Optimal Transport for Domain Adaptation through Gaussian Mixture Models
Authors: Eduardo Fernandes Montesuma
Fred Maurice NGOLE MBOULA
Antoine Souloumiac
Venue: Accepted by TMLR ()

Abstract: Machine learning systems operate under the assumption that training and test data are sampled from a fixed probability distribution. However, this assumptions is rarely verified in practice, as the conditions upon which data was acquired are likely to change. In this context, the adaptation of the unsupervised domain requires minimal access to the data of the new conditions for learning models robust to changes in the data distribution. Optimal transport is a theoretically grounded tool for analyzing changes in distribution, especially as it allows the mapping between domains. However, these methods are usually computationally expensive as their complexity scales cubically with the number of samples. In this work, we explore optimal transport between Gaussian Mixture Models (GMMs), which is conveniently written in terms of the components of source and target GMMs. We experiment with 9 benchmarks, with a total of $85$ adaptation tasks, showing that our methods are more efficient than previous shallow domain adaptation methods, and they scale well with number of samples $n$ and dimensions $d$

Published in Transactions on Machine Learning Research (01/2025)
Optimal Transport for Domain Adaptation through Gaussian Mixture Models
Eduardo Fernandes Montesuma
edumfern@@mail.com com
Université Paris-Saclay CEA, List, F-91120 Paloiseau France
Fred Ngolè Mboula
fred.ngple-mbonlooecn.ff
Université Paris-Saclay, CEA, List, F-91120 Palaiscau France
Antoine Souloumiac
antoinee.ooulouiiaccccac.
Université Paris-Saclay, CEA, List, F-91120 Palaiseau France
Reviewed on OpenReview: /ttp::///pperevttte net/ forum? id- DCAeXuLen
Abstract
Machine learning systems operate under the assumption that training and test data are sampled from a fixed probability distribution. However, this assumptions is rarely verified in practice, as the conditions upon which data was acquired are likely to change. this context, the adaptation of the unsupervised domain requires minimal access the data of the new conditions for learning models robust to changes in the data distribution. Optimal transport is thooretically grounded tool for analyzing changes in distribution, espocially as it allows the mapping between domains. However, these methods are usually computa- tionally expensive as their complexity scales cubically with the number of samples. In this work, we explore optimal transport between Gaussian Mixture Models (GMMs), which is conveniently written in terms of the components of source and target GMMs. We experi- ment with benchmarks, with a total of 85 adaptation tasks, showing that our methods are more efficient than previous shallow domain adaptation methods, and they scale well with mumber of samples and dimensions d.
1 Introduction
Supervised machine learning models trained with significant amounts of labeled data, constituting training set. The theory of generalization (Redko et. al.) 2020), provides theoretical background that guarantees accurate predictions on unseen samples from the same distribution. Nonetheless, these models are often forcod to predict on related, but different data samples (Quinonero-Canden et al.] 2008 This distinction modeled by a shift in the probability distributions generating the data (Sugiyama et al, 2007), which motivates the field of transfer learning (Pan Yangl (209)) and more specifically the problem of Unsupervised Domain Adaptation (UDA) in which models are adapted from labeled source domain, towards unlabeled target domain, following different distributions]]
In this context, Optimal Transport (OT) Villani et al. 2009: Peyré et al. 2019 is powerful, theoretically grounded framework for comparing and manipulating probability distributions [Montesuma et al. 2024a This framework works by computing a transportation strategy, that moves one probability distribution into the other at least effort. Based on this core idea, different methods have been proposed for [UDA such as Courty ct al. (2017): El Hamri et al. (2022): Chuang al. (2023) and Struckmeier al. 2023).
Our code available https://github. http:a/aa.ccmeeec///caaeaaaaccee siddaardd/gnn-ottdde

---

Published in Transactions on Machine Learning Research (01/2025)
However, this methodology faces few challenges. For instance, OT maps computed between discrete distributions are only defined for samples in the training set. Extrapolating these maps to new samples is the subject of intense research Perrot et al. (2016); Seguy et al. (2017). A possible workaround consists of using Gaussian approximations (Flamary et al. 2019; Struckmeier ct al.) 2023). While this approach effectively defines mapping over the whole ambient space, its hypothesis do not reflect the possible sub- populations within the data, which are common in classification problems.
A natural solution to tackle multi-modality in data distributions is using Gaussian Mixture Models (GMMs) A further advantage of this approach is considering the recently proposed GMM OT (GMMOT) by Delon & Desolneux (2020), which establishes an efficient, discrete problem between the components in the GMMs Furthermore, recent works have established the effectiveness of this idea for multi-source domain adapta-
tion Montesuma et al. 2024b), notably through the use of mixture- Wasserstein barycenters.
Nonctheless, some questions on the use of GMMs for UDA remain open. For instance, Delon & Desolneux (2020) propose different mapping strategies between GMMs but these either fail to map one GMM inlo
the other, are subject to randomness when sampling transportation maps between GMM components.
Furthermore, if we label the components of [MMs]] it is natural to propagate these labels, in the sense of Redko et al. (2019) towards the target domain. This paper tackles these two questions.
Summary of contributions. In this paper, we propose 2 new strategies for UDA based on GMMs First, we use basic rules of probability theory for propagating the labels of source domain GMM towards the target domain GMM We do so, by interpreting the [MMOT] plan as the joint probability of source and target map samples from the source domain into the target domain based the GMMOT plan. For point in the source domain, our strategy consists of first estimating the component, in the source MMM most likely to have generaled the sample. We then transport this point to components in the target domain, while assigning importance weights based on the GMMOT plan.
GMM components. Second,
Paper organization. The rest of this paper is organized as follows. Section works on OT for [UDA] Section 3 covers the preliminaries on OT and GMMOT theory. Section 0 covers our methodological contributions. Section details our experiments and discussion on UDA Finally, section D concludes this paper.
2 presents a few related
Notation. We use uppercase letters P and Q to denote probability distributions, and Ps and Pr to denote source and target domain distributions. More generally, we use Pr to denote probabilities. For instance, Pr(Y = y = x) denotes the conditional probability of label Y given a feature vector X =xx. Let P be a distribution over feature vectors. We denote samples from P as xP). We reserve yP) for categorical labels (i.e., 1,.,, for na classes), and (P) for its one-hot encoding.
2 Related Works
Optimal transport for domain adaptation. Optimal transport has been extensively employed for the design of algorithms (Courty et al.] 2017], well as analyzing the domain adaptation problem (Redko) et a.. 2017 The key idea of this method is to use the Kantorovich formulation to acquire a matching, known as transport plan, between source and target domain distributions. This matching defines a map between points in the source domain, towards the target domain, called barycentric mapping. Based on this idea, different methods have proposed improvements. For instance, Perrot et al. (2016) proposed learning linear and kernelized extensions of the barycentric map through convex optimization. El Hamri et al. (2022) uses clustcring for learning matching with additional structural dependencics. Flamary et al. (2019) uses optimal transport between Gaussian distributions for estimating an alfine mapping between source and target domains. More recently, Chuang et al. (2023) proposed method that leverages kernel density estimation for defining a new optimal transport problem based on information maximization.
Gaussian-mixture based optimal transport. An optimal transport problem involving Gaussian mix- tures was initially proposed by Chen et al. (2018), in which a linear program between the components of the two mixtures is solved. This setting was further studied by Delon & Desolnnux (2020), who proved an interesting connection to a continuous optimal transport, when the transport plan is constrained to the set of Gaussian mixtures. Based the framework of Delon & Desolneux (2020), Montesuma et. al. (024bb
2

---

Published in Transactions on Machine Learning Research (01/2025)
proposed the extension of multi-source domain adaptation algorithms of Montesuma & Mbbula 2021a b
and Montesuma et. al. [2023] However, these authors focused on performing adaptation through Wasserstein barycenters. Although based on the same framework, our work focuses on mapping samples and propagoting labels of Gaussian mixtures, especially for singlo-source domain adaptation.
3 Theoretical Foundations
3.1 Optimal Transport
Founded by Monge [1811, optimal transport field of matheraticc concerned with transporting mass least effort. Let X be set and P(X) the set of probability distributions on X. For P,Q P(X), the Monge formulation of the optimal transport problem is,
T arginf / c(a, (x:)) T:T:P=Q fx
(1)
where TyP denotes the pushforward distribution (Santambrogio 2015 Problem 1.1) of P by the map T, and c: X' X + R, called the ground-cost, denotes the cost of sending x to position T(x).
Although equation 1 is formal description of the optimal transport problem, the constraint T,P- poses technical difficulties. An alternative description was proposed by Kanlorovichi (1942). in lerins of an optinal transport plan
arginf / c(21, (2)),11)) TET(P.)) xxxx
7
(2)
where I((, Q) the set of joint distributions with marginals P and Q. This formulation is simpler to analyze because the constraint 7 E I(P.Q) Q) is linear with respect to the optimization variable ?,
When dX is a metric space, it is possible to define distance on P(X) in terms of d. Let E (0,+
and c(.,-) )a, One then has the so-called a- --sssersseii distance;
(P.Q))) inf / c(a1, 20)d vET(P..) Jxxx
In the following, we do optimal transport on Euclidean spaces, i.e., X' Rd. In this case, it is natural to use d(x1, x2) = ||x] X2|12- Furthermore, we set a
Equations and are linear programs, where the optimization variable the joint distribution 7. In the following, we discuss 3 particular cases where optimal transport either has closed for, or is approximated by finite problem, thus tractable by computer.
Empirical Casc. If we have samples {x! (P)
(x)2) p with probabilitics pæ and qj respoctively, we li=1 and fx)
can make empirical approximations for P and Q
11 P(x)
Σ -xo(x-x(f)) i=1
(4)
The approximation in equation 4 is at the core of discrete optimal transport Peyré el. al. (2019) If we plug equation into equation the optimal transport problem bocomes computable, i.c., it turns into lincar programming problem with variables,
7" arg min ()
Ro Cij,
where G2 c(x(") , 10)). As a linear program, one should keep in mind that solving equation has a complexity One²² logm). This complexity can be reduced by regularizing the problem in terms of the OT plan entropy,
m
2" = arg min (ET(P.Q)
my Ci +
(lo my - 1),
(6)
1=1
3

---

Published in Transactions on Machine Learning Research (01/2025)
Σ
Wkynk2
E(P)
1 1
(P)
T (xc
(xx
(PP)
Vis
(a) Empirical OT
(b) Gaussian OT
(c) GMM-OT
Figure 1: Comparison of different ways of solving OT
In empirical OT P and Q approximated
non-paramelurcally through their samples. In Gaussian OT P and Q are Gaussian distributions, and the
mapping between these distributions is affine. In GMM-OT, P and Q are assumed to be GMMs and an OT plan between components, w, defines an OT plan between samples, ?.
which can be solved through the celebrated Sinkhorn algorithm (Cuturi 2013 A solution can be found with
complexity O(Tn²), where I is the number of iterations of the algorithm. From the Kantorovich formulation, we can recover a correspondence between distributions through the barycentric map,
(x(x
min xEX
m)c(x, (
Gaussian Case. When P p= N(((),, (PP) (resp., Q), equation has closed and affine form Takatsu 22011) (x) Ax + b, where,
A (E(P))= (21))))))
and b (0) Ap
(8)
and the Wasserstein distance takes the form,
W2(P. 11 11 + TT(E(P) + ( (
(9)
In contrast with empirical optimal transport, under the Gaussian hypothesis, computing the Wasserstein distance and mapping between P and Q has sample-free complexity. Indeed, the complexity of equations and is dominated by computing the square-root of the covariance matrix with complexity O(ds).
Furthermore, in high dimensions and when a only a few data points are available, estimating full covariance matrices is challenging. these cases, it useful to assume axis alisnal Gaussians, ie., is a diagonal matrix with diagonal clements Σn o? In this case, equations [ and 9 can be further simplified,
A f b 1==1 (9) Ap and W2(P.Q)² = (1 1 + - (
(10)
Gaussian Mixture Case. A Gaussian mixture corresponds to P Ex
(P)
Pk. where Ps
T
N( (P) (P)).. As in Delon & Desolneux (2020), we denote by GMMa(K) the of distributions PE PE P(Rd) written mixture of at most K components. In this framework Delon & Desolneux (2020) explores the optimal transport problem 2 under the constraint that y is GMM as well, i.c., T(P. Q) M2alo.. This formulation is interesting because it is equivalent to a discrete and hierarchical problem Delon &
Desolneux 2020 Proposition 4) in terms of the GMMs components
Kp Kg
w GMMOT(P,O) Q) = arg min
Whikk Wo(Ps, Qu.)
(11)
(P)( (()
A
k
We call this problem GMMOT This latter equation is an hierarchical optimal transport problem, i.e., a problem that involves itself an inner optimal transport. As in the previous cases, we have a notion of distance related equation of different. strategies for solving
M)2(P.Q)2 E Wky W, Q2,)². We show an overview OT under different assumptions, in Figure
4

---

Published in Transactions on Machine Learning Research (01/2025)
Remark 3.1. (Computational Complerity) The overall complexity of the GMMOT in equation 11 is O(K3 DoK K). However, one should keep in mand that the ground-cost matris must be computed beforchand, as given by equation g which involves a complerity that scales with the dimension of the ambient space, due the matrir inversions and square-roots. The complexity of these operations is O(K2d)) in general. However, assuming dingonal covariance matrices, the compulational complexxly is drastically reduced to O(K2d), i.e., the complerity of computing K² Euclidean distances between d dimensional vectors. We refer readers to our oppendix for a running time analysis of our method.
Remark 3.2. (Parometer Estimation) Besides the computational advantage, using diagonal covoriance matrices yields simpler estimation problem for GMMS This is pivotal in Domain Adaptation (DA)
the target domain likely does not have enough samples for the accurate estimation of complete covoriance
matrices. Furthermore, data is oftentimes high-dimensional, as feature spaces commonly incolve thousands of fealures. This is sharp contrust. with previous works in GMMOT such as Delon & Desolneus (2020)
and Chen et al. (2018), which involved a few dimensions. In our empirical validation (c.f., section 5.3) we show that using diagonal covariances gields better adaptation performance high dimensions.
3.2 Learning Theory and Domain Adaptation
In this paper, we deal with DA for classification. This latter problem can be formalized mathematically, as the learning of function h + V. from a feature space X' (e.g., R4) to a label space, y = {1,.... ne) through samples of a probability distribution. As reviewed by RRekoo al.) (2020), from the point of view of the Empirical Risk Minimization
probability, there multiple ways of formalizing this problem. Here,
[ERM) framework of Vappik (2013). For a probability distribution P, loss function L x 3 + R,, and family of classifiers H. one may define a notion of disagreement between pairs h, h' E H.
Rp(h, h²) = x-p _ h(())))
(12)
Equation 12 defines the risk of h with respect N. Given a ground-truth labeling function ho : X + y classification can be phrased in terms of minimizing the risk of with respect the ground-truth ho, i.e.,
h- = =arg min R (h, keH
Henceforth, we adopt (h) = Rr(h, in short. This formalization equates the problem of learning classifier with optimization problem. Nonetheless, in practice one does not have access to a priori knowledge from P nor hg. In a more realistic scenario, one has samples fx!?) v/ I where x! and )/ ho (x()))) Based these samples, one may estimate the risk empirically, by resorting to the approximation in equation 4
itd P.
1 n Σ [(((x(P)), ).1; (.v()))) (P)) and h arg min R
Rp(M)
(13)
In machine learning literature, the true risk P. (h) is called generalization error, i.e., the error that classifier h makes on samples from the distribution P. In contrast, algorithms usually minimize the training error, Rp),), i.e., the error of h on the particular examples available during training.
A key limitation of the presented theory is its assumption that data originates from a single probability distribution P. As discussed by (Quinoncro-Candee et al. (2008), this seldom happens in practice. For instance, in fault diagnosis, process conditions influence the statistical properties of measured signals (Mon- tesumna et al. 2022). As result, generalization must be carried to new, related probability distribution. This problem is known in the literature as DA a sub-field within transfer learning.
As discussed by Pan & Yang 2009),, in transfer learning, a domain is a pair D = (X' P) of a feature space and a probability distribution over X. Likewise, a task a pair T= ho)ho) of a label space and a ground- truth labeling function. Transfer learning characterized by different source and target domains and tasks, i.c., (Ds. Ts. Dr Tr), where at least one element from the source different from the target. In this work, we assume different domains Ds F Dr but the same label space, Yss Yr (1,... ,Ne).
5

---

Published in Transactions on Machine Learning Research (01/2025)
In this paper, we deal primarily with distributional shift. In this case, we assume X' Xr = RA, that source and target domains are characterized by different probability distributions Ps * Pr. Further- more, we place ourselves in the unsupervised DA setting, that is, we assume labeled source domain data, 1j=1-
(Ps)in {x, (x)()), yy, Ti=1 and unlabeled target domain data (x) 13 Our goal is to use these samples to learn
classifier h that works well on the target domain, I.e., that achieves small target risk R.yy.
From the point of view of DA the quality and similarity of source domain data plays a prominent role, since supervision comes from this distribution. For instance, if source domain data contains noisy labels, these can be transferred to the target domain, leading to poor results. Likewise, the success of domain adaptation is correlated with the distance, in distribution, between these two domains. We refer readers (Ben-David et al. 2010 Redko al. 2017) for further discussion.
Optimal Transport for Domain Adaptation was proposed by Courty et al. 2017 and primarily tries
to match samples from P to those of Q based on the empirical OT problem in equation 5 After acquiring
7", the authors propose mapping samples from P towards those of Q via the baryncetric map. This strategy
(x("") u1"" (1)) yr where T.- is defined by equation 7 Note, here, that under T the source domain points carry their labels, to the target domain. This operation is valid, as long as the conditionals (s(Y = Pr(Y)T(())) which is restrictive, but reasonable under the covariate shift hypothesis. A few problems plague this stratogy. First, TT. is only defined on the support of P. The mapping of new points has been extensively studied in the literature (Perrot et al. 2016 Seguy et al.) 2017 2023). Second, it is desirable to have mappings with additional structure with respect the
effectively constitutes new dataset {T+,-xx ),
Chuang et al. classos in DA This problem is partially solved by considering special regularization schemes, covered in Courty et al. (2017). Third, this method is not scalable with respect the number of samples n, due its prohibitive complexity O(n logn). In this paper, we offer a solution for the aforementioned problems through the GMM OT framework of Delon & Desolneux (2020).
4 Domain Adaptation via Optimal Transport between Gaussian Mixtures
Algorithm 1: Fitting procedure for GMMs.
Algorithm 2: Fitting procedure for labeled GMMs.
function (M() Kp)
1 function ConditionalEM Y(P),
for it 1,.... .Riter do
11 Expectation Step TE
2 cpc + Kp/nei
(P) N(x) (P) 1
(P) x((P)
3
for / 1, 1,... ng do
Gi,, En Th N ( 1p
// Samples from y-th class
(P) >(0))
(Ps) + -x v, (P)
// EM on conditionals y): ) =
11 Maximization Step
n&= Σm= Gi,Ai
((P)), (P))) (Po) EM(X cpc);
Mk + E Gax!!! (P)
(P)
5
6
(Pw) + fone_hot(f hot cpc
n
(P)
Ei= (x) 14 (x:
(P)
(P) (P)
(P)
yr
1/ Concatenates all parameters ((P)) = 111k (P2
k nik
(P3) Inc 1k=1
7
(P) + Tlk
(P) (1 k
(P&) ope n
J Jk=l Jysi
7
k
) = ( (p)) cepe
9
return ( (P))
(Pa)
cpc
(P) cpe
(P₂)
11 return / (P) k=1 Ey= T
k=
y=1
(P)) (.P)) (P);
In this section, we develop new tools for DA betwowee MeriitooenGGMMM GMMs As we discussed our preliminaries section, we are particularly interested in DA for classification. In this context, data is naturally multi-modal,
forDA
which justifies the mixture modeling. As we previously defined in Section 3
GMM is mixture model
6

---

Published in Transactions on Machine Learning Research (01/2025)
with parameters (uk)) s(P) (P) T ===1- These parameters can be determined through maximum likelihood:
(ng (()) 1
Σ log p(x P)), Du x
arg max
(14)
where P = Σ k== K Ps. A practical approach for optimizing equation 14 was proposed by Dempster et al. (1977, and is known as Expectation Maximization (EM) We show a pseudo-code for this strategy in Algorithm We refer readers to (Bishop M. 2006 Chapter 9) for further details on GMMs
In our approach, we need to define labels for the components of GMMs in the transportation problem. We do so through an heuristic, that is, we model each Pg = P(XYY = y) through a GMM for y = -, Rg.
As result, fit GMM to the data ( (x; (
im: [n my² using epc Kr ///nc components. Here, we
conveniently choose P as a multiple of ne, for ensuring that epe is an integer. We present in Algorithm 2 pseudo-code for this strategy. Furthermore, we create a one-hot encoded vector of component labels, denoted (P) = if the k-th component comes from the y-th class, and 0 otherwise. We v(P) E (An,)*, where Vkyy
interpret the vector v/2) as the conditional probability Pr(Y k))
In the following, we discuss two strategies for domain adaptation. The first, based on label propagation, leverages the optimal transport plan between components to define pseudo-labels for the components of the target domain GMM The second, based on mapping estimation, leverages the hierarchical nature of the GMMOT problem for defining a map between source and target domain. These methods are summarized in Algorithms 3 and
Algorithm 3: Pseudo-label target GMM.
Algorithm 4: Tweight.
function
1 function Tweight (x (Ps), (Ps)) Ps. w)
Y(Ps), ((Pr)
// Using Equation
P < CondtionalEM(((()))) Y(Pe));
k1 + estimate components(x*) (Ps), Ps);
2
Q + EM(X((T));
3
for j such that Whit 2 r do
4
w + GMMOT(Ps, Pr);
// Using eq. 8 or 10
(PP)) = /
x; x
(Ps) Thi (x('s));
return ((Pr);
(Ps) t (Pss)
Dkz
6
Wk2 + Wki
(P) (P)
7 return (Wkar Xb2 Vkz 27
Limitations. In this work, we assumee that data is multi-modal, which is often the case in classification. Furthermore, assume that it can be modeled accurately through [MMs i.e., data is well separated into clusters. While we can expect estimation and inference to be difficult in high-dimensions, we show in our experiments that our method outperform previous basclines based on empirical OT Finally, we show in our appendix that our methods are robust to the over eslimation of GMM components.
4.1 Label Propagation and Maximum a Posterior Estimation
Recall that, in equation 11 the result of the GMMOT problem is a transportation plan w, between compo-
nents of the GMMs P and Q. As result, has as marginals the probability vectors (PP) and =!Q), Fur-
thermore, in the original probabilistic view of GMMs Th (Ps) Pr(Ks k1), whereas Th
(Pr) = Pr(Kr k2).
Given this interpretation, it is natural to see the transportation plan as Why,k, = Pr(Ks = k Kr = kz). Note that we can estimate the probability Pr(Y]Kr) using the law of total probabilities,
Ks
Pr(Y Kg = k2) (=)= = Σ
Pr = ki, Kr = -22) = Kr =k))
7

---

Published in Transactions on Machine Learning Research (01/2025)
12.6
125-
12.5-
10.0
1000
10.0
7.5
7.5
75-
8
25
25-
25-
D-
00-
-2.6-
-25-
-25-
-5
-5
-5
0
-5.0
-2.5
0.0
25
(a)
(b)
(c)
(d)
Figure Illustration of the label propagation strategy. (a) Shows the source and target GMMs
GMMs are represented through their means (stars) and covariance matrices (ellipses). The labels of com- ponents are represented through colors. Since the target domain is unlabeled, their means are gray colored. (b) Shows the OT plan between components, w. allows us to propagate the labels of source domain GMM towards the target. (c) Shows the obtained target GMM Finally, (d) shows the MAP classifier.
Here, assuming that Y and K2 are conditionally independent given K1, we have,
Pr k2)
Σ Pr(Y]Ks ka)Pr(Ks lo K2 = K2),
This assumption plays the same role as covariate shift hypothesis (Sugiyama et al. 2007 in conventional DA works. On an intuitive level, our assumption explicit the fact that Ky is redundant with respect to Ks. The conditional Pr(Ks]KT) (r)p(()) can be computed through the optimal transport plan, ie..
K
or, D(Pr) w (Ps)
(T) lbg
(Ps) Wky he ky
(15)
(P Fka
(QT)
k1=1
where the division should be understood elementwise. Equation 15 is known in the OT literature as label propagation (Redko al., 2019), and, as we discussed in the related works section, has been used extensively in the context of empirical OT Given the estimated labels (PPr) we effectively defined a labeled GMM for
the target domain. Based on this GMM we can perform Maximum A Posteriori (MAP) cstimation to define a classifier in the target domain,
KT
Pr(Y = = = = x, K, k)Pr(K x), hMAP(x) = arg max Pr(Y = y|X = x) = arg max g. Me
y. me
1=
(Pr) (x) T
arg max
(Pr)
(16)
Kr (Pr) EK= T Pr.
where, from the second to the third equality we assumed that Y and K are conditionally independent given X. This hypothesis is intuitive, as classes and components are representing the same structures within the data points. We show an illustration of these idcas in Figure 2
4.2 Mapping Estimation
In this section, we propose new mapping ostimation technique between two GMMs Ps and Pr. As discussed in (Delon & Desolneux] 2020 Section 6.3), this problem is not straightforward. Indeed, since 7 is GMM it cannot be, in general, written as (Id, P) Thus, these authors consider two strategies: mean map and random map. First, samples mapped T Tmean (x(ss) (x(())))(()()))) (xpeee)((((()))) [x(r)], but it may end up not actually matching Ps with Pr (see, e.g., DDelnn & Desolneux 2020 Section 6.3)).
8

---

Published in Transactions on Machine Learning Research (01/202))
14-
14-
Q
12
12-
12-
10
10
322
8-
6-
6-
4-
0
10
2.5
5.0
7.5
10.0
2
11
21
21
(a) Trean-
(b) Trand-
(c) Tweight-
Figure 3: Mapping estimation using GMMOT In (a) and (b), show the Teean and Trand strategies of Delon & Desolneux (2020), respectively. In (c), show our strategy Tweight. Our mapping reduces randomness by first estimating the component k1 most likely to have generated x x(P), Then, we weight the
importance of transported samples by W22
Second, Delon & Desolneux (2020) defines
N ((x(Pe)(ub))
(Ps) x((P)))
T (x = Thi T2 (x( (x()s with probability Pky (xx(Ps) = Whi.k Σn PaN (x((ss) pk 2
(Ps) (PPs)
which has the advantage of matching Ps with Pr. However, as noted by Delon Desolneux (2000) Trand usually leads to irregular mappings due the sampling procedure of indices (k1, k2) as shown in (Delon & Desolneux 2020. Figure 7).
We put forth third strategy for mapping Ps into Pr. Our intuition is twofold. First, can increase the regularity of T Trand by estimating the component kn that most likely originated x( that is,
(Ps) Ps.((x(Ps) "E (Ps) Eks E Th (x
h arg max Pr(Ks = k Xs x(Ps))
(17)
k=1.-.,K
Second, we map x (Ps) into the components of Pr. Note that, since the marginals (Ps)) and (Pr) are different, the optimal transport plan may split the mass of Ps.A, into several Pr,ag As result, we produce (Tk, (x (x(*s)))
( ka i.e, we map (Ps) to all components Pro,, such that Whikk 2 T 2 0. In principle, one may choose T = and filter only the components that are not matched with Ps Third, we further weight. the importance of generated samples, by using Wky as sample weights. At the end, generate weighted dataset {(Wk), Tky (x. (Ps))) vi (P)))] where is the total amount of samples generated. We call our overall mapping T Feigght. for which pseudo-code is presented in 4
The mapping we just defined has a few interesting properties. First, it is piece-wise affine map, as each Thy is affine. This property contrast with the Gaussian hypothesis, which defines an affine map between Ps and Pr- Second, with respect. the transportation of samples, our mapping strategy is naturally group-spurse, in the sense of Courty al. (2017). This claim comes from the fact that samples in Ps are transported based the Gaussian component they belong to. Third, our mapping defined the whole support of the GMM Ps. In contrast, empirical OT only defined on the samples {x[ (Ps) p of Ps.
5 Experiments
In this section, we present our experiments with DA We consider a wide range of benchmarks computer vision and fault diagnosis. In the first case, we consider Caltech-Offic (Gong al. 2012), ImageCLEF (Ca- puto al. 2014), Office31 (Saenko et al. 2010), Office-Home (Venkateswara al. 2017), (MNIST, USPS.
9

---

Published in Transactions on Machine Learning Research (01/2025)
SVHN) (Seguy et al.,
2017 and VisDA Peng et al. 2017 In the second case, we consider the CWRU CSTR (Pilario & Cao 2017 Montesuma et al. 2022 and TE process benchmarks (Montesuma et al.) 2024c). Further details on these benchmarks are available in the appendix. In our experiments, an adaptation task is a pair (5, of a source domain S and a target domain T. To summarize our experimentation, there are in total 9 benchmarks, and 85 domain adaptation lasks. Our experiments with Cross-Domain Fault Diagnosis (CDFD) are available in our appendix.
For computer vision benchmarks, follow previous research (El Hamri et al., 2022 Chuang et al. 2023 and pre-train ResNet (He et al. 2016) networks the source domains. We then use the encoder branch as feature extractor, and perform shallow DA on the extracted features. These feature serve as the basis for each domain adaplation algorithm. With the exception of GMM-OTDAMAP performance on the target domain is based on the generalization of 1-layer neural network trained with transformed data. For GMM- OTDA MAP we use the MAPsstraten described in equation ( For the MNIST, USPS, SVHN benchmark, we follow Seguy et al. (2017) obtain comparable results. For M + U and U + M we downsize MNIST to the resolution of USPS, i.e., we downscale images to 16). For - S, we upscale MNIST images to match the resolution of SVHN, i.e. (32, 32), then we use features extracted from the last layer of a LeNet5.
We refer readers to Seguy et al. 2017 and Struckmeier et al. 2023 for further details on these benchmarks.
Our main point of comparison is with other OT based DA methods. We compare our GMM-OTDA strategies with other OT methods for DA namely, we consider the OT for DA (OTDA) (OTDA]sttraegy of Courty et al. (2017) (Exact and Sinkhorn), the linear mapping estimation of Flamary et al. (2019), and the InfoOT strategy of Chhang et al. (2023) (barycentric and conditional mappings). For the scalability experiment using the MNIST, USPS, SVHN benchmark, we consider the large scale OT methods of [Seguy et al. (0117), denoted
as Alg. 1 and 2. Furthermore, for completeness, we consider the Linearly Alignable Optimal Transport (LOTT) strategy of Struckmeier et al. 2023).
5.1 Scalability with respect d
In this section, our goal is to evaluate how our method scales with the data dimensionality d. To do so, we evaluate methods based on their performance on visual adaptation benchmarks. The goal is to classify images into categories, based on 2048 dime-dimansoonal vectors from ResNets (He al. 2016) fine-tuned on the
source domain of each adaptation task. We summarize our results in Figure The detailed results may be
found in the appendix, i.e., Table 4
Over Caltech-Offit and ImageCLEF, our methods outperform other state-of-the-ar methods. For Office 31 and Office-Home, Info-OT, and OTDAaffen proved to be more effect than our methods, but ours still ranks second. For Office 31, the density estimation strategy of Info-OT, proves effective in finding a better map between the domains. For Home, the baseline is already one of the best performing methods. An affine transformation is therefore sufficient for an effective adaptation.
Nonetheless, our method surpasses empirical OT over all tested benchmarks, especially preventing negative transfer in the Office-Home benchmark. Likewiso, approximating class conditional distributions P through GMMs proves effective over empirical OT Indeed, our method improves over HOT-DA of Hamri et al. 2022), which is based on empirical distributions.
These experiments prove that our method can effectively perform UDA
between high-dimensional distri- butions. Note that, for these benchnarks, we use 2048-dimensional vectors, which is by far the largest dimensionality values considered in this study.
5.2 Scalability with resepct n
In this section, we use the MNIST, USPS, SVHN benchmark. Our goal is to evaluate how our method scales with the number of samples n. As we discussed in Remark 3.1 a major advantage of the GMM formulation is reducing OT complexity from O(n logn) to O(K3 log K), i.e, we replace number of samples
htttp:://engineerring htt.://eww...ccceeee////aaaaaaaa
10

---

Published in Transactions on Machine Learning Research (01/2023)
Ofirel-Offre 1I
ImrreCCLE
Offre:::
Otfer Home
non
heom
OTE
OTDAutio
OTD
OTDA
0IDN
GTM
Figure 4: Average adaptation performance over visual domain adaptation benchmarks.
by number of components, which are orders of magnitude inferior. To give a practical comparison, while MNIST has 6 x 10t samples, represent its probability distribution through GMM with K 10a components. This modeling choice improves scalability, especially when the components of GMMs have diagonal covariance matrices.
With respect performance, table shows that GMM-OTDA has a similar performance to LaOT iee, per- formance degrades on M + U, but increases on U + M and M + S. Curiously, this corresponds a more complex dataset (e.g., SVHN).
the case where simpler dataset (e.g., MNIST) is transferred The performance similarity is not surprising, since LaOT and GMM-OTDA work under similar principles. However, the GMM modeling again, proves superior to the Gaussian hypothesis, are able to improve performance on U M and M + S tasks.
Algorithm
M +
U
M
Baseline
73.47
36.97
54.33
OTDA EMD
57.75
52.46
OTDA sink
68.75
57.35
Alg. of Seguy et al. (2017) with H
68.84
57.55
58.87
Alg. of Seguy et al.
(2017) with l2.
67.80
57.47
60.56
Alg. + 2 of Seguy et (207)) with H
77.92
60.02
61.11
Alg. 1 + of Seguy al.
2017 with lz.
72.61
60.50
62.88
LaOT
72.57
62.28
60.36
GMM-OTDA (ours)
71.83
63.11
87.19
Table 1: Large scale OT experiment. We consider the adaptation between 3 digit recognition benchmarks, namely, USPS, MNIST and SVHN, Overall, GMM-OTDA largely outperforms other methods on harder adaptation tasks, i..., U M and M S.
5.3 VisDA-C Benchmark
We experiment with the VisDA benchmark (Peng et al. 2017). a large scale DA dataset containing 152397 and 55388 source and target domain samples. As in the previous benchmarks, we pre-train the feature extractor using source domain data, then proceed to perform adaptation over the extracted features. This experiment stresses the scalability of our strategy comparison with empirical OT methods, since solving OT problem over this benchmark would lead to linear program with ns NT 8.44 x 10 variables. More dramatically, for running the barycentric map this benchmark one would need to store leading to floating point coefficients, that is, approximately 270.11 GB of memory.
To cope with the sheer volume of data, we run empirical OT methods on sub-sample of ns nr = 15000 samples. Parametric versions of OT methods, such as OTDAaffine and GMM-OTDA are run with the full datasots, which illustrates the advantage of having compact representation for distributions. Besides, we explore how these methods improve performance over different feature extractors. Hence, besides ResNet 50 and 101, we also consider ViT-16-b (Dosovitskiy et al. 2021 Our results are shown in table 2
11

---

Published in Transactions on Machine Learning Research (01/2025)
A Algorithm
ResNet 50
ResNett101
ViT-b-16
Source-Only
47.93
53.90
56.70
OTDAEMD
53.69 (A + 5.76)
57.42 (A 3.552)
63.25 (A + 6.55)
OTDA
53.02 (A 55.09)
10.54 (A 43.36)
66.75 (A 10.05)
OTDA
7.46 -0.47.77
11.82 (A 42.08)
6.75 (A 19.95)
OTDAAAine-DiD
51.41 (A 4.448)
56.91 (A +3.01)
59.94 (A + 3.24)
HOTDA
47.51 (A - 0.42)
47.64 (A - 6.26)
62.55 (A + 5.85)
GMM-OTDAT
58.35 (A + 10.42)
56.57
(A 4.677)
74.10 (A + 17.40)
GMM-OTDAMAP
57.36 (A + 9.433
58.77 (.87)
74.44 (A + 17.74)
Table 2: Comparison of domain adaptation performance different feature extractors pre-trained with source domain data. We report the classification accuracy (in %) and the difference A over the source-only baseline. Our methods GMM-OTDA T and MAP consistently outperform other OT-based methods.
Magel ooroo peats
Teppt poiss
Mappsi scace poids
Tups pials
Maped oxpro jiclo
Deigt pobds
(a) OTDA
(b) OTDA Sink
(c) O
Soxr poirs
Togs priclo
Mazed
Trapt wents
(d) GMM-OTDAMAP
(e) GMM-OTDAT
visualization OT map-based DA techniques on the VisDA-C benchmark with ViT-16-b features. Colors represent different classes. In (d), stars represent the components of source and target obtained through the conditional EM and the EM algorithm respectively.
Figure
SNF
GMMs
From table 2 note that GMM-OTDA methods consistently outperform other OT-based methods. Fur- thermore, we ablate on the choice of using diagonal covariances, by comparing the performance of estimating
an OT map between Gaussian measures with full (equation {) and diagonal covariances (equation 10 Hence,
using diagonal covariances provide a regularizing effect, improving the estimation of an OT map between Gaussian measures when the full-covariances are singular.
Finally, we analyze how the different mapping strategies match source and target domain measures. We summarize this analysis in Figure 5 where show the t[Stochastic Neighbor Embeddings (SNE) of the concatenation of mapped source and target domain data. In contrast with OTDAEMD, OTDAsinkhorn and GMM-OTDAy: OTDAaffine does not manage to match source and target data, mainly due the simplicity of the Gaussian assumption. Furthermore, GMM-OTDA7 manages to map source domain data in a way that does not mixes the classes (for instance, compare Figure 5 (a) with (e)). The label propagation approach is also discriminative of target domain classes, as is evidenced Figure 5 (d). These considcrations explain the superior performance of GMM-OTDAz with ViT-16-b features. We provide similar analysis for the CWRU benchmark in the appendix.
12

---

Published in Transactions on Machine Learning Research (01/2025)
6 Conclusion
In this paper, we consider the GMMOT
framework of Delon & Desolneux (2020)
a candidate for UDA
Based on probability and on theory, devise 2 new effective strategies for UDA The label propagation interprets the OT plan between GMM
components as joint probability distribution over source-target component pairs. This modeling choice allows us to predict the label of target GMM components through a label propagation equation similar to Redko et al. (2019). Furthermore, we propose a mapping strategy that transports samples from the same component together through an affine map, which has 2 advantages. First, it enforees group sparsity (Courty et al.) 2017)- Sccond, has an analytical form in terms of GMM parameters. We show through a series of 85 [UDA] lasks that our methods outperform, are compelitive with the state-of-the-art. in shallow domain adaptation, while being scalable with both number of samples n, and number of dimensions d. Our work further confirms previous studies the intersection of GMMs and [UAA such as Montesuma et al. (2024b), showing that the GMMOT is powerful candidate for shallow domain adaptation.
References
Shai Ben-David, John Blitzor, Koby Crammer, Alex Kulcsza, Fernando Percira, and Jennifer Wortman Vaughan. A theory of learning from different domains. Machine learning, 79(1):151-175, 2010.
Christopher Bishop M. Pattern Recognition and Machine Learning. Information Science and Statistics. Springer, New York, NY, edition, August 2006.
Barbara Caputo, Henning Müller, Jesus Martinez-Gomez, Mauricio Villegas, Burak Acar, Novi Patricia, Neda Marvasti, Suzan Üsküdarh, Roberto Paredes, Miguel Cazorla, et al. Imageclef 2014: Overview and analysis of the results. In Information Access Evaluation. Multilinguality, Multimodality,, and Interaction: 5th International Conference of the CLEF Iniliaaive, CLEF 2014, Sheffield, UK, Seplember 15-18, 2014- Proceedings 5, pp. 192-211, Springer, 2014.
Yongxin Chen, Tryphon T Georgiou, and Allen Tannenbaum. Optimal transport for gaussian mixture models. IEEE Access, 7:6269-6277 2018.
Ching-Yao Yao Chuang, Stefanie Jegelka, and David Alvarez-Melis. Infoot: Information maximizing optimal transport. In International Conference Machine Learning, pp. 1728-6242 PMLR, 2023.
Nicolas Courty, Rémi Flamary, Devis Tuia, and Alain Rakotomamonjy. Optimal transport for domain adaptation. IEEE Transactions on Pottern Analysis and Machine Intelligence, 39()):183311855, 2017. doi: 10.1 10.1109/TPAMM. 2T.11..........1111111
Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. Advances in neural information processing systems, 26, 2013.
Bharath Bhushan Damodaran, Benjamin Kellenberger, Rémi Flamary, Devis Tuia, and Nicolas Courty. Deepjdot: Deep joint distribution optimal transport for unsupervised domain adaptation. In Proceedings of the European conference on computer vision (ECCV), pp. 447-463, 2018
Julie Delon and Agnes Desolneux. A wasserstein-typee distance in the space of gaussian mixture models. SIAM Journal on Imaging Sciences, 13(2):936-977) 2020.
Arthur P Dempster, Nan M Laird, and Donald B Rubin. Maximum likelihood from incomplete data via the em algorithm. Journal of the royal stotistical society: series B (methodological), 39(1):1-22, 1977.
Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A largo-scale hicrarchical image database. In 2009 IEEE conference on computer vision and pottern recognition, pp. 248-255. Icce, 2009.
13

---

Published in Transactions on Machine Learning Research (01/2025)
Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Un- terthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszkoreit, and Neil Houlsby, An image is worth 16x16 words: Transformers for image recognition at scale. In International Conference on Learning Representations, 2021. URL http://:/ppaameiisss..////oooooa net//forunniddiicoaaanam
Mourad El Hamri, Youncs Bennani, and Issam Falih. Hicrarchical optimal transport for unsupervised domain adaptation. Machine Learning, (11(11):4159 4182, 2022.
Rémi Flamary, Karim Lounici, and André Ferrari. Concentration bounds for linear monge mapping estima- tion and optimal transport domain adaptation. arXiv preprint 1011511905.1015, 2019.
Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, Mario March, and Victor Lempitsky. Domain- anveraaadversarial training neural networks. Journal of machine learning research, 17(5)): 35, 2016.
Muhammad Ghifary, W Bastiaan Kleijn, and Mengjie Zhang. Domain adaptive neural networks for object recognition. In PRICAI 2014: Trends in Artificial Intelligence: 13th Pacific Rim International Conference on Artificial Intelligence, Gold Coust, QLD, Australia, December 1-5, 2014. Proceedings 13, pp. 898 904. Springer, 2014.
Boqing Gong, Yuan Shi, Fei Sha, and Kristen Grauman. Geodesic flow kernel for unsupervised domain adaptation. In 2012 IEFE conference computer vision and pattern recognition, pp. 2066-2074 IEEE, 2012.
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770-778, 2016.
L Kantorovich. On the transfer of masses (in russian). In Doklady Akademii Nauk, volume 37, pp. 227-229, 1942.
Gaspard Monge.
Mémoire sur la théorie des déblais et des remblais. Histoire de l'Académie Royale des Sciences de Paris, 1781.
Eduardo Fernandes Montesuma and Fred Maurice Ngole Mboula. Wasserstein barycenter transport for acoustic adaptation. In ICASSP 2021 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp- 8405-3409, May 2021a. doi: 1000///CAASS3302222200..1191
Eduardo Fernandes Montesuma and Fred Ngolè Mboula. Wasserstein barycenter for multi-source domain adaptation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 16785-16793, 16793, June 2021b.
Eduardo Fernandes Montesuma, Michela Mulas, Francesco Corona, and Fred-Maurice Ngole Mboula, Cross- domain fault diagnosis through optimal transport for cstr process. PrAC--,apesOnnLine, 55(7):946 951, 2022.
Eduardo Fernandes Montesuma, Fred Ngolè Mboula, and Antoine Souloumiac. Multi-source domain adap- (ation through dataset dictionary learning in wasserstein space. In 26th European Conference on Arlificial Intelligence (ECAI), pp. 1739 1746. IOS Press, 2023.
Eduardo Fernandes Montesuma, Fred Maurice Ngole Mboula, and Antoine Souloumiac. Recent advances in optimal transport for machine lcarning. IEEE Transactions Pattern Analysis and Machine Intelligence, 2024a.
Eduardo Fernandes Montesuma, Fred Ngolè Mboula, and Anloine Souloumiac. Lighter, beller, faster multi- source domain adaptation with gaussian mixture models and optimal transport. In Albert Bifet, Jesse Davis, Tomas Krilavičius, Moelis Kull, Eirini Ntoutsi, and Indré Zliobaité (ods.), Machine Learning and Knowledge Discovery in Databases. Research Track, pp. 21 21-38, Cham, 2024b. Springer Nature Switzerland. ISBN 978-3-031--0365-2.
14

---

Published in Transactions on Machine Learning Research (01/2025)
Eduardo Fernandes Montesuma, Michela Mulas, Fred Ngolè Mboula, Francesco Corona, and Antoine Souloumiac. Benchmarking domain adaptation for chemical processes on the tennessee eastman process. In Machine Learning for Chemistry and Chemical Engineering (ML4CCE) Workshop, 2024c.
Sinno Jialin Pan and Qiang Yang. A survey transfer learning. IEEE Transactions knowledge and data engineering, 22(10):1345-3355 2009.
Xingchao Peng, Ben Usman, Neela Kaushik, Judy Hoffman, Dequan Wang, and Kate Saenko. Visda: The visual domain adaptation challenge. arXiv preprint ar Xr1i::1710. 06924, 2017.
Xingchao Peng, Qinxun Bai, Xide Xia, Zijun Huang, Kate Sacnko, and Bo Wang. Moment matching for multi-source domain adaptation. In Proceedings of the IEEE/CVP international conference on computer vision, pp. 1406-1415 2019.
Michaël Perrot, Nicolas Courty, Rémi Flamary, and Amaury Habrard. Mapping estimation for discrete optimal transport. Advances in Newral Information Processing Systems, 29, 2016.
Gabriel Peyré, Marco Cuturi, et al. Computational optimal transport: With applications to data science. Foundations and Trendsl in Machine Learning, (5-66)):355 607, 2019.
Karl Ezra Pilario and Yi Cao. Process incipient fault detection using canonical variate analysis. In 2017 23rd International Conference on Automation and Computing (TCAC), pp. 1-6. IEFE, 2017.
Joaquin Cuinonero-Candela. Masashi Sugiyama, Anton Schwaighofer, and Neil D Lawrence. Dolasel shift in machine learning. Mit Press, 2008
levgen Redko, Amaury Habrard, and Marc Sebban. Theoretical analysis of domain adaptation with optimal transport. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases. pp. 737-753. Springer, 2017.
levgen Redko, Nicolas Courty, Rémi Flamary, and Devis Tuia. Optimal transport for multi-source domain adaplation under target shift. In The 22nd International Conference on artificial intelligence and slatislics, pp. 849-858, PMLR, 2019.
levgen Redko, Emilie Morvant, Amaury Habrard, Mare Sebban, and Younès Bennani. A survey domain adaptation theory: learning bounds and theoretical guarantees. arXiv preprint arXiv:900,,11829, 2020.
Christopher Reinartz, Murat Kulahci, and Ole Ravn. An extended tennessee eastman simulation dataset for
fault-detection and decision support systems. Computers & Chemical Engineering, 149:107281, 2021.
Kale Saenko, Brian Kulis, Mario Fritz,, and Trevor Darrell. Adapting visual calegory models to new domains.
In European conference on computer vision, pp. 213 226. Springer, 2010.
Filippo Santambrogio. Optimal transport for applied mathemattcians. Birkäuser, NY, 55(58-63):94, 2015.
Vivien Seguy, Bharath Bhushan Damodaran, Rémi Flamary, Nicolas Courty, Antoine Rolet, and Mathieu Blondel. Large-scale optimal transport and mapping estimation. arXiv preprint mrXii:1711.082 2017.
Jian Shen, Yanru Qu, Weinan Zhang, and Yong Yu. Wasserstein distance guided representation learning for domain adaptation. In Proceedings of the AAAT conference artificial intelligence, volume 32, 2018.
Oliver Struckmeier, levgen Redko, Anton Mallasto, Karol Arndi, Markus Heinonen, and Ville Kyrki. Learn- ing representations that are closed-form monge mapping optimal with application to domain adaptation. net/
Transactions Machine Learning Research, 2023. ISSN 2835-8856. URL
morum?id--OIGf
Masashi Sugiyama, Matthias Krauledat, and Klaus-Robert Müller. Covariate shift adaptation by importance weighted cross validation. Journal of Machine Learning Research, 8(5), 2007.
Asuka Takatsu. Wasserstein geometry of Gaussian measures. Osaka Journal of Mathematics, 48(4):1005 1026, 2011.
15

---

Published in Transactions on Machine Learning Research (01/2025)
Laurens Van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. Journal of machine learning research, 9(11), 2008
Vladimir Vapnik. The nature of statistical learning theory. Springer science & business media, 2013.
Hemanth Venkateswara, Jose Eusebio, Shayok Chakraborty, and Sethuraman Panchanathan. Deep hashing network for unsupervised domain adaptation. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 5018 5027, 2017.
Cédric Villani et al. Optimal transport: old and new, volume 338. Springer, 2009.
16

---

Published in Transactions on Machine Learning Research (01/2025)
A
Additional Details about Benchmarks
Benchmack
Domncis
Backbone
Sumples
Classes
Caltech
Bing (I3)
600
Beachmanke
Domalns
Backbone
Samples
Classes
ImageNet (1)
ResNor, 50
600
12
1772rpm (A)
8000
Paacal (P)
Total
2400
600
CWRU
1730ppm (C)
1700ppm (R)
MLP
5000
10
'httll
Amazon (A)
958
Caltech Office
Webcamm (W)
dSLR (D)
ResNet
200
157
Mode
Mode
2845
2900
Caltech (C)
Total
2533
1123
TEP
Mode
Fully Convolutional
2665
29
Amazon (A)
2517
Mode Total
2897
dSLR
49H
31
17289
Webcam (W)
795
N L00.c = 0.10
260
Art [Ar]
2:27
N 0.15
Ollice-Home
Product (Pc)
Clipart (CI)
BeeNet 101
1139
65
CSTR
N 15,
0.5.c 0.15
- 0.15
200
200
13
Roal World (Rw)
0.15
Total
15598
Total
2800
Visual Domain Adaplation Benchmarks
(b) Cross-Domain Fault Diagnosis Benchmarks
Table 3: Overview of Visual Domain Adaptation and Cross- Domain Fault Diagnosis benchmarks
Tn table 3 we show an overview of the used benchmarks. We run our experiments on 4 visual DA datasets, and 3 CDFD datasets. For vision, we use Residual Networkss (ResNets) pre-trained on ImageNET as the backbone. For each adaptation lask (e.g., C - B in ImageCLEF) we fine tune the network using labeled source domain data. For all methods, we extract the features of source, and target domain, using the fine- tuned checkpoint. The size of the ResNett is used to agree with previous rescarch, such Peng et al. (2019) and Montesuma et al. (002).. We refer readers to for further technical details on the fine-tuning of vision backbones.
For CDFD we considered the same sotting as previous works using these benchmarks, such as Montesumal
ct al. (2023), Montesuma et al. (2024c) and Montesuma et al. (2022), for Case Western Reserve University
(CWRU) Tennessee Fastman (TE) process and Continuous Stirred Tank Reactor (CSTR) respectively.
For CWRU we extract windows out of raw signals of size 2048, then get the frequency representation for these windows using fast Fourier transform. These are treated as 2048 feature vectors, that are then fod to a neural network. In the TE process, we consider the same setting of Montesuma et. al. (2024c), i.e., fully convolutional neural net. For the CSTR directly the signals, concatenated into 1400 dimenssia vector as the features. The data for these benchmarks is publicly available here and herd
Noto that, for each datasct, there are Adoomain (aaiiims@@ddaaaaa (Adomains 1) adaptation tasks, except for CSTR which is a multi-target benchmark (i.e., a single source, and 6 largets). As result, we have 12 x 3 + x 2 30 + adaptation tasks.
3 http:://gtttiioooo////dddddadaaa https://www. http://www.gggg..aaaaaaaaaaataas adaptatiion
17

---

Published in Transactions on Machine Learning Research (01/2025)
B Additional Experiments
B.1 Detailed Results
Benchmark
Task
Beseline
OTDAgaD
OTDA.f/uu OTDA
InfoOT,
InfoOT
HOT-DA
GMM-OTDA
GMM-OTDAr
+
77.42
97:10
93.55
93.55
93.55
4 A-
91.53
93.22
96.61
96.61
96.61
94.92
96.61
93.22
91.53
88.44
91.56
73:33
91.11
$7.11
90.67
74.67
88.44
89.44
D W
98.31
93.22 22
93.22
91.92
100,00
91.53
91.92
100.00
98.31
D
73.36
67.11
17.03
71.56
68.44
75.11
51.11
86.67
71.56
Ofliech-Olliee
85.94
60.94
81.90
W -+ D
100.00
03.55
93.55
96.77
90.32
93.55
90.32
96.77
93.55
C
84.89
87.56
87.
88.44
S7.
87.56
56
88
C
98.44
98.44
98.44
98.44
96.88
98.44
98.44
98.44
C- D
83.55
87.10
90.33
93.55
87.10
87.10
100.0
911.55
80.66
91.53
91.92
91.99
94..92
93, 23.22
93.22
98.31
96.61
91.92
Avg.
90.32
87.95
78.41
91.11
57.87
89.46
87.72
93.22
88.90
B
92.50
95.00
96.00
91.17
97.50
96.67
96.67
95.00
91.17
B
90.00
89.17
89.17
91.67
94.17
91.67
95.00
95.00
93.33
68.33
69.17
70.00
71.67
73.33
75.83
74.17
72.50
74.17
C- B
65.00
65.83
66.83
66.00
51.67
62.50
62.50
65.00
66.67
C-
96.67
92.50
C-PP
71.67
74.17
73.33
71.67
75.00
75.83
72.50
70.83
75.83
InageCLEF
1-
68.33
70.00
68.33
70.00
66.
69.67
61.67
67. 50
70.00
1-C C
1-2 P
71.67
74.17
75.00
13.33
73:33
71.67
72.50
73.33
75.00
P-
67.50
69.17
66.87
68.33
57.50
65.88
62.50
62.50
64.17
P
85.00
95.53
95.00
96.67
96.67
96.67
95.00
95.00
P
90.83
02.50
92.50
90.83
95.83
95.83
95.00
94.17
95.00
Avg.
80.28
82.29
82.01
81.88
50.69
82,78
81.74
81.67
82.99
75
69.64
75.89
76.79
69.66
4- W
76.02
74.27
80.12
80.12
79.53
79.53
13.08
76.61
80.70
-
65.68
67.77
66.90
67.
66.20
61.15
29
73.52
Office 31
D
94.15
96.32
98.25
98.25
95.91
97.08
84,50
98.83
95.32
W + A
63.41
66.90
67.42
65.51
67.60
67.60
61.67
66.35
65.68
D Avg.
96.13
90.18
92.86
95.54
$7.50
91.96
81.25
91.90
91.07
76.96
76.88
79.34
79.32
79.00
79.86
72.48
78.62
79.77
Ar + Cl $5.19
51.98
51.87
56.24
17.
53.95
47.88
53.95
57.96
Ar + Pr
70.95
69
71.
71
30.97
70.27
-23
74.10
Ar Rae
79.68
79.68
80.53
80.71
40.53
80.25
76.00
77.96
82.43
C1 + Ar
63.51
61.62
633.19
62.68
31.44
62.68
53.81
59.79
64..33
C1 Prr
68.81
70.72
CI Jiw
72.68
69.92
71.18
72.33
38.81
71.18
67.97
68. 66
74.63
Offoe-Home
Pr - Ar
66.80
02.47
64.12
66.39
32.16
64.33
56.88
57 773
62.89
30.770
Py Rar
78.76
77.84
79.22
79.22
47.99
78.30
71.64
73.50
80.94
Ro -> Ar
Rw 72.99
71.96
72.37
73.81
51.13
70.72
62.47
66.39
60.69
57.85
57.39
56.93
00
47.65
50.63
56.36
Rw Pr 62,32
80.86
81.67
82.21
64.86
81.31
72.30
80.41
82.00
Avg. 66.81
63.88
65.99
07.67
36.58
65.79
59.17
63.73
67.40
Table 4: Single-source domain adaptation results. We compare 8 methods over benchmarks, with a total of 42 adaptation tasks.
B.2 Ablations and Visualization
A Ablating the number of components and entropic regularization. In this experiment, we ablate the two parameters of our methods, namely, the number of components K, and the entropic regularization Recall that we normalize the ground-cost by the maximum value, i.c., Č₁j = C../(max,,Cp) which improves the numerical stability of the Sinkhorn algorithm. We summarize our results in Figure b
For the number of components K, the relationship with performance is mostly clear. Indeed, except for 10 on the mapping stratogy, using more components enhances performance. Note that, even though
this implies a more complex GMMs we still have far less components than samples (n 600 per domain, i.e, 5 times more samples than components).
18

---

Published in Transactions on Machine Learning Research (01/2025)
GMM-OTDAMAP
GMM-OTDAT
Average adaptation performanie
82
80
78-
10-1
76-
10-2
71-
10
72-
10-2
12
24
36
18
60
81 96 108 120
12
21
36
48
60 72
96 108 120
Number components K
Number of components K
Figure 6: Ablation on number of components K, and entropic penalty 6, for the MAP estimation strategy based on labeled propagation (left), and the mapping estimation strategy (right).
For the entropic penalty e, we have two drastically different scenarios. For the MAP estimation, using higher entropic regularization coefficients improve performance, whereas the mapping strategy works better for smaller regularization coefficients (or exact (TT) While this may seem contradictory, we note that, in the mapping strategy, we actually filter out irrelevant matchings between components based on a parameter т. However, the entropic regularization is known to generate smoother couplings, which means that more entries of w are non-zero, or possibly greater than a fixed т. As a consequence, the mapping strategy ends up behaving like Trand, which causes bad reconstruction for the target domain. Naturally, this effect gets amplified with more components in both GMMs there are more possible matchings.
Source: A
Target: C
Figure 7: Source and target domain samples alongside the centroids (denoted by stars) found through EM colors reflect the different classes.
Visualizing components and mapped samples. In this experiment, we use the CWRU benchmark adaptation task A - C. We start by embedding the source and target domain data with the tSSNE technique of Van der Maaten & Hinton (2008). We do this in separate plots, where we concatenate the source domain features with the centroids obtained by running the EM algorithm (resp. target). This visualization is shown in Figure 7 Next, we map samples from the source to the target domain, with the various strategies described in this section, with the exception of InfoOT, which did not had a reasonable running time. These are shown in Figure
Overall, while the exact OT solution provides measure that better reflects the feature positions, mixes the classes, as evidenced by the orange and blue classes being mapped the same place, well as the green and violet classes. This phenomenon does not happen for other methods, the cost of having mapped
19

---

Published in Transactions on Machine Learning Research (01/2025)
(a) OTDAAMD
(b) STDA
(c) OTDAAffine
(d) GMM-OTDA7
Figure 8: t SNE visualisation of mapped samples to the target domain, on the CWRU task A + C.
points distributed in a different way. However, for OTDA sink and GMM-OTDAT, the mapped points better respect the class boundaries. For OTDAaf fine, note that the mapped distribution does not actually match the target. Overall, we achieve better mapping through the GMM modcling.
B.3 Cross- Domain Fault Diagnosis
For CDFD benchmarks, we follow the experimental settings of Montesuma et al. (2023). Montesuma et al. (002)) and Montesuma et al. (2024c), which roughly follows a similar idea visual adaptation tasks. This means that we pre-train a neural network with source domain data, then use its encoder for feature extraction. We refer readers to the original papers, and our appendix, for further information. To summarize our experimenlation, there are in total 8 benchmarks, and 84 domain adaptation tasks.
Case Western Reserve University Benchmark. With respect other benchmarks, the CWRU has the most number of samples per domain, i.c., 8000. In this case, InfoOT was intractable due its computational complexity. Furthermore, this benchmark illustrate the advantage of employing a grouping technique for enforcing the class structure in OT
Task
Baseline
OTDA
OTDA.ffine OTDA
InfoOT, InfoOT,
HOT-DA
GMM-OTDA GMM-OTDA-
51.12
72.00
75.19
78.12
79.75
79.75
A->C C
94.12
100.00
95.62
100.00
99.94
100.00
A
42 2550
76.12
78.50
75.88
79.75
80.00
80.00
++
37.44
77.62
78.88
75.38
79.81
79.56
79.94
C+
52.81
98.38
99.25
94.12
98.75 75
99.12
99.88
C - 55.62
70.25
74.50
75.50
83.12
79.75
80.00
Avg.
50.40
81.42
84.39
82.44
85.22
86.35
86.59
Table 5: Experimental results on the CWRU benchmark. For each task (i.e., each row), we highlight the best performing method in bold. Overall, InfoOT did not have a reasonable running time due the large mumber of samples on each domain.
Continuous Stirred Tank Reactor. As covered in Montesumna et al. (2022), the CSTR process carries an exothermic reaction > B. The reactor jacketed, and an inflow of coolant is controlled by a aPPrportional, Integral, Derivative (PID) controller described in PPllario & Cao (2017). From this reactor, a set of 7 variables measured throughout simulation, corresponding to different temperatures, concentrations and flow-rates. We refer readers to Montessma et al. (2022) for further details. Associated with this process, there are set of 12 different faults, ranging from process and sensors faults, and input disturbances. On top of these 12 faults, there is the no-fault scenario, characterizing a classification problem with 13 classes.
The different domains this benchmark correspond changes the process conditions. These are of 2 kinds. First, one introduces noise, 17, the process parameters (e.g., reactor or jacket volume), reflecting the possible uncertainty in the mathematical modeling of the reactor. Second, one changes the reaction
20

---

Published in Transactions on Machine Learning Research (01/2025)
Target Domain
2
3
5
6
Reaction Order (N)
1.0
1.0
1.0
0.5
1.5
2.0
Score
Parameter Noise (n)
10%
15%
20%
15%
15%
15%
Baseline
69.23
67.30
73.07
53.84
63.46
57.69
64.10
OTDAEMD
71.15
71.15
71.15
61.53
57.69
50.00
63.78
OTDA
67.31
67.31
69.23
55.76
51.92
53.84
60.89
OTDAAffine
65.38
71.15
71.15
61.54
65.38
53.84
64.74
InfoOT,
67.31
67.31
67.31
50.00
51.92
40.07
58.65
InfoOT,
71.15
67.31
71.15
53.84
51.92
48.07
60.57
HOT-DA
55.77
40.38
44.23
55.77
40.38
42.31
46.47
GMM-OTDAMAP
78.84
73.07
73.07
61.54
57.69
55.77
66.67
GMM-OTDAT
76.92
73.07
73.07
63.46
53.84
50.00
65.04
Table Average classification accuracy with confidence intervals over a 5-fold cross-validation.
order, N, of the reaction A + B, which drastically changes the dynamics of the system. As Montesuma ct al. (022)). we consider 77 0.15, 0.2), and N (1,0.5, 1.5,2). The source domain corresponds to Table 0.0, whereas the 6 different targets correspond to combinations of and N. These are shown in
N
The CSTR benchmark poses a few challenges. First, has a small number of samples on domains. While the source composed of 1300 samples, each target only has 260. Second, each sample lies in a 1400-dimensional space. Third, target domains noisy, due the parameter noise y). As result, most methods have difficulty in adapting, and performance usually degrades for more intense shifts (e.g., N 2.0 and rj 0.15). However, GMM-OTDAMAP and GMM-OTDAy outperform other methods.
Tennessee Eastman Process. Our last, experiment consists of the TE process, a benchmark widely used by the chemical engineering community (Reinartz et al.) 2021 This benchmark has the largest number of domains, i.e., Each of these domains is characterized by different mode of production for the products of chemical reaction, which affects the measured signals from the chemical plant. We summarize our results in Figure which comprises the 30 adaptation tasks.
For this benchmark, compare 11 methods. We divide those into shallow DA methods, and deep DA meth- ods. Shallow methods try to cope with distributional shift by transforming or re-weighting the samples in a feature space. In the case of this benchmark, we use the encoder's activations as fcatures. In contrast, deep methods cope with distribution shift by learning discriminative, domain incariaan features, by penalizing the encoder's parameters 0g so that, after encoding the data points, the domains are indistinguishabll from each other. Besides the shallow methods compared throughout this paper, we also consider classic deep methods, such as Domain Adversarial Neural Network (DANN) Ganin et al. 2016), Domain Adaptive Net- work [DAN) Ghilary et al. 2014), Wasserstein Distance Guided Representation Learning (WDGRL.) (Shen et al. 2018) and Deep Joint Distribution Optimal Transport (DeepJDOT) (Damodaran et al. 2018).
In comparison with other benchmarks, the TE feature vectors have fewer dimensions (i.c., 128). As a result, OTDA EMD the best performing method. However, GMM-OTDA manages to improve over OTDA and HOT-DA. Overall, our methods are especially better on harder adaptation tasks, such as - and + 2. We refer readers to the exploratory data analysis of Montesuma et al. (2024c) for further insights on why these adaptation tasks harder.
Furthermore, note that shallow methods are comparatively better to deep methods. Indced, the deep neural nets use considerably less data than, for instance, the image benchmarks considered in our experiments section. In this latter case, previously to the fine-tuning step on the source domain data, ResNets and ViTs are pro-trained on the ImageNet benchmark (Deng al. 2009), which provides good starting model for natural image classification. In the context of the TE process benchmark, this is not possible since data are time series of specific chemical process.
21

---

Published in Transactions on Machine Learning Research (01/202))
7U
75
85
7.4
6.3
Teto Teforeara www.-/100reaa Dww.ran Dadte
5.
6.7
6.6
Fettraa Difrraee P3:: Dariaa
-46.79 7.
S7
22
25
16
44
15
12
-31
15
17
46
-21
12
17
-20
Searr Mode
13555
Supa
97
68
-7.5
9.5
0.1
2.4
1.5
0.0
2.6
28
4.2
1.5
96
0.0
1.3
10
Apg
A0k
Ang
6 Ang
Ang
Avg
Targe
Middl
Tarye Morke
Tergst Mards
Targei
Modi
Baseline
(b) OTDAEMD
O'DDAsinkhrr
(d) O'DAAAfiee
10
3:540
Podee memmann Dibrreereee MOA. Senker
Soate Mode
33
0
221277
9.5
Seate Made:
16
11
Shate Mode
15
30
4.4
9.
-077-11 -23
-2.8
12
1.5
-0
6.5
76
71 17
-0
13
9.1
0
18
4.0
11
-0
Searce
-0.6100 16
122
-0
-15
9-3
-30
-30
1.5
Ang
5.6
32
11
A0gg.
6.
Targer Mode
Torget
Tirget
Target
(e)
HOTDA
(I) GMM-OTDAM
(g) GMM-OTDAP
(h) DANN
93
1.9
Declubeen Exfiree RIC. Becke
225
52
39
4.
-2
0.
2.3
0.
12
23
-0
12
37
53
32
-20
Soave
3.5
Sterre Made
-57
8.6
03
Source Mode
1.5
0.5
12
9:4
0.4
5.5
8.6
3.6
8822
0
3,3
11
3.4
Avg.
2.3
5/4
2.5
5
Avgg
40
2
A.A
6
Ave.
Targe: Mode
Torget Mode
Tiaget Mode
(i) DAN
(i) WDGRL
(k) DeepJDOT
Figure 9: Domain adaptation results the Tennessee Eastman Process. In (a), we show the baseline adaptation tasks, where each row represents source domain, and each column represents target domain. From (b) to (k), we show the performance offsct with respect (a) of adaptation algorithms. Note that (i) through (k) are deep learning-based algorithms.
B.4 Running Time Analysis
Bosides our remark is follows. We use the adaplation task A + W of the Office 31 benchmark. In this case, ng 2817, nr = 624, Re = 31 and d = 2048. We ran each algorithm 10 independent times, then computed the mean and standard deviation of their running time. Our results are reported on tables and g
also running time analysis of the tested algorithms. Our experimental setting
Algorithm
Running Time (seconds)
Accuracy (%)
OTDAEMD
0.775 + 0.007
74.27
OTDA
14.119 + 0.125
80.12
OTDA Affine
8.503 + 0.097
80.12
InfoOT
105.407 + 0.716
79.53
HOTDA
2.508 = 0.023
73.68
Table 7: Running time (in seconds) and classification accuracy (in %) of different OT based strategies.
22

---

Published in Transactions on Machine Learning Research (01/2025)
Starting from table the fastest algorithm is OTDAEMD: which has complexity O(n' logn). In comparison, OTDAsink has complexity O(n²) per iteration. Here, it is important to note that we run the Sinkhorn algorithm until convergence, for 1000 iterations, which explains its superior running time. It is noteworthy that O also has a higher running time, since its complexity dimension-deeendent,, i.c., O(d)). Due the high dimensional character of the data at hand, this results higher rumning time.
Another example of higher running time comes from HOTDA, which solves nē -n. 465 sub empirical OT problems, resulting in higher running time in comparison with OTDAEMD- Finally, it is noteworthy that InfoOT is considerably slower than other methods, due to its O(n²) complexity by iteration.
In comparison with previous methods, we show the running time and classification accuracy of GMM-OTDAT and GMM-OTDAMAP for K {31,66 62, ... ,17]. We do so for e = 10-2, which in practice yielded the best empirical performance. As a result, the running time should be directly compared to OTDAsInk. For all number of components, our algorithm has an inferior running time almost all methods, with the expection of OTDARMD and HOTDA.
Number of Components
Running Time (seconds)
GMM-OTDAT
GMM-OTDAMAP
31
0.742 + 0.074
78.94
64.91
62
1.329 + 0.015
79.53
70.76
93
1.995 + 0.035
76.02
70.17
124
2.679 + 0.051
80.70
74.85
155
3.517 + 0.025
78.36
75.44
186
4.343 + 0.090
80.70
70.17
217
5.069 + 0.059
77.77
76.02
Table 8: Running time (in seconds) and classification accuracy (in %) of GMM-OTDAT and GMM- OTDAMAP as function of number of components.
23


Reviews:
[{"review_id": "45wxCGRP0x", "reviewer": "Reviewer_hmXt", "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": "Summary of strengths\n\n- The paper is well-organized, clear and easy to follow. \n- The proposed GMM-OTDA method demonstrates better scalability than previous OT-based domain adaptation methods.\n- Theoretical analysis is provided for the proposed approach.\n\nSummary of weaknesses\n\n- The method appears to have limited scalability on larger datasets, as demonstrated in Table 5. Specifically, it does not achieve optimal performance on datasets with a large number of classes, such as Office-31 and Office-Home. More complex and widely used UDA datasets like VisDA and DomainNet were not addressed.\n- The related work section is somewhat limited. Although the paper claims state-of-the-art performance, it only compares against a few OT-based UDA methods, lacking a more comprehensive evaluation.\n- The practical contribution of optimal transport for domain adaptation in this work is limited for complex datasets, as it relies heavily on feature extraction from pretrained models. Additionally, when classes are imbalanced, the effectiveness of optimal transport can be compromised.\n\n- Some unclear points:\n  - In Table 1, further introduction of the compared methods would enhance clarity.\n  - In the Baselines and Backbones section, $GMM-OTDA_{MAP}$ is described, but there appears to be insufficient detail on the other proposed method, $GMM-OTDA_{T}$.", "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 builds on the GMMOT framework, applying it to domain adaptation and proposing two strategies. Compared to previous approaches, the method reduces time complexity and demonstrates performance improvements on multiple benchmarks.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- Please consider comparing your method with the following OT-based works [r1, r2] and other state-of-the-art UDA approaches. To my knowledge, CoVi [r3] is a relevant option.\n\n  [r1] Optimal transport for conditional domain matching and label shift, Machine Learning 2021.\n\n  [r2] Mapping conditional distributions for domain adaptation under generalized target shift, ICLR 2022.\n\n  [r3] Contrastive Vicinal Space for Unsupervised Domain Adaptation, ECCV 2022.\n\n- In the paper, the authors mention that proposed method has a smaller computational complexity and that InfoOT does not exhibit reasonable runtime. Could you provide specific runtime comparisons for clarity?", "broader_impact_concerns": "No concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "4fTfncszUX", "reviewer": "Reviewer_MxMx", "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- Optimal transport is a theoretically grounded tool for analyzing distribution changes.\n- The result of $M\\to S$ is impressive.\n\nWeaknesses:\n- I am not an expert in OT. However, most of the contents seem to come from previous works. Therefore, the novelty and contributions are not well elaborated regarding the scalability and generalization towards novel samples out of the source support.\n\n- The experimental comparisons are only conducted within OT-based methods, far below SOTA, while non-OT baselines like FixBi [1] achieve much better results, e.g., 72.7\\% in Office-Home and 91.4% in Office-31.\n\n-  In visual DA tasks, the proposed method's advantage is marginal except for Caltech-Office 10, which only contains 10 classes. Therefore, I doubt this proposal can be generalized to larger label space cases.\n\n[1] Fixbi: Bridging domain spaces for unsupervised domain adaptation, CVPR 2021", "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 work proposes an optimal transport algorithm between Gaussian Mixture Models to address domain shift. The algorithm can efficiently scale with both the number of samples and dimensions.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- The writing, especially regarding notations, requires an extensive revision. E.g.,\n  - In Eq.(2) and Eq.(3), the right sides are the same while regarding the left sides, one is a joint distribution, and the other is a distance metric. \n  - The distributions over $X$ and $K$ are denoted by the same $P$. \n  - The line above Eq.(12), where $\\mathcal{H}\\ \\subset \\mathcal{Y}^{\\mathcal{X}}$ is confusing. \n  - $\\omega^\\star$ in Eq.(11) is inconsistent with the notation in the equation on Page 8. \n  - Some equations are not numbered.\n\n- Assuming the same true labeling functions $\\mathcal{T}_S=\\mathcal{T}_T$ is apparently not feasible for DA.\n\n- Is the alignment achieved in feature space $g:\\mathcal{X}\\to \\mathcal{Z}$? There is no such explanation throughout the paper.\n\n- What is the cost of the diagonal covariance matrix assumption? Does it harm the performance?\n\n\n- Euclidean distance is no longer discriminative due to the dimensionality curse. Therefore, I can hardly believe the proposal can work for an increasing label space.\n\n- The details of training a source GMM and target GMM are not explained. Pseudo code describing the entire algorithm is encouraged for readability.\n\n- The intention to assign multiple Gaussian components to each class for the source domain is unclear. E.g., mapping source data into Gaussian clusters can be easily achieved by a ResNet backbone with a Bayesian linear classifier.\n\n- What is the difference between setting $\\tau=0$ and previous work? How do we choose $\\tau$ in practice? If a transport plan $\\omega$ splitting $P_k$ into several $Q_k$ is discouraged, why not add some constraints to make it sparse?\n\n- The author claims the contribution of generalization towards novel samples out of the source support. However, judging from the $T_{rand}$ on Page 8, I do not see why the existing estimation is only defined on empirical source data.\n\n\n- The author claims the contribution on scalability, but the results on large-scale datasets, e.g., VisDA and DomainNet, are not provided. In addition, comparisons regarding the actual time consumption are not presented.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "0uYoiQVH3W", "reviewer": "Reviewer_zrmE", "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. The proposed work is well-grounded in OT theory and GMMs, which provides a solid theoretical basis for the proposed method. \n\n2. Using label propagation to explain the OT between the source domain and the target domain is interesting.\n\n3. The proposed work addresses the computational challenges associated with traditional OT and GMM methods, particularly in high-dimensional spaces, by demonstrating a reduction in complexity when applying GMMs.\n\n4. The paper is well-written with clear motivations. The experiments are also well-structured and clearly presented.\n\n### Weaknesses:\n\n1. While the theoretical contributions are strong, implementing the proposed method might be complicated, particularly with larger datasets. This complexity might make it challenging for practitioners who are not familiar with GMMs and OT.\n\n2. Since OT requires working in spaces of the same dimension and GMMs have their own limitations, the paper could benefit from discussing these drawbacks in more detail. To be specific, it should discuss more about the assumptions made about data distributions, like the challenges that can arise if the number of components is misestimated. Moreover, the paper should consider scenarios where the methods might not perform well, such as with high dimensionality, sparse data, or distributions that do not fit well with GMMs.\n\n3. The performance of GMMs can be sensitive to hyperparameters like the number of components and the covariance structure. The authors should conduct a more detailed ablation study to assist practitioners in selecting the appropriate hyperparameters.\n\n4. GMMs can be prone to overfitting, especially if too many components are used relative to the amount of data available, which could lead to poor generalization on unseen data. Could the authors conduct the experiments on a larger dataset, like VisDA-2017-C?\n\n5. The effectiveness of the proposed methods heavily relies on the quality and representativeness of the source data. If the source data is biased or unrepresentative, the adaptation may fail to generalize well to the target domain.\n\n6. While the paper emphasizes efficiency, implementing GMMs and OT still demands considerable computational resources. Could the authors conduct a runtime complexity analysis and compare it with other UDA methods?", "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 using optimal transport within Gaussian mixture models (GMMs) for solving the UDA problem. To achieve this, the propose method has two key aspects: 1. label propagation: propagate the ground truth knowledge of the source domain's GMM to that of the target domain using the optimal transport. 2. Sample Mapping: to ensure the effective transport of samples from the source domain to the target domain, a mapping estimation is proposed. The authors validate the proposed method via extensive experiments across 8 benchmarks. They demonstrate that their proposed method is competitive with existing state-of-the-art methods while maintaining efficiency in terms of both sample size and dimensionality.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Providing a clear guidelines or instructions for practitioners on how to implement the proposed method, especially for those who might not be familiar with GMMs and OT. \n\n2. Including a more detailed discussion on the limitations of OT and GMMs: (1) The assumptions made about data distributions; (2) The issues that can arise from misestimating the number of GMM components. (3) cases where the methods might not perform well, such as in high-dimensional spaces, with sparse data, or with distributions that do not fit well with GMMs. \n\n3. Conducting a thorough ablation study to analyze how different hyperparameters (e.g., the number of components and covariance structure) affect performance.\n\n4. Conducting experiments on larger datasets, such as VisDA-2017-C, to see how well the model performs and generalizes. This will provide insights into its scalability and robustness against overfitting. \n\n5. Discussing the importance of the quality and representativeness of the source data. Suggesting ways to evaluate the quality of the source data and its potential impact on adapting to the target domain. \n\n6. Conducting a runtime complexity analysis of the proposed method and compare it with other UDA methods.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The submission meets the bar for both the Claims and Evidence and the Audience criteria. Minor revision requested:

The scope of the claim about outperforming or being competitive with the state-of-the-art is well explained in the Experiments section of the updated manuscript, but the abstract and conclusion remain ambiguous on that distinction. The authors should make sure that this (i.e., the fact that the SOTA claims pertain to shallow domain adaptation approaches) is clearly explained in the abstract and conclusion before publication.
Decision: Accept (poster)