for domain adaptation. Int. J. Comput. Vis., 132(10))4768-733) 4791, 2024.
Chunmei He, Xiuguang Li, Yue Xia, Jing Tang, Jie Yang, and Zhengchun Ye. Addressing the overfitting in partial domain adaptation with teafnnraining and contrastive learning. IEEE Trans. Circuits Syst. Video Technol., 34(3):1532-1544 2024.
Kaiming He, Xiangyu Zhang. Shaoqing Ren, and Jian Sun. Identity mappings in deep residual networks. In European conference on computer vision (ECCV), pp. 680-645, 2016.
Nhat Ho, XuanLong Nguyen, Mikhail Yurochkin, Hung Hai Bui, Viet Huynh, and Dinh Q. Phung. Multilevel clustering via wasserstein means. In Proceedings of the 34th International Conference on Machine Learning (ICML), pp. 1501-1509, 2017.
Jiaxing Huang, Dayan Guan, Aoran Xiao, Shijian Lu, and Ling Shao. Category contrast for un- supervised domain adaptation in visual tasks. IEEF/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2022.
L. Kantorovitch. On the translocation of masses. C. R. (Dokl.) Acad. Sei. URSS, 37:199-2000 1991201,,,, 1942.
Fabian Levihn. On the problem of optimizing through least cost per unit, when costs are negative: Implications for cost curves and the definition of economic efficiency. Energy, 1 11::1155-116 2016.
Jian Liang, Yunbo Wang, Dapeng Hu, Ran Hc, and Jiashi Feng. balanced and uncertainty-awar approach for partial domain adaptation. In European Conference Computer Vision (ECCV), volume 12356, pp. 123-140. Springer, 2020.
Matthias Liero, Alexander Mielke, and Giuseppe Savaré. Optimal new HellingerKanntrovvich Kantorovich distance between positive measures. Inventiones mathematicae, 211: 969 - 11 1117, 2017.
problems and a
Mingsheng Long, Zhangjie Cao, Jianmin Wang, and Michael I. Jordan. Conditional adversarial domain adaptation. In Neural Information Processing Systems (NeurIPS), 7a
Mingsheng Long, Han Zhu, Jianmin Wang, and Michael I. Jordan. Deep transfer learning with joint adaptation networks. In Proceedings of the 34th International Conference on Machine Learning (ICML), pp. 2208-2217, 2017b.
Jifí Matoušek and Bernd Gärtner. Understanding and using linear programming. 2007.
Gaspard Monge. Mémoire sur la théorie des déblais et des remblais. Histoire de I' l'Académie Royale des Sciences de Paris, 1781.
Khai Nguyen, Dang Nguyen, The-Anh Vu-Le, Tung Pham, and Nhat Ho. Improving mini-batch optimal transport via partial transportation. In International Conference Machine Learning (ICML), pp. 16656-16690, 2022.
Xiaocao Ouyang, Yan Yang, Wei Zhou, Yiling Zhang, Hao Wang, and Wei Huang. Citytrans: Domain-adversarial training with knowledge transfer for spatio-temporal prediction across cities. IEEE Trans. Knowl. Data Eng., 36(1)6662766)) 2024.
François- PPerre Paty and Marco Cuturi. Subspace robust wasserstein distances. In Proceedings of the 36th International Conference on Machine Learning (ICML), pp. 5072-5081, 2019.
12

---

Published as a conference paper at ICLR 2026
Xingchao Peng. Ben Usman, Neela Kaushik, Judy Hoffman, Dequan Wang, and Kate Saenko. Vis- da: The visual domain adaptation challenge. arXiv preprint arXiv:1710.06924, 2017.
Xingchao Peng, Qinxun Bai, Xide Xia, Zijun Huang. Kate Saenko, and Bo Wang. Moment matching for multi-source domain adaptation. In IEEE/CVF International Conference on Computer Vision ICCV, pp. 1406-1415. IEEE, 2019.
Kate Saenko, Brian Kulis, Mario Fritz, and Trevor Darrell. Adapting visual category models to new domains. In European conference on computer vision (ECCV), pp. 213-226. Springer, 2010.
Justin M. Solomon, Gabriel Peyré, Vladimir G. Kim, and Suvrit Sra. Entropic metric alignment for correspondence problems. ACM Transactions on Graphics (TOG), 35:1 13, 2016.
Zhongxiang Sun, Weijie Yu, Zihua Si, Jun Xu, Zhenhua Dong, Xu Chen, Hongteng Xu, and Ji- Rong Wen. Explainable legal case matching via graph optimal transport. IEEE Trans. Knowl. Data Eng., 36(6):2461-2473 2024.
Chris Xing Tian, Haoliang Li, Yufei Wang, and Shiqi Wang. Privacy-preserving constrained domain generalization via gradient alignment. IEEE Trans. Knowl. Data Eng., 36(5):2142-2150, 2024.
Laurens Van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. Journal of machine learning research, 9111), 2008.
Hemanth Venkateswara, Jose Eusebio, Shayok Chakraborty, and Sethuraman Panchanathan. Deep hashing network for unsupervised domain adaptation. Proceedings of the IEEE conference on computer vision and pattern recognition (CVPR), pp. 5018-50222 2017.
Cédric Villani. Topics in optimal transportation. 2003.
Garrett Wilson and Diane J. Cook. A survey of unsupervised deep domain adaptation. ACM Trans- actions on Intelligent Systems and Technology (TIST), 11(5):51:1-51:: 2020.
Jingjing Xu, Hao Zhou, Chun Gan, Zaixiang Zheng, and Lei Li. Vocabulary learning via opti- mal transport for neural machine translation. In Proceedings the 59th Annual Meeting of the Association for Computational Linguistics (ACL), pp. 7361-7373, 2021.
Delvin Ce Zhang and Hady W. Lauw. Topic modeling on document networks with dirichlet optimal transport barycenter. IEEE Trans. Knowl, Data Eng.. 36(3):1328-14) 1340, 2024.
A
APPENDIX
A.1 THE PROOFS OF THEOREMS
A.1.1 PROOF OF THEOREM II
Proof We get rid of the inequality constraints of the elastic optimal transport problem defined in Equation by adding the slack variables:
Im b= p - 'n +a-aa- =
(23)
(24)
where E IRT and b E 1RTx1 are slack variables. Denote the transported mass by
ΣΣ Tij. Note that s is alaptively learned instead of user-defined or fixed. Based on the above
=1
two equations, we have
+ ||b||
||
+ ||lli |||||
Denote the extended transport plan by
7 =
(25)
a s
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Next, show that both the elastic optimal transport problem and its reformulation have the equiv- alent optimal transport plan.
5" = arg min
sell(a,))
m Ceftes o (Ialll + |[6|[1 2s}
arg min sell(p.P) i-1
m
arg min {ElT(p.0)
Coge σ (1ell + 11llls)
m Σ
arg min FEll(p,P) i=1
Ciyies + Const
f=1
m
ΣΣ Cay
arg min SEll(P.))
suggests that the optimal transport plan 5 for the reformulated problem is controlled by the cost matrix C rather than the augmented one C. Assume To is the matrix 5 with the last row and column removed. Since 5 E [[,, D), we have % E (I,( according Equations 23 24 and 25 Following the last equation of the above derivations, we prove the equivalence of the optimal transport plans:
arg min
ΣΣ
Cing Cin-- -
(E.) ==
Meanwhile, according the above derivations, have the result that the difference between the optimal transport distances is a constant:
W D) W) = + (1111l)) Const
which be ignored in optimization.
A.1.2 PROOF THEOREM 2
Proof 2 Let xh be the optimizer for the elastic optimal transport problem. We first derive two propositions:
For the positive cost Cij have Rij 0.
b For the negative cost Cij have
E or Σ, Vly
(26)
We prove the proposition b by reduction to absurdity, and the proposition @ can be proved in similar way. Assume that both equations Equation 26 do not hold for C.j Let's set
Am = minkue us - 30
One can increase the total transport mass by Am such that at least one of the equations in Equation 26 holds. Then the objective of the elastic optimal transport problem will decrease by -Cij Am. Note that -Cij Am > O, therefore we obtain a better optimal transport plan, which contradicts with the premise that x" is the optimizer.
Then we divide the elastic optimal transport problem in Equation [3 into two sub-problems ac- cording the sign of ground cost, and show the equivalence between the elastic optimal transport
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Algorithm The ELOT Algorithm
Input: Source data {==)]== with label set 3 {yr target data { trade-off parameters a and 8, batch-size and maximum epoch T.
Output: The learned extractor y and classifier f.
1: Randomly initialize the classifier f-
2: for t= 1 T do
Randomly sample mini-batch samples from source data ((),) and target data {z,]z=1- re- spectively.
4: Fix extractor and classifier f-
5: Compute the ground cost matrix C.
Obtain the optimal transport plan y by solving the ELOT problem defined in Equation transport plan 7/
8: Compute the total loss.
9: Update extractor and classifier f via SGD algorithm.
10: end for
problem and its corresponding constrained worst transport problem as follows:
>" arg min
Cigrss
1
arg min
arg min
Cis
(27)
(ET'(u.))
1
arg min vEl'((u.))
Cijytss
i,
arg max
7EL'c(a.)) 2.5
where the third and fourth equations follow from the propositions @ and ( respectively.
A.2 THE ELOT ALGORITHM
Algorithm [ adopts a mini-batch sampling strategy to solve the elastic optimal transport problem, enabling scalable large datasets. In each mini-batch, iteratively updates the transport plan 7. and the feature extractor g and the classifier f. First, when the extractor g and the classifier are fixed, the problem defined in Equation 21 reduces the standard elastic optimal transport. According to Theorem [ the optimal transport plan can be obtained by solving the reformulation of the elastic optimal transport problem. We adopt the linear programming or the Sinkhorn- Knopp algorithm (Cuturi (013) to solve the reformulated problem. Second, fixing the transport plan the extractor 9 and the classifier f can be updated with the stochastic gradient descent algorithm (SGD).
A.3 RELATIONS WITH EXISTING OPTIMAL TRANSPORT
We clarify that ELOT is fundamentally different from unbalanced optimal transport (UOT) and partial optimal transport (POT), highlighting the novelty of ELOT.
A.3.1 ELOT vs UOT
We would like clarify that EL.TT is not a limiting case of unbalanced optimal transport. As the penalty coefficients approach zero, i.e., - 0 and 72 + 0. allowing relaxed inequalities, UOT will lead to the degenerate solution with zero transport plan due to the nonnegative constraint of costs. Even though UOT is allowed accept the mixed-sign cost (which means the nonnegative constraint on costs is violated), it will lead to the solution with infinite mass flowing on negative-com entries + 7
due to the overly relaxed inequalities 71 Po(yIm
in the case of тι 0 and T2 + 0. In contrast, ELOT depends on the combination of the marginal inequality constraints and mixed-sign cost to achieve adaptive-mass transport.
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Therefore, ELOT is certainly not the limiting case of unbalanced OT. Instead, ELOT is fundamen- tally distinctive from unbalanced optimal transport. First, the problem formulations are different. Second, the mass transportation mechanism is different, indicated the theoretical analysis. ELOT can achieve adaptive-mass transport, relying on the combination of the marginal inequality con- straints and mixed-sign cost, while UOT depends users to specify the hyper-parameters of soft constraints.
A.3.2 ELOT Vs POT
We go deep into the relation and difference between ELOT and partial optimal transport, and high- light the distinctive advantages ELOT against POT.
The continuous formulations of optimal transport are used in this subsection. Suppose X, Z C R are domains in Euclidean space. Let c(x, 2) be the bounded continuous cost function which tells X to location plan (z, z) measures the amount of mass transferred from location æ to location 2. Without loss generalization, let's assume that both μ and v are probability measures such that fx p(x)dz
how much it costs to move unit of mass from location
Z. The transport
Iz (==)d= here for simplicity.
Partial optimal transport (Caffarelli & McCann 2010) is then formulated as
J
min SEn's (11M) fxxx do(o,a))m
Jxxz ct (x))))d7(0,
(28)
where the cost function et (ar, 2) non-negative, and m is the user-specified mass budget. The set of admissible transport plans is denoted by
Tg( 7 2 0/ / yl,, 2(2, s u(xx) / / (a, 22, s u(a)
(29)
Caffarelli and McCann introduced Lagrange multiplier Am 2 conjugate to the fixed-mass con- straint Jxx2 d'( 2) = m and reformulated the POT problem as
min (prg(p))) xxz
-x) Am]dr(us,
(30)
Likewise, elastic optimal transport defined in continuous formulation can be reformulated as
min TE > xxz [[((E))) Ae) 2e (x
/
(31)
where the cost function is mixed-sign and A (-c(2, 2)]. is worth noticing that Am corresponds which is user-defined, while Aa is selff-dttrminned by the cost structure. The reformulation provides insights into how elastic optimal transport is essentially different with partial optimal transport.
First, the fundamental difference is that ELOT preserves adaptive-mase mass while POT transports fixed- mass. For the POT problem, the goal of introducing the Lagrange multiplier Am is remove the fixed-mass constraint fxx2 (x) = m, making it easier to solve the POT problem. However, this does not eliminate the limitation that it needs to specify the mass budget m (or equivalently find the appropriate value of the Lagrange multiplier Xm). which is challenging because usually have no prior knowledge on how much mass should be transported. For ELOT,, we have no such fixed-mass constraint, thus there is no need to introduce an extra Lagrange multiplier.
Second, ELOT determines the mass according to the task structure, while POT relies on the user to specify the mass budget. The reformulation gives some insights into how AOT attains adaptive- mass transport. According to Caffarelli & McCann (2010), for each mass there is unique À corresponding to the m, and m increases continuously as is increased. For POT, the specific value of the Lagrange multiplier Am is irrelevant to the task structure itself. On the contrary, ELOT self- determines the total mass, relying on the native structure the ground costs. Specifically, a definite A in ELOT results in definite mass, while larger X₂ leads to the more to be transported.
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Last but not least, ELOT provides much larger capacity than POT by exploring the whole spectrum of mass instead of the fixed-mane mass. Consider the optimal transport problems with parameterized cost functions Paty & Cuturi 2019). Denote the parameterized cost function by ca(2, 2) where
the learnable parameter. The elastic optimal transport with parameterized cost function can be formulated as
Sx co(z, 2)),2)
min regg(e.))
(32)
Likewise, it can be reformulated
I Ixxz [co(()))) Aco) deo dr((0)z)
min ser (n:0)
(33)
where Aep max,,= max,,=l-co(t; 2)]. The total transport mass of ELOT could increase continuously from 0 to as Ae increases. Therefore, ELOT could attain the adaptive-mass ranged continuously across the whole spectrum of mass, thus offering a much larger capacity to search for the learn- able parameters. In contrast, POT sticks the user-specified mass budget no matter how the cost functions are varying, which is likely to be trapped into local optimums.
In summary, the distinctive advantages of ELOT against POT lie in three aspects: a) adaptive- mass preserving, b) self-determining according to task structure, c) larger capacity by exploring the spectrum of mass. Since the classical optimal transport with full mass constraints can be viewed the special case of partial optimal transport by setting m = 1, the claims hold for the classical optimal transport too.
A.4
MORE EXPERIEMENT RESULTS
We report the implementation details, as well as more empirical evaluations.
A.4.1 IMPLEMENTATION DETAILS
For a fair comparison. we basically follow the settings of JUMBOT (Fatras et al. 2021) and m- POT Nguuenn al. 2022) for the experiment setups.
Networks. Similar to Fatras et al. 2021 Nguyen et al.) 2022), we use ResNet-50 (He et al. 2016)
pretrained on ImageNet our extractor and one fully connected (FC) layer our classifier for all
datasets.
Sampling. We adopt class-balanced sampling
domain where each class has the same number samples in a mini-batch, and random sampling used on target domain since labels are unavailable in training, which the same with (Fatras et 2021 Nguyen et 2022).
Data Augmentation. Following (Fatras et al. 2021 Nguyen et al. 2022). we use the same data
pre-processing for all three datasets. The images first resized into 256 256 and then randomly cropped with size of 224 x 224. Random translation/mirror and normalization are also applied for training. For testing, we adopt the ten-crop technique (Fatras et al. 2021 Nguyen al. 2022 for
robust results, Note that these settings are commonly used and the same as previous works (Fatras
al. 202T Nguyen al. 2022 for fair comparison.
Training Details. Following the settings of (Fatras et al. 2021 Nguyen et al. 2022). we adopt SGD optimizer with 0.9 momentum and 5e weight decay for training, and the learning rates are set with the same strategy as (Ganin et al. 2016). Note that the learning rate of the classifier is set to be 10 times that of the extractor as the classifier is trained from scratch.
Syyper-parametery For all three datasets, the weight of feature-wise cost is set to 0.01. The weight of label-wise cost 3 is set to 2.3, 9, and 4.5 for VisDA, Office-31, and Office-Home re- spectively. The entropy-regularized coefficient is set to 0.1, 1, and 1 for VisDA, Office-31, and Office-Homm respectively. The batch size is set to 72, 62, and 65 for VisDA, Office-31, and Office- Home respectively, which are consistent with JUMBOT (Fatras al. 2021 and m-POT (Nguyen ctt al. 2022). The experiments are trained for 2000, 5000, and 5000 iterations for VisDA, Office-31, and OfficeHHome respectively.
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Published as a conference paper at ICLR 2026
A.4.2 SENSITIVITY ANALYSIS
We report the sensitivity analysis with a, 3 and c on VisDA dataset. Figure shows the average ac- curacy and standard deviation varying with the parameters. It verifies the robustness of the proposed method on the hyper-parameters For comparison, the best performance of m-POT (gguyen et al.) 2022) is also plotted as base line.
The coefficient a is used to adjust the weight of the feature-wise cost. The optimal range for a is between 0.004 and 0.012. ELOT yields the best accuracy VisDA dataset when 0.011 So fix a = 0.01 for the VisDA dataset the experiments.
we can see that Note that is the weight to control the impact of label-wise cost. From Figure 2 ELOT attains the satisfactory performance when 3 is greater than 2, and exhibits the high accuracies in wide range of B.
The rugraiy-regularized coefficient e is to control the sparsity of transport plan. It shows that cernopy-regulrizze term helps getting better accuracies when increases from 0 0.1. The curacy of ELOT reaches its maximum around 0.1. However when c too large (say = 1), the accuracy falls. Nevertheless when varies in wide spectrum from 0 to 0.5, ELOT beats the second best method m-POT (Nguyen et al. 2022).
6.9
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de ndoe t/ea olonoris di whes insons
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Figure 2: Sensitivity analysis with parameters o, 3, and e.
A.4.3 T-SNE VISUALIZATION
We plot the T-SNE (Van der Maaten & Hinton 2008) embeddings of four methods including Deep- JDOT (Damodaran et al. 2018), JUMBOT (Fatras et al. 2021), m-POT (Nguyen et al. 2022). and ELOT on VisDA dataset, in order to intuitively demonstrate the superiority of ELOT.
From Figure 3 we can see that the clusters of ELOT are more separate than those of the comparison methods. Mcanwhile, the source samples and target samples with the same class label are relatively well aligned by ELOT. It suggests that proposed method is able to better maintain the discrim- inative information in the course of domain alignment. simultaneously aligns the domains and categories.
A.5 RELATED WORK AND DISCUSSION
In real applications, the training and test data usually do not follow the independent and identical- ly distributed assumption. Domain adaptation is the critical task in real-world machine learning applications since distribution discrepancy is ubiquitous the data.
Most existing works on domain adaptation can be roughly classified into two categories: discrepancy-basee methods and adversarial-learning based methods. The discrepancy-bant meth- ods explicitly minimized the domain distance using discrepancy metrics such as optimal transport distance or Maximum Mean Discrepancy (MMD) (Borgwardt et al., 2006). The domain adaptation
methods based on optimal transport include OTDA (Courty al. 201 7b), ROT (Balaji et al. 2020
DeepJDOT (Damodaran et al. 2018). J) JUMBOT (Fatras et al. (021). I. POP (Nguyen et al.
2022
etc. The typical methods based on MMD are JAN (Long et al. 2017b), PPDG (Tian et al. 2024
DeepONet (Goswami et al. 2022), to name a few. The eeeesarial-eaamnnng based methods aim to learn donain-invariant representations via adversarial training. The typical methods include DAN-
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Published as a conference paper at ICLR 2026
Deep]DOT
JUMBOT
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Figure 3:
T-SNE embeddings of 2000 samples of VisDA for different methods. Samples are colored based on their class labels.
N (Ganin al. 2016), CDAN (Long al. 2017 ALDA (Chen al. 2020). DrugBAN (Bai al.
2023). CityTrans (Ouyang et al. 2024). etc. Please refer to the survey paper (Wilson & Cook 2020) for more details.
We demonstrate the effectiveness of the instantiation of ELOT in domain adaptation tasks. However, widely applicable various branches of machine learning due the fact that the training and text data in real applications usually do not follow the independent and identically distributed assumption.
Furthermore, although we focus on machine learning applications here, it is worth noting the appli- cations in wide variety areas beyond artificial intelligence can opt for elastic optimal transport whenever the adaptive-mass preserving is preferred for the alignment of probability distributions. This is indeed the case since full-mass or fixed mass preserving is too restrictive many scenarios.
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Reviews:
[{"review_id": "EYZbygAhQi", "reviewer": "Reviewer_sLP5", "summary": "This paper proposes Elastic Optimal Transport (ELOT), a novel formulation using marginal inequalities and a mixed-sign ground cost. The authors present two key theoretical results: an equivalence to an equality-constrained problem whose plan is invariant to a parameter σ (Thm. 1), and a \"mass transport mechanism\" (Thm. 2) showing that mass flows only on negative-cost entries, which enables automatic outlier filtering. Empirically, the method demonstrates consistent improvements over domain adaptation baselines on VisDA, Office-31, and Office-Home.", "questions": "**Q1**: Under what specific assumptions on the cost C or marginals (μ,ν) do ELOT and UOT provably yield different couplings?\n\n**Q2**: Does elastic solver can be considered in entropy-based regularized settings? Given that negative costs can cause instability in entropic solvers (e.g., Sinkhorn's exp(−C/ε)), what stabilization techniques are used? Does the plan invariance (Thm. 1) hold exactly under this entropic regularization?\n\n**Q3**: The paper defines the resulted W and calls it an “optimal transport distance,” but fails to discuss its metric properties as identity, symmetry. Does the resulted solution value actually provide some sort of Wasserstein distance?  I the cost matrix C has no negative entries, the zero plan (γ=0) is feasible and optimal, yielding W(μ,ν)=0 by construction. This seems to violate the identity property of a distance and is consistent with Theorem 2, which states mass only flows on negative-cost entries.\n\nminors:\nAdd runtime/memory measurements or complexity analysis to justify statements.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "In general the method is interesting and promising, avoiding limitations of the current OT solvers. Theoretically justified formulation with consistent gains in UDA and partial DA using a uniform backbone and setup.", "weaknesses": "While the formulation is clean and intuitive, my primary concern is the unclear formal relationship to Unbalanced Optimal Transport (UOT). The authors mentioned UOT in the background but did not consider these methods in more detail. Although UOT is a well-known approach to a similar problem to that stated in the paper, a more detailed analysis of its relationship to existing solvers is necessary. The  The authors assert a connection, but do not analyse it sufficiently. Experimental comparison to UOT is completely ignored. Why? To strengthen the contribution, please consider addressing the following questions:", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JPquo8iwDD", "reviewer": "Reviewer_y8JA", "summary": "The paper introduces Elastic Optimal Transport (ELOT), a novel formulation of optimal transport that relaxes the full-mass or fixed-mass constraints present in classical OT, partial OT, and unbalanced OT. ELOT allows for adaptive-mass preservation and supports mixed-sign cost matrices, making it more flexible for real-world applications where noise, outliers, or distribution shifts are present. The authors provide theoretical analysis, an equivalent reformulation solvable with standard OT solvers, and apply ELOT to unsupervised and partial domain adaptation tasks. Experimental results on standard benchmarks (VisDA, Office-31, Office-Home) demonstrate that ELOT outperforms several state-of-the-art OT and non-OT baselines.", "questions": "1.  Potential Degeneracy When Transport Mass is Zero: \nThe formulation of ELOT allows for the possibility that the total transported mass s=0, which occurs when all ground costs are positive. While this may be desirable in some outlier-rich scenarios, it could also lead to degenerate solutions in applications where some degree of alignment is necessary.\nSuggestion: The authors should discuss this property and its implications for the general applicability of ELOT. A discussion on whether this poses a limitation in practice and if potential modifications or safeguards could mitigate this issue would strengthen the paper.\n2. Ambiguity in Theoretical Relationship Between ELOT and Unbalanced Optimal Transport: \nIn the current formulation, ELOT is presented as distinct from unbalanced OT. However, in unbalanced OT, the marginal divergence penalties act as soft constraints. when τ1,τ2→0 the penalties vanish, allowing relaxed inequalities γ1_m ≤μ,γ^T 〖1〗_n≤ν —which closely resembles the ELOT formulation. The paper currently does not analyze this limiting behavior, making it unclear whether ELOT can be regarded as a limiting case of unbalanced OT.\nSuggestion:\nTo further strengthen the theoretical contribution, the authors may consider discussing the limiting behavior of unbalanced OT as the penalty coefficients approach zero, and clarifying how ELOT relates to this case. This would help position ELOT more clearly within the broader OT framework.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1.\tNovel Formulation: The idea of adaptive-mass transport is well-motivated and addresses a clear limitation of existing OT methods. The introduction of a mixed-sign cost matrix enhances its applicability to real-world problems.\n2.\tStrong Empirical Performance: The paper provides extensive and convincing experiments on multiple domain adaptation benchmarks, showing consistent and significant improvements over a range of strong baselines.\n3.\tTheoretical-Practical Bridge: The work offers valuable theoretical insights and a practical reformulation that enables the use of standard OT solvers, facilitating wider adoption.", "weaknesses": "Proofs of Theorems 1 and 2 Lack Rigor: \nThe proofs of Theorems 1 and 2, while intuitively appealing, lack rigor in their current form. In particular, the argument that two transport plans are equivalent because they minimize the same cost ignores the issue of non-uniqueness of solutions in ELOT—unlike in classical OT where uniqueness often holds under mild conditions. The authors should revisit the proofs to account for potential multiple optima. The theorems could be rephrased to state that there exists an optimal plan for the reformulated problem that matches an optimal plan of the original ELOT, rather than implying equality of all such plans.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "5HfiLN9jjv", "reviewer": "Reviewer_B4zg", "summary": "The paper proposes Elastic Optimal Transport (ELOT), an optimal transport (OT) formulation designed to address practical limitations of classical OT, partial OT, and unbalanced OT. In standard OT, all mass from the source distribution must be transported to the target distribution. Partial OT transports only a fixed budget $s$ of mass, which the user must choose. Unbalanced OT relaxes the marginal constraints but requires tuning divergence penalties. ELOT instead allows the optimal plan to decide how much mass, depending only on the cost. ELOT explicitly allows the cost matrix $C$ to include negative entries, and thus adaptively determines the transported mass. The authors apply ELOT to domain adaptation and compare it with OT-based approaches.", "questions": "- Could the authors explain how to define the cost for a general task, as the commonly utilized costs are often non-negative in most tasks?\n- Can the private class data be detected and omitted by ELOT in partial DA?\n- The compared methods are not soTA. Could the authors include more recent SOTA? For example, MOT[1], ARPM[2], both are OT-based.\n\n[1] Mot: Masked optimal transport for partial domain adaptation\n\n[2]Adversarial Reweighting with α-Power Maximization for Domain Adaptation", "limitations": "", "rating": 4, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The motivation is strong. The authors aim to adaptively match datapoints without hand-tuning transport mass, which is really useful for practical applications if realized.\n- The authors consider a signed cost to realize the adaptive amount of transported mass, which is reasonable.\n- The computation is transformed into an OT formulation, and thus can be solved by existing tools.\n- The paper is easy to read.", "weaknesses": "- Novelty relative to partial OT / unbalanced/robust OT needs to be strengthened.  Partial OT already allows transporting only part of the mass. In classical formulations, one can introduce a Lagrange multiplier that effectively shifts the cost matrix and induces an “automatic” choice of how much mass to transport. Prior work (the paper cites Caffarelli & McCann, 2010) can create a situation where negative effective cost encourages matching only for “good pairs,” and the transported mass adapts as a function of that shift. \nELOT is described as more general because it directly allows mixed-sign costs and only imposes marginal $\\le$ constraints, and it claims to “automatically finds the optimal mass to be transferred without setting a priori budget.” However, ELOT can still look like “partial OT with a specific cost shaping and slack embedding.” The paper should give a crisper mathematical argument for why ELOT is fundamentally different and not just a repackaging.\n\n- Calling $W(\\mu,\\nu)$ a ‘distance’ may be misleading. Because $C$ can be non-symmetric and can contain negative values, the induced objective $W(\\mu,\\nu)$ is not guaranteed to be a metric or even a divergence: non-negativity, symmetry, and triangle inequality can fail.\nThe paper uses “optimal transport distance” language, which is standard OT terminology, but here it risks overstating the geometric meaning. This should be toned down or clarified. Meanwhile, the commonly used costs, e.g., L1 L2, are not suitable. How could ELOT be applied to partial transport under such widely utilized costs?\n\n- The paper aims to avoid hand-tuning transport mass, a hyperparameter. However, the costs in the domain adaptation task and the partial domain adaptation task involve hyperparameters, to which the performance is sensitive, as shown in the experiments. So I feel like the authors avoid tuning one parameter but instead tune several other parameters. For DA, the parameters are set to the values in the compared methods. But for other applications, how these parameters are determined.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "siOmja2pxH", "reviewer": "Reviewer_qzrK", "summary": "This paper considers the mathematical aspects of optimal transport (OT), where a variant of existing OT formulation is proposed. Specifically, the authors focus on the strict equality or inequality constraints on transport mass, and try to propose a new mechanism to achieve a relaxation on the strict hard constraints, i.e., adaptive mass. Empirical validations are conducted on standard domain adaptation datasets compared with other OT counterparts.", "questions": "Q1. The elastic optimal transport (ELOT) is not rigorously defined and analyzed. Specifically, the notation $\\mathbb{R}_{±}$ in Eq. (6) for the cost function is not defined. If it implies that the cost function could be both positive and negative, the ELOT could not be well-defined, e.g., does it still satisfy the metric property? A rigorous and complete theoretical analysis is necessary to ensure the validity of ELOT.\n\nQ2. The ELOT seems to be an incremental work of partial OT. Specifically, the basic conclusions and technical details (i.e., Thm. 1 and its proofs) of ELOT are almost the same as partial OT [27, Prop. 1], while the proofs in line 54-56 manuscript seem to be problematic, i.e., there are no rigorous proofs to ensure that the submatrix of augmented OT plan is equivalent to the OT plan of ELOT.\n\nQ3. Moreover, the formulation of ELOT in Eq. (6)-(7) still cannot ensure the adaptive learning of mass value $s$. Specifically, the technique based on dummy variables still induces the parameter $\\sigma$ in Eq. (8), which is basically equivalent to adjusting $s$.\n\nQ4. In the claim below Eq. (8), the authors set $\\sigma$ as 0 in the experiment. However, in such a formulation, the ELOT problem seems to be exactly the original Kantorovich OT problem, as the augmented cost matrix is block-diagonal where the $n\\times m$ block and $1\\times 1$ block are independent. Based on this problem, it would be hard to understand the performance improvement of ELOT in experiments.\n\nQ5. The comparison experiment with SOTA methods is not significant. This experiment is conducted on 3 DA datasets, where the performance is significantly lower than the SOTA methods in recent years. Some further verifications on the latest proposed datasets and advanced baselines are necessary, e.g., DomainNet dataset and ViT backbone.\n\nQ6. The quality of OT plan learned by ELOT is not sufficiently validated. For example, if ELOT can adaptively adjust the mass of plan, a quantitative/qualitative comparison between the OT plans of different OT formulations should be considered.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "S1. A new OT formulation with elastic constraints and theoretical analysis.\n\nS2. The organization is clear and the writing is easy to follow.\n\nS3. The experimental performance is superior to other standard OT variants.", "weaknesses": "W1. The rigor of the claim and theoretical analysis should be improved.\n\nW2. The empirical validation could be improved, where the SOTA comparison methods should be considered.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)