Wasserstein Transfer Learning
Authors: Kaicheng Zhang
Sinian Zhang
Doudou Zhou
Yidong Zhou
Venue: NeurIPS 2025 poster ()
Keywords: data aggregation
multitask learning
non-Euclidean output
optimal transport
Wasserstein metric
Primary area: general_machine_learning
TL;DR: We propose a novel transfer learning framework for regression where outputs are probability distributions residing in the Wasserstein space.

Abstract: Transfer learning is a powerful paradigm for leveraging knowledge from source domains to enhance learning in a target domain. However, traditional transfer learning approaches often focus on scalar or multivariate data within Euclidean spaces, limiting their applicability to complex data structures such as probability distributions. To address this, we introduce a novel framework for transfer learning in regression models, where outputs are probability distributions residing in the Wasserstein space. When the informative subset of transferable source domains is known, we propose an estimator with provable asymptotic convergence rates, quantifying the impact of domain similarity on transfer efficiency. For cases where the informative subset is unknown, we develop a data-driven transfer learning procedure designed to mitigate negative transfer. The proposed methods are supported by rigorous theoretical analysis and are validated through extensive simulations and real-world applications.

Wasserstein Transfer Learning
Kaicheng Zhang' Sinian Zhang?* Doudou Zhou Yidong Zhou*i
SSchool of Mathematical Sciences, Zhejiang University, China Division of Biostatistics and Health Data Science University of Minnesota, USA
3 Department of Statistics and Data Science, National University of Singapore, Singapore *Department of Statistics, University of California, Davis, USA 32101020332zju edu. cn, zhan9381 @umn edu, ddzhou@nus. wwwpunuonuuueuuu..... ag. ydzhou@ucdavis. edu
Abstract
Transfer learning a powerful paradigm for leveraging knowledge from source domains to enhance learning in target domain. However, traditional transfer learn- ing approaches often focus on scalar or multivariate data within Euclidean spaces, limiting their applicability to complex data structures such as probability distribu- tions. To address this limitation, we introduce a novel transfer learning framework for regression models whose outputs are probability distributions residing in the Wasserstein space. When the informative subset of transferable source domains is known, we propose an estimator with provable asymptotic convergence rates, quantifying the impact of domain similarity on transfer efficiency. For cases where the informative subset is unknown, we develop data-driven transfer learning pro- cedure designed mitigate negative transfer. The proposed methods are supported by rigorous theoretical analysis and are validated through extensive simulations and real-world applications. The code available at https://github.comm// wwt.:/ithhub.co/////i//ioaam TL.
Introduction
In recent years, transfer learning [32] has emerged as powerful paradigm in machine learning. enabling models leverage knowledge acquired from one domain and apply related tasks in another. This approach has proven especially valuable in scenarios where data collection and labeling can be costly, or where tasks exhibit inherent similarities in structure or representation. While early successes focused on conventional data types such as images 300, text [28). and tabular data (38). there growing interest in extending these methods to more complex data structures. Such data often reside in non- non-Euclidean spaces and lack basic algebraic operations like addition, subtraction, or scalar multiplication, posing challenges for traditional learning algorithms. A key example is probability distributions 27 where for example the sum of two density functions does not yield valid density.
Samples of univariate probability distributions are increasingly encountered across various research domains, such as mortality analysis [14], temperature studies [44), and physical activity monitoring 23 among others (27] Recently, there has been a growing focus on directly modeling distributions as elements of the Wasserstein space, geodesic metric space related to optimal transport [36 24).
The absence of linear structure in this space motivates the development of specialized transfer learning techniques that respect its intrinsic geometry.
To address this gap. introduce Wasserstein Transfer Learning (WaTL), novel transfer Iearn- ing framework for regression models where outputs are univariate probability distributions. WaTI.
"Equal contribution.
'Corresponding authors. Doudou Zhou and Yidong Zhou contributed equally.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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effectively leverages knowledge from source domains to improve learning in a target domain by intrin- sically incorporating the Wasserstein metric, which provides a natural way measure discrepancies between probability distributions.
1.1
Contributions
The primary contributions of this work are summarized as follows:
Methodology. We propose novel transfer learning framework for regression models with distribu- tional outputs, addressing the challenges inherent in the Wasserstein space, which lacks a conventional linear structure. Our framework includes an efficient algorithm for cases where the informative subset of source domains is known, and a data-driven algorithm for scenarios where the subset unknown. To the best of our knowledge, this is the first comprehensive transfer learning approach specifically designed for regression models with outputs residing in the Wasserstein space.
Theoretical analysis. We establish the asymptotic convergence rates for the WaTL algorithm in both the case where the informative set is known and the more challenging scenario where it must be estimated. In the latter case, we also prove that the informative set can be consistently identified. In both settings, we demonstrate that WaTL effectively improves model performance on the target data by leveraging information from the source domain. The proofs rely heavily on empirical process theory and a careful analysis of the covariate structure. Our key theoretical results extend beyond responses lying in the Wasserstein space, offering potential applications to other complex outputs.
Simulation studies and real-world applications. We evaluate WaTL through simulations and real data applications, demonstrating its effectiveness in improving target model performance by leveraging source domain information. The benefits become more pronounced with larger source sample sizes, underscoring its ability to harness transferable knowledge.
1.2 Related Work
Transfer learning. Transfer learning aims improve performance in target population by leverag- ing information from a related source population and has seen wide application across domains [c.g..
16 15 8 41 40). Recent theoretical developments have focused on regression in Euclidean settings, including high-dimensional linear [21] and generalized linear models [31], nonparametric regression 22), and function mean estimation from discretely sampled data [5]. In parallel, optimal transport has been used to measure distributional shifts for domain adaptation [10, (2)]. However, to the best of our knowledge, no existing work has investigated transfer learning in regression models where outputs are probability distributions residing in the Wasserstein space. This represents a significant gap in the literature, highlighting the need for novel methodologies that address this challenging yet. important setting.
Distributional data analysis. The increasing prevalence of data where distributions serve funda- mental units of observation has spurred the development of distributional data analysis [27]. Recent advancements in this field include geodesic principal component analysis in the Wasserstein space D. autoregressive models for time series of distributions 39 44 and distribution- -n--o--triidtttionnnn regression [4] 77) Leveraging the Wasserstein metric, regression models with distributional outputs and Euclidean inputs can viewed as a special case of Fréchet regression [26]. which extends linear and local linear regression outputs residing in general metric spaces. In practical scenarios where only finite samples from the unknown distributional output are available, empirical measures have
been utilized as substitutes for the unobservable distributions in regression models 4
2 Preliminaries
2.1 Notations
Let (0, 1) be the space of square-integraie functions over the interval (0, 1), with the associated I norm and metric denoted by llz and dzz, respectively. To be specific, ||d||2 = (S' (2)d22)11 and (g2( |lg1 - g2ll2- For a vector Z, |||||| denotes the Euclidean norm. Given matrix Σ, we define its spectrum as the set of its singular val-

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ues. For a uub-Gaussian random vector X, we define the Sub-Gaussian norm as |X i inf { ( 0: E(e((,v)) < 2
We write On I bn if there exists positive constant C such that an Cbn when is large enough and On x bn if On S bn and bn I an. The notation On = -p(bb) implies that P( /b S C) +1 for some constant C O, while an op (b,,, implies that P(jan/b, >> c) for any constant > O. Superscripts typically indicate different data sources, while subscripts distinguish individual samples from the same source.
2.2 Wasserstein Space
Let W denote the space of probability distributions on with finite second moments, equipped with the 2- Wasserstein, or simply Wasserstein, metric. For two distributions Mi M2 W, the Wasserstein metric is given by (1,, (2) = inf xnf SRYR {s -1,2 dT(s,, 1), where [[ (22) denotes the set of all joint distributions with marginals μ41 and μl2 [8]. For a probability measure E W with cumulative distribution function Fys we define the quantile function F--1 as the left-continuous inverse of Fu. such that F (w) inf(t E F,tt) 2 u). for (0, 1). has been established 36 that the Wasserstein metric can be expressed as the L metric between quantile functions:
di = S. (F) du.
The space W, endowed with the Wasserstein metric, forms complete and separable metric space, commonly known as the Wasserstein space [36].
Assuming (1, < x for all W, the Fréchet mean K of a random distribution E W given by Me = arg min E(dey (v))). Since the Wasserstein space W a Hadamard space [20] the Fréchet mean is well-define and unique. Moreover, from (T) it follows that the quantile
function of the Fréchet mean, denoted F-1, satisfies F (u) = E F (u)), u E (0,
2.3 Fréchet Regression
Consider random pair (X, with joint distribution F on the product space R2 W. Let X have mean 0 = E(X) and covariance matrix Σ = Var(X), where D is assumed to be positive definite. To establish regression framework for predicting the distributional response from the covariate X, we employ the Fréchet regression model, which extends multiple linear regression and local linear regression to scenarios where responses reside metric space (26). The Fréchet regression function is defined as the conditional Fréchet mean of given X =x,
m(x) = arg min (( neW
For a detailed exposition Fréchet regression, we refer the reader to 266. Given n independent realizations {Xx,, + we define the empirical mean and covariance of X as X -m X;
and E 1 Σi (X, X)(X- - X)=.
The global Fréchet regression extends classical multiple linear regression and estimates the conditional Fréchet mean as
mG(x) arg min ((ss(()( neW
where the weight function is given by sc(x) = 1 (X -)- (xr -). The empirical estimator formulated
mc(x) arg min
(
NEW
where (X,
Similarly, local Fréchet regression extends classical local linear regression to settings with metric space-valued outputs. In the case of a scalar predictor X E R, the local Fréchet regression function is
(1,((x) arg min (8 (v,
NEW
3

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where the weight function is sz h)) = K^(X - - (-1((ux))/od -2)]/00. E( x)(X =x}}. O, 1, 2, and oj = u u³. Here, (n(-) = h k-((/h()) is a kernel function with bandwidth h. The empirical version given by
1
(L,((1) arg min WEW
SiL (x. h) h (k),
i=1
where (z, h) - - 01 (X, <)//0g. with üg = nn "ii Kn(X, - z)
xy.j = 1, and 03 = ū0 u..
3 Methodology
3.1 Sctup
We consider a transfer learning problem where target data {(X; M
dently from the target population (X("), (())) ~ Fo, and source data {(X, Ji= are sampled indepen-
(0) }
(1)) th are sampled independently from the source population (x((), v(6) ~ FR. for k = ,,..., K. The goal is to estimate the target model using both the target data and source data from K related studies.
For k = O,..., K, assume x() has mean Og and covariance Ex. with Es positive definite, Define the empirical mean and covariance of {x, j as X m E x. and Σg m ΣE (x, (k)
(x) XR)". For fixed æ R2, the weight function is s (I)
02) (x)) (x-
0s), with the sample version Si (x) = (x -Xa) " 2 (xx) The target regression function (k)
(0) (z)dly(( pi)}. In the following, we present details on transfer learning for global Fréchet regression. where the key difference in the local Fréchet regression setting is the use of a different weight function. The technical details for transfer learning in local Fréchet regression are therefore deferred to Appendix D
for a given x E Rº is then mg (x) = arg ww Eisg
The set of informative auxiliary samples (informative set) consists of sources sufficiently similar the target data. Formally, the informative set defined as Av {1 k: K (1 (x) -1()(x)) (x)lla S
e for some > 0, where (k) (x) E(sg
(k) (x) }. For simplicity, let n. Ek=1 7k-
K
3.2 Wasserstein Transfer Learning
We propose the Wasserstein Transfer Learning (WaTL) algorithm, which combines information from source datasets under the assumption that all source data are informative enough. This assumption implies that the discrepancies between the source and target are small enough to enhance estimation compared to using only the target. When this condition is met for all source datasets, the informative K}. The detailed steps of WaTL are presented in Algorithm |
set is given by As =
Algorithm I Wasserstein Transfer Learning (WaTL)
Input: Target and source data (xx (0) v. Hm U UiSKSK (xx wu )( regularization
(0)
parameter X, and query point x E IP.
Output: Target estimator m (a:).
Weighted auxiliary estimator: f(x)
notn Ex=o nu (z). where (()
(k) " Ein SG PT)
2: Bias correction using target data: Fo(x) arg min, giinger (2(001)
1 Σ i=1
Fr)
g11
X||g~ - f(x)]22-
(0) (x) arg min - Fo(z))12- 3: Projection to Wasserstein space: m
In Step 1, the initial estimate ʃ aggregaaess information from both the target and source, weighted by their respective sample sizes. While this step incorporates valuable auxiliary information, the resulting

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Algorithm 2 Adaptive Wasserstein Transfer Learning (AWaTL)
Target and source data ((x;") // (00) p u ( UigeSK (ox y ) ). regularization
Input:
parameter X, prespecified number of informative sources L, and query point IR..
Output:
Target estimator m (z).
Compute discrepancy scores. For each source dataset k L, ....., K, compute the empirical
discrepancy: 4/x- = [1 (x) (1))l,, where (1)) "kk Dim1 GG (x) FT Construct
(A)
the adaptive informative set by selecting the L smallest discrepancy scores À = (k:1sk< K and th is among the I. smallest values)..
2: Weighted auxiliary estimator: f(x) Eeeāuro th E n4. (x).
3: Bias correction using target data: fo(z) arg ge no Ei= 5ic F g12
(0)
Algg f(z)]12-
4: Projection to Wasserstein space: @ (0) (z) = arg min NeW new 14 - fo(x))l2-
estimate may be biased due to distributional differences between the target and source populations. In Step 2, the bias in I is corrected by focusing on the target data. The regularization term Algg Allg-S(a)) (1))]l ensures balance between target-specific precision and auxiliary-informeed robustness. Theoretical guidelines for selecting À are provided in Theorem 2 The final step projects the corrected estimate
fo onto the Wasserstein space, ensuring the output m (a)) respects the intrinsic geometry of W.
(0)
This projection exists and is unique because W is closed and convex subset of (0,
3.3 Adaptive Selection of Informative Sources
In many practical scenarios, the assumption that all source datasets belong to the informative set Ag. may not hold. To address this, we extend WaTL with an adaptive selection procedure to identify the informative set. The discrepancy for each source dataset defined as the = (z) f(k) (() (x)]la. which measures the distance between the target distribution and the auxiliary distribution. Since (z)) and f(*) (z) are unknown, compute empirical estimate Vh for v/x, which is used to adaptively estimate the informative set Ay. To implement this approach, an additional input parameter T., which specifies the approximate number of informative sources, is required. In practice, I. can be treated as a tuning parameter and selected through cross-validation or other model selection techniques. The full procedure is formalized in Algorithm
The proposed algorithm adaptively identifies the informative set À in Step by evaluating the empirical discrepancy scores th. The selected set is then used to compute the weighted auxiliary estimator in Step 2, ensuring that only the most relevant source datasets contribute to the final target estimator. Steps 3 and 4 follow the same bias correction and projection procedures as described in Section 3.2] This adaptive approach enhances the robustness of WaTL by excluding irrelevant or highly dissimilar source datasets.
4 Theory
In this section, we establish the theoretical guarantces of the proposed WaTL and AWaTL algorithms using techniques from empirical process theory [34). For WaTL, we present the following lemma, which characterizes the convergence rate for each term contributing to the weighted auxiliary estimator f(x) computed in Step 1.
Condition 1. For k O...., K, the covariate ((k) is sub- Gaussian with ( E Jai, 02) the mean vector satisfies [10xl] < R2, and the spectrum of the covariance matrix Eg lies within the interval [R2, Ra). Moreover, (k) is supported on a bounded interval.
5

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Lemma 1. Let (10 " Σi= iG v/
(k) (2) and its population counterpart be defined as
(M) Bisg (1) (x) for 0, ..... ... K. Then under Condition ! 1/²)
(x) (x)1l=
To derive the convergence rate of (x)) rely on the following condition.
Condition 2. There exist positive constants C1, C2 C3. C4 such that
K
2e
nh ng
+ C2e
Cs -V = =(1),
k=
where Rg is as in Condition 7
In addition,
Vno + nA
Vno + nA o(1)
o(1) To0
SinI ni
Remark 1. These conditions typically satisfied in practice as they Condition T requires that covariates and covariance matrices are bounded in a specific way, which is standard in the transfer learning literature and generally holds in real-world scenarios [5]. In particular, this assumption is common when dealing with high-dimensional data where regularization necessary [21]. Condition 2 assumes that the number of samples is significantly larger than the number of sources K, which is reasonable since K is usually fixed in practical settings, and we often have sufficient source data compared target data. In practice, Condition 2 may be slightly violated if there exists a source k with relatively small ng. such that vng + RA/nn is not o(1). In such cases, the kth source can simply be excluded from Step of the WaTL algorithm.
overly restrictive.
Theorem 1. Suppose Conditions RP that
7 and hold. Then, for the WaTL algorithm, it holds for fixed
Ek= T
[1f(x) f(x)]l2 Op
(na + mA)-1/2)
no nA
where f(x) (no + n1)=1 Ek E (z))
The proof of Theorem T iinolless detailed analysis of the sample covariance matrix and leverages M-estimation techniques within the framework of empirical process theory. The result extends beyond responses in the Wasserstein space, applying other metric spaces that meet mild regularity conditions. Consequently, Theorem provides a versatile framework that can be applied to transfer learning regression models with responses such networks [42], symmetric positive-definite matrices 255, or trees (2). The following theorem establishes the convergence rate for the estimated (0) regression function mi (1)) in the WaTl. algorithm.
Theorem 2. Assume Conditions I and hold and the regularization parameter satisfies À H. no -1/2+c for some O. Then, for the WalLL algorithm and a fixed RP, it holds that
ΣK Ek
(m (0) (I)), m (z)) = Oz (n0'111/0 -1/24e (15
V + (no + = -/221 +
(z) (x)(2 quantifies the maximum discrepancy between the target where v = maxi (1) -
and source.
Theorem can be compared to the convergence rate global Fréchet regression 26 applied solely to the target data, for which the rate is d (x). m (z)) (n 1/2). The WaTL algorithm achieves a faster convergence rate when there are sufficient source data and the auxiliary sources are
informative enough, satisfying v S ny 11/2-c This result highlights that knowledge transfer from auxiliary samples can significantly enhance the learning performance of the target model, provided the auxiliary sources are closely aligned with the target data.
For the AWaTL algorithm, require the following condition.
Condition 3. The regularization parameter satisfies X no > 6, there exists a non-empty subset A C {1,..., ......,K) K} such that
11/24c for some e > and for some
70e 1/2 +
maxKEA Uh.
o(1),
0(1,,
AO the
1/2 + no
11/2-e
11.

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where A = [L....,KKAAAn, n.
miniskgK ni and Vh - (x)))
-
Remark 2. We allow A to be the whole set {1....: K}, in which case the condition becomes
1/2-c' O().
1/2
74
no
Besides, if A exists, then it is unique since
(IskSk: n 1/2 + no
11/2
4/
1/2 = 0(1) = 0. + no
4
71
Condition ensures that the source datasets can be effectively partitioned into two groups: infor- mative ones and those sufficiently different from the target data. This separation guarantees that the informative set can be accurately identified. Under this condition, by setting the number of informative sources to = AA, we establish the following rate of convergence for the AWaTI. algorithm. In practice, T. can be decided by cross- validation.
Theorem 3. Under Condition 3 and the conditions of Thhorem for the AWaTl. algorrihm with a fixed number of sources K we have P( = A) + and
ΣnEAUHO; ViN ExEALHO) Ti
(0) (x), ma (x)) - 0, 11/2+e d
11/22/
max + REA
+
nx) 1/2) 112-1/21) kEAU(D)
Σ
The convergence rate of the AWaTL algorithm simplifies to no m--''t when the informative source data is sufficiently large. This rate surpasses that of global Fréchet regression (66 applied solely to the target data, offering a theoretical guarantee that AWaTI. effectively mitigates negative transfer by selectively integrating relevant auxiliary information.
While the above analysis assumes that each probability distribution is fully observed, an assumption commonly adopted in the distributional data literature [26] 71, real-world applications often provide only independent samples drawn from underlying distributions. In such cases, this limitation can be overcome by replacing unobservable distributions with their empirical counterparts, constructed from sample observations [43]. Additional details and theoretical justification for this extension are provided in Appendix
5 Numerical Experiments
In this section, we evaluate the performance of the proposed WaTL algorithm, alongside two baseline approaches: the global Fréchet regression using only target data (Only Target) and using only source data (Only Source). Consider K = source sites. The data are generated as follows. For the target population, we sample (0) ~ U(0, 1) and generate the response distribution, represented by its quantile function,
Fr (u) = (0 u x + x00) (u) u E (0, 1).
where ((0) (0..5, 1) (0.1) and (0) ~ N(0, 1) 10.-00000. Here, N(,,o2)
denotes normal distribution with mean μ and variance o², truncated to the interval (a, b). For source populations, define 2hh 0.1k for k 1...., K, and generate (a) U(0, 1). The corresponding response distribution is generated as
F.b (u) = w (8) -
(1 (x)) (2) ( (u), " E (0,11)
where (k) (0.5,, v and (k) ((0, Consequently, for each predictor
I, the true regression function is m (8) (x) (1-x)u +)u (u), for & 1. K.
We vary the target sample size no from 200 to 800, while the source sample size is set as n4 kr,
where E (100, 200} and 4: = 1,..., K. The regularization parameter À in Algorithm [] selected via five-fold validvaai,,,inn, ranging from 0 to 3 in increments of 0.1. To evaluate performance, sample 100 predictors uniformly from the target distribution. Using Algorithm I compute
7

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mgg (x) and compare it with the corresponding estimates obtained from global Fréchet regression using only target or source data. Performance is assessed using the root mean squared prediction
(0)
risk RMSPR = V 100 Ei= (me (xz), m (I:)), where 24 denotes the sampled predictor, (0)
(0) me (a:) is the estimated function, and m (x:4 represents the ground truth. To ensure robustness, we repeat the simulation 50 times and report the average RMSPR, as shown in Figure
As shown in Figure aa), WaTL consistently outperforms global Fréchet regression trained solely on target or source data. When the target sample size nop is small, the Only Target method exhibits high RMSPR due to the instability of models trained on limited data. In contrast, WaTL significantly reduces RMSPR by effectively incorporating auxiliary information from the source domain. As no increases, the performance of Only Target improves and gradually approaches that of WaTL, which is expected as larger sample sizes lead to more stable and accurate estimators. Nevertheless, WaTL maintains consistent advantage across all fo. suggesting that it leverages complementary information from the source. The performance of the Only Source estimator remains nearly unchanged across different ng values, it does not benefit from additional target data.
Comparing two panels of Figure [)), we also observe that WaTL improves as source sample size increases, confirming its ability to effectively integrate information from source domains. This demonstrates the benefit multi-source transfer learning, where WaTL balances knowledge from both target and source domains to achieve improved prediction.
To better understand when negative transfer may occur, we conduct an ablation study with K 1 source and vary the similarity parameter t/1 from 0.01 to 1 in increments of 0.01, with fig = 100 and n1 = 200. Our method outperforms the Only Target approach when 1/1 0.9, while for 4/1 2 0.9, the Only Target method becomes preferable. This confirms that negative transfer arises when the source too dissimilar the target.
We further evaluate the effectiveness of AWaTL in selecting informative sources. In this experiment, set L = 2 and 74 100 for 0, ... The similarity parameters specified th 0.1 for k = 1, 2 (informative sources), and 4/x = 16, increasing from 0.2 to 1 in increments of 0.1, for 3, 4, (uninformative sources). Each configuration is repeated 100 times, and the corresponding selection rates are reported in Figure [b). The results show that AWaTL successfully identifies informative sources, with selection rates for sources and rapidly increasing and reaching perfect accuracy once zb > 0.6. These findings demonstrate the robustness of AWaTL in distinguishing useful sources under varying similarity levels.
Method
Only Source
Only Target
WaTL
Source Source Source Sourte
1.00
0.0040
0.0040
0.0035
0.0035
Selection Rabe
RASSP
0.0030
0.0030
0.0025
0.0025
0.25
0.0020
0.0020
0.00
200
400
600
800
200
400
600
800
0.2
0.4
0.6
0.8
1.0
Target Sample Number
Sargle Number
1
(a)
(b)
Figure 1: (a) Root mean squared prediction risk (RMSPR) of WaTL, only Source, and Only Target methods under varying target sample sizes, with source sample sizes = 100 (left) and 7 200 (right); (b) Selection rate of each source site as v increases.

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6 Real-world Applications
We evaluate the WaTL algorithm using data from the National Health and Nutrition Examination Survey (NHANES) 2005-2006 focusing on modeling the distribution of physical activity intensity. NHANES is a large-scale health survey in the United States that combines interviews with physical examinations to assess the health and nutrition of both adults and children. The dataset includes extensive demographic, socioeconomic, dietary, and medical assessments, providing a comprehensive resource for health-related research.
During the 2005-2006 NHANES cycle, participants aged 6 and older wore an ActiGraph 7164 accelerometer on their right hip for seven days, recording physical activity intensity in I-minute epochs. Participants were instructed to remove the device during water-based activities and sleep. The device measured counts per minute (CPM), ranging from to 32767, capturing variations in activity levels throughout the monitoring period.
we analyze them separately. Physical activity intensity is influenced by multiple factors, and we consider body mass index (BMI) and age as key predictors [19 To accommodate potential nonlinear relationships, we implement local Fréchet regression within the WaTL algorithm, treating the distribution of physical activity intensity as the response and BMI and age as predictors.
Since female and male participants exhibit distinct physical activity patterns 12
Following the data preprocessing steps in [23,, we remove unreliable observations per NHANES protocols. For each participant, we exclude activity counts above 1000 CPM equal to zero (as zeros may correspond to various low-activity states such as sleep or swimming). Participants with fewer than 100 valid observations missing BMI, age, gender information are also excluded. The remaining activity counts over seven days are concatenated to form the distribution of each participant's activity intensity.
To evaluate the WaTl., we set White, Mexican Americans, and other Hispanic individuals as sources and Black as the target. For females, the source data include 1308 White people. 884 Mexican Ameri- cans, and 108 Other Hispanic individuals. For males, the source data include 1232 White participants, 805 Mexican Americans, and 92 Other Hispanic individuals. We set 200 Black participants as the target data for both genders. During evaluating. we perform five-fold cross- valss-vali,,tin, using four folds for training and one for testing, cycling through each fold. For comparison, we also apply local Fréchet regression using only the target data. The results, summarized in Figure 2a),, show that WaTT. improves performance over local Fréchet regression for both females and males, demonstrating its ability to leverage information from other demographic groups to enhance modeling for Black participants.
Method
Only Target WaTL
Method - Ground Truth WaTL - Only Target
3.00
1.00
45
RMSPR
-0
Probability
8.50
0.50
35
0:35
30
0.00
0.00
250 Physical Activity Intensity
300
750
250
500
2000
Female
Male
P Physical Activty Intensity
Gender
(a)
(b)
(a) Root mean squared prediction risk (RMSPR) of WaTL and Only Target methods for females and males, evaluated using five-fold cross-validation; (b) Cumulative distribution function of physical activity levels for one selected female (left) and one selected male (right), along with estimates from WaTL and Only Target methods.
Figure 2:

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To further illustrate the effectiveness of WaTL, we visualize the cumulative distribution function of physical activity levels for one female participant and one male participant in Figure[[b) along with estimates from WaTI. and local Fréchet regression, using only the target data. The results indicate that Wa'T. provides better to the true distribution, outperforming the estimate obtained using only the target data. We further evaluate the robustness of WaTL on human mortality dataset in Appendix A] where WaTL continues to demonstrate strong performance.
7 Conclusion
We introduce Wasserstein transfer learning and its adaptive variant for scenarios where the informative is unknown, addressing the challenges posed by the lack of linear operations in the Wasserstein space. By leveraging the Wasserstein metric, the proposed algorithm accounts for the non-Ruclidean structure of distributional outputs, ensuring compatibility with the intrinsic geometry of the Wasser- stein space. Supported by rigorous theoretical guarantees, the framework demonstrates improved estimation performance compared methods that rely solely on target data.
This paper focuses on univariate distributions, which arise frequently in real-world applications such the NHANES and human mortality studies discussed in Section 6 The proposed framework, however, is not limited to the univariate case and can be extended to multivariate distributions by incorporating metrics such as the Sinkhorn [T]] or sliced Wasserstein distance [4]. Extending the methodology to higher dimensions is an exciting future direction that entails addressing both computational and theoretical challenges specific to high-dimensional Wasserstein spaces.
Acknowledgments and Disclosure of Funding
We thank the Area Chair and the reviewers for their constructive feedback. Doudou Zhou was supported by the NUS Start-Up Grant (A-0009985-00-000 and the MOE AcRF Tier I Grant (A- 8003569-00-00).
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We recognize that providing effective safeguards is challenging, and many papers do require this, but we encourage authors to take this into account and make best faith effort.
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Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
Question:
Answer: [Yes]
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involve new data collection through crowdsourcing
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Guidelines:
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A Human Mortality Data
We further assess WaTL using the age-at-death distributions from 162 countries in 2015, compiled from the United Nations Databases and the UN World Population Prospects 2024 The dataset provides country-level age-specific death counts, which we convert into smooth age-at-death densities using local linear smoothing; see Figure for an illustration. For this analysis, we define the 24 developed countries as the target site and the remaining 138 developing countries as the source site,
We evaluate the performance of WaTL against several key baselines on this dataset. These include models trained only on the target data (Only Target), only on the source data (Only Source), and on a naive pooling of both (Target + Source). Furthermore, for these baselines, we compare the performance of our underlying Fréchet regression framework against Wasserstein Regression 7 another state-of-the method for distributional data.
Table 1: Performance and Training Time with Varying Target Sample Sizes.
Number of Target Samples
RMSPR
Training Time (ms)
14
0.028
0.598
19
0.025
0.597
24
0.022
0.694
The comprehensive results are presented in Figure 3 (b). The proposed WaTL method achieves the lowest Root Mean Squared Prediction Risk (RMSPR) of 0.022, demonstrating a marked improvement over all alternatives. Notably, it significantly outperforms models trained solely on the 24 target samples (RMSPR of 0.027 for Fréchet Regression and 0.025 for Wasserstein Regression) and the naive data pooling approach (RMSPR of 0.033). This highlights that simply combining datasets is insufficient to overcome the domain shift between developing and developed countries, validating the necessity of our bias-correction mechanism.
To further assess WaTL's robustness and practicality, especially in common scenarios with limited target data, we analyze its performance with a varying number of target samples, As shown in Table [ the model's predictive accuracy consistently improves the target sample size increases from 14 to 24. It is also worth noting that the method maintains exceptional computational efficiency, with training times remaining under one millisecond. These findings confirm that WaTL is not only highly accurate but also robust and efficient solution for real-world demographic studies where target data can be searce.
Developed
Developing
Wasserstein Regression Frichet Regrossion
0.05
0.06
0.04
RMSPR
0.03
0.04
Deneiyy
0.02
0.02
0.01
Only Source
Target Source
Only Terget
Walz
0.00
0
100
Age
Method
(a)
(b)
Figure 3: (a) Age-at-death densities of developed and developing countries; (b) Root mean squared prediction risk (RMSPR) of WaTL and Only Target methods for human mortality data.
lttpp:://data.uu.org
https://populationn. un wwt.:or/ppp//ooo.aa..o
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B Additional Notations
For any 91, 92 E (2 1), the L² inner product is defined as
(9 22.92)22 S.¹ 91 (1)))2(2))
The total variation of a function g on the interval (a, b) is
V
lim <.
sup
[9(z,) g(
The space of bounded variation functions is given by (V((0, 1), H) (o: (0, - (-H, s H),
which is equipped with the 1,? metric.
For a matrix E, let ||a|| and |||| denote its operator norm and Frobenius norm, respectively, and let (xE) be its ith smallest singular value.
We denote the characteristic function of a set A by LA and use |A| to represent its cardinality. In metric space (52, d), the covering number W(K, d, e) represents the smallest number of closed balls of radius required to cover a subset K Q. The notation [ is used to denote the disjoint union of sets.
For notational simplicity, we will omit (I) from f(z) when the meaning clear from the context.
C Proof
Lemma 2. Consider U W (0,1) + bR R where U(v, := =(v and W x (0, 1) is endowed
with the product o-algebra generated by the Borel algebra on W and the Borel algebra (0, 1). The first Borel algebra B(W) generated by open balls (W,dw) and the second ([ 1)) is generated by Euclidean open balls. Then U is measurable.
Lemma 2 enables simplified expression for the conditional Fréchet mean of v given X = x by applying Fubini's theorem. Specifically, we have
m m(x) arg min (fdiy(r, = x}
NEW
arg min Efd9,2(, F | = =}
NEW
arg min (F x) FT
NEW
where the last equality follows from Fubini's theorem, since EVP, 1X x} measurable. A similar argument applies to mz( and (z(
C.1 Proof of Lemma 2
Proof. Without loss of generality, It suffices to prove (F (x) > 0) is measurable. Note that any quantile function is left continuous, hence {(u, x) F, (x) 0} = U, ((,, z) F, (g) > 0}.
Then without loss of generality, we suffice to prove ((v, 0,5) > 0)) measurable. Since W is separable, we can select a dense countable subset K. We then define A = {v E K|F > 0}. Assuming A {v, N). we have {r|F, (0.5) > 0} {v) lim, dw(p,, = 0}
lim,-+oo .i B1 (14) where (v(v) is the ball centered at y with radius Here we remark that both left continuous and monotone increasing are used in the first equation. Hence is measurable.
21

---

C.2 Proof of Lemmal
Proof. First, it easy to check that f(l) (V( 1), Ho) for some Hg since f(h) Elsg (k) (T) (1)2o) - Ellsa (1) (z) (o (x)<0) F and Elsg (z) < co. By taking
Hu large enough, We also claim that P( ((f()) (V((0, 1), Ho)) + 1 since f()) also has a similar decomposition and
(z)
(1$
1G (x) 5/ (x)) 1s (x))
(8)
SG
1
S op) + H1
for some H1 with high probability, where 8; (M) (x) = (x. 0z) (x 0g). The last
inequality holds because of the result given by the proof of Theorem 2 in 26 that 1 (s:) = (1) and the assumption that (8) is subguassian.
Ex 18 1i
nn
(x)-
1==1
To use Theorem 2 in 126], taking Q to be ((0,0, 1), Mo), M x) to be ( (g,4) Elsg (k) (F) (()) g|{3 for any I E ((0, Ho). it suffices to check that the other as- sumptions of the theorem:
S. VI (oBo BV 1), Ho), dc)d = 0(1)
as + and
-M (f(0),, ([(( g11.. (g, M
Example 19.11 in 33 gives bound of the covering number of the space (BV((0, 1), Ho), dze),
((BV((O, 1), H.,HH),,d dga,e) S K
for some K > O where K is independent of e.
The rest is similar to the proof of Proposition in 26 Forg BV 1), Ho). (g) denotes the L? ball of radius 7 centered at g. Let C, (f(()) {ou u U} such that [U] (O, Ho) n Bi (f), de=,e)] S ke and the balls (g.)) covers Bi (V)))B((O,, 1), Ho). For å > 0, we define Tu (M) +8gg- f())). Then the balls (āu) covers (f) (V 1), Ho). Hence
Jo V 1 ((g n BV (VO 1), Ho), dej, < [
V -e- s 1+2VK <0.
For the last assumption, just note that
( (g. a)) - M (k) (k))) (f))) = -1((() -112.
Then according Theorem 2 in 26), [[ f(k) 112 Op 1/2).
C.3 Proof of Theorem
To prove Theorem T we first establish a lemma that quantifies the bias introduced in Step of Algorithm[ [ This lemma is formulated for a general metric space, making it broadly applicable and potentially useful for extending transfer learning algorithms other metric spaces.
Here are some notations, most of which are similar to our original setting. The only difference is that we use Y instead of represent a random object in general metric space (R,,). We define K
n. mini K 7i, n = E Nis an = nu/ + nAA),
K
mi (w) arg min M (w,,)), M (w,xx) wel
Σ
a El (8)
(w,
K=0
and
n
my (x:) arg min (w, M.(w))) no + nA
ΣΣ
(o)
een
k=u
22

---

We impose the following mild condition on the metric space (Ω, d), which is widely used in the literature on non- non-Euclidean data analysis [26. This condition holds for various metric spaces commonly encountered in real-world applications, including the Wasserstein space, the space of networks, and the space of symmetric positive-defintee matrices, among others.
Condition 4. (i) my m1(x) uniquely exists and mi (ar) uniquely exists almost surely. Additionally for any > 0, inf inf d(m, (a)w)>> M (w,x) > M (m (m)()),
(ii Let (8(mi (z)) C 1t be the ball of radius 8 centered at mi (x). Then
J(6) = 1//
O(1)
as 8+00 O.
(iil) There exist 1 > 0, C 0 and B > 1. possibly depending on x, such that whenever
d(ml (x),w) < 7, we have M (w,, x)) Mm (z), (m) 2 (w, m1 C mz)))
0g(1
Lemma 3. Under Conditions 1
2
and 4 d² (fz(z))
+ (no +
notua
C.3.1 Proof of Lemma 3
Proof. Denote Vn(w,x) .)- = MM x-- MM (w; and D. (w,x) - d(Y, (8) () -x(,
(K) (x))
and recall sf (k) (x) 1+ (x.(*) 2x) (x 0z). Then
1 nEe
[V)(wx)) - (z), s Σ
as
ll
Σ
30 (2) 3/ 12) (z)) (w))
T1k:
k=
+
(R) D (w) - Elst (k) D. (2) (w))
(A.1)
n2k
i=1
For any 8 >>0 0,
n2. 2d)
s0 (1) (00) (x)) (2) (w)
(b)
w + w/1 (z) x
sup d(u.m,
S
Tkk iml
T4k
where w/(h) (x:) (x))
(X T5 -1 xx²)) 1 02) and (x) = Ex (s 02)
Ex x ) Then
(z) 8: (x) = ( (z) + (,))
(A.2)
We neglect the superscript for the sake of notation simplicity.
Denote BL,M == [11 | Ei |Xl|| - } where |X,| represents the element-wise absolute value of
X₁. Let B2,15
-115-111 S M}. BM BI.M n B2,M and E' 1 n E(X, OX, -)". In BI.
S| - Σ' <
| E'IlF
1 Σ [X, - X) -X)"" (X, - 0)(X 0)7
< 12p - 02 + - 012)
< (M + 10)]|X 0112.
Using Theorem 6.5 in 37
leads to
P( Σ|| > e) 2e
2e Cien + Oze
Cslfe-v 4
23

---

for any c C (2(2 (2 n: 1). Note that
|| 1 || +| 1 ---1
|Σ + 1m(2) (1)1
71 (E)r1
|Σ - Σ
$ - 11 Σ (n (Σ)
|
where the third inequality is from the Weyl's inequality. Then there exists M > 0 such that
P(I 2 M) S 2e
Ci
ng""
C- /
Consequently,
101 "
-
P(B < 3e
+
In BM.
Wo(z) = - -0----000000-00000--------0-000-
$ (X
18| M)
Hence in By K B we have
P
Ok (o(()) >1
2e
< 2e
Y of 1o
C
< 2e
-x-0
S 2e (co(nna))
(A.3)
for some constants Cs Cs > 0.
In the first inequality, general Hoeffding's inequality (Theorem 2.6.2 in 35 and in the second inequality, we use Proposition 2.6.1 in [35].
In addition,
(X- (Wi(z)) & - --------
s MM[|= M[) 1R -
I (||[=+ + M Ei1 + MR3 -X-.
Hence in B'
P
Σ
ak
1 Rk
(X. (K)
W (z) (x))
< 2c
C212 MA
+ P
Σ
anCy Ex
Exel >
Tik
k=0
(A.4)
for some constant Cg, Cy > Using Markov inequality, the second term is bounded by
MI k=o De Collf S4 <e- -t 4pk + - Cio
which is from Theorem 6.5 in 37 for any À satisfying < Cii,, for some constants Cio, Cn > O. We can choose X
t(ng+tma) 2C1o
which does satisfy the theorem's assumption if
t t= O ((no + 1/2). Here we utilize Condition
Then
AmggnnA) 4dK 4012
P
a4 Co1E
Ed1l >
Se
(A.5)
24

---

Note that K = o(no + nA). hence
K
- (x)) = Op (5(mo + n n12))
sup (2))
since P(BM) + 1 due to Condition 2
To bound the second term of (A.1), following the proof of Theorem 2 in 26 for each k. we define
the functions (k) RP 52 IR as
(1) (z, 1) [1+ (= 02) 0x)"2x"(( (xc - Ox]]d& (v.w)
and the function class
M (k) = (0th (k) - 9m (u) d(w, (1()) < 6J.
We have
E sup
G (x) (1) (g) { (M) (x) (0)) S J(E (
(
(())
n& i=1
where Cr8 (z) = 2diam(())([ + (z 0x) =z 1x (x 0x)] the envelop function.
Note that
E(G (x)3) C2
for some constant C14 > O, which does not depend on kt Hence
E
sup d(w.m,
(k) Si le (x) (g) - Eis; (k) D. (1)
-
=0 -0
muk
(6)) k=0
Ykn + TIA
t1 + TA
i=1
k=0
(1.6)
Define
DR =
sup
a6
GG (1) $6 (z)) (w) < Ro(no + n2) 11/2
((l:mi()))< k=
1=1
We have
E(ID D
VV((w) - Vm S aRd
Σ k=0
Vng
+ (no + n2) 11221 -12).
sup (s)
Yin + n1A
for some constant > 0.
Next we show
M x) - (w,,))) = op(():
and for all Z and for any K > 0, 4 O, there exists 8 > such that
lim sup P( sup
(w1, (, > k) S.
(A.7)
n d(w))ag))
To prove the first assertion, we denote
K (1))
(w,
nA Then (M,(w,,) MM. (wi, x) and Var Var(M (a,,
k= 2m1
S 2 2 C' [motnaja for some constant
C' > O. Hence (w, (w (1).
Besides,
K
(k)
K + Σ
(W (k) yr
(w - My(W) = Σ
d
((
(k)
w)
x (k) d (Y, .w)
nik
k=0
1=1
k=0
1=1
(1).
25

---

The last equation is true since Ex=o ak 1 Ei (1).
(Wi())) '2l = (1) and Ex= a [o(=)) =
To prove A.7p. note that for any n, 12 E S
K
& [M, - (,1,,))) -M,M)
1
+ W0 + +
no nIA
k=0 i=1
= =0p(d()))2)))
The last equation is true since Ek=o ah 1 Di=l (1), and we can prove
= (1)(() and ΣR=0 an ag Wo(x))
a
= 1 + op(1)
(A.8)
km
similar way of (A.3). It follows that d((m2(x) (x), m2 = (1)
Set rm = (Ex= k= tnt + (no + nna) nA)-112
1/2 1/ and
(z) fw 2 rnd m1 (z)) s 2¹).
Choose 7, 0 to satisfy (iii) in Condition and also small enough that (ii) in Condition hoold for all 8 < 17 and set == 1/2 For any integer T,2
P(rndP/2 (mhn (c), mn (x)) > 24x) S P(Di)) + ((2d(mm (m), 21)
(V(w) (m(m C 22(5-1) WES,,n
+ Σ
P(( sup
DD)
12
The second term converges to since d( (x), ma (x)) = op (1). For each j in the sum in the third term, we have d(w,m1(1)
m1(x)) s (2) n,. so the sum bounded by
4aC'1 Σ
2
S 4
21-88
J2L2
Tn
+ (n. + aa))
12h.
S'r
+
Choose L2 large enough, this probability can be small enough.
Hence we have
Ex Vnn + (no +1 12/3
(c), =Op
n1 nd
C.3.2 Proof of Theorem Given Lemma 3
Proof. The proof is similar to that of Lemma some large H2 > since
First, it easy to check that f BV 1), H2) for
== E
a 0
(2) (x)
EE an (2)20) F-1 E
(k) Σ ause (z))
of 1 (x)<0) Fuod
1 1
and E ER ag (x) < 2o. Taking H2 large enough, we also claim that P(
BV( 1), H2)) 1 since To also has similar decomposition and
1 au
no n2 (x) 8
(k) (x)) + 10, (k) (x)))
no + 12 1
I (1) + H3.
26

---

for some H3 > 0 with high probability. The last inequality follows from A.2 (A.4). Thus. (i) in Condition Sundition[hhodds.
A.88
A.5
and
(ii) in Condition 4 follows from the same arguments used in the proof of Lemma which holds since
1 It remains to
verify (iii) in Condition +
- -
Hence the result follows by using Lemma
C.4 Proof of Theorem 2
Proof. Recall that f(o (x) = E'(sg (0)
(2) Using the definition of Fo.
1 no
11o - 1
10 (x) - T. |1
20o 112)
YA0
2 no
(0)
(F) p jo )(0))2 1,00
i
no
< - fl
flle + (11 fll2
Σ
(0)
(F)(F - -0 Jo - 22
no i=1
< - fl
fll2
x11(((0) fll2
+ +
1(0 ||a||
1 Σ
(z) 1100 3d
==1
We define the event E,, == {1lm Ei=
(0)
(z)
f( S X/2). According to Lemma
we
have P(E,)) + 1 since À x T2o -1/2+e
Under E, for n large enough,
I1Fo , || < f (0) ~112 S 2X(1 - -1112 + +115 +15--1112)
holds. Hence we have
(0) (z), m d (m
(0) (1)) S 1150 11 Or n10 -1/2tc (1F S112 + v)).
Then using Theorem it follows that
Ex= TiA
dn (0) (x), me
- 1/2tt((b (0) (z)) O0 (060 (100
+ (no + 1122 no + nd
C.5 Proof of Theorem 3
Proof. We first prove P( A) + Note that
P(Ā= A) 2 P(max Eh < min v4))
REA
keAc
> P( max
10xx v/ll >
Imax the REA
min thel)
I
REAc
K max
P((
vhal
max REA
thk
min
v/xl)
ISRSK
kEA4
According to Lemma
|W4- Jyl = (p 1/2 + 12
1/2). According to Condition
maxxE U/- min l'kl
- 0o.
n_ 1/2
1/2
+ Tig
27

---

Hence
P(Ā= A)= = o(1).
Then using Theorem where we substitute A for f1...., K we have
max 22 + ExEAUçO) VTA REA
( (0) (x), ma (x)) (~
+ Σ nk) nx2-1/2)
ΣREALHO) Thk
REAU(O)
D Transfer Learning for Local Fréchet Regression
For simplicity, we consider a scalar predictor (() E RR for k = 0. O..... K. For predictors in IR?, the explicit form of the weight function can be found in Section 2.3 of [17] data setting, we define
Under the same source-target
(x) = arg min Efd (v xx] = =x],
m
NEW
m I.h (x)
arg min E(s L
(z, h) hd ,1)},
NEW
m L.h
(k)
(x) = arg min
Σ
19 (x, 1)),
NEW
Tik
where St (x) and SiL (z) represents the population and sample weight functions of local Fréchet regression for the kth source.
(k)
For notational simplicity, define
f (x) (x)
(k) E(F (( (x) x),
5 (1)) E(s) (K) (I, h) b
K
fn(x) = Σ anhh (x).
k=0
Similar to Section 3) we introduce the Local Wasserstein Transfer Learning (LWaTL) algorithm in Algorithm [ assuming that all sources are informative. When the informative set is unknown, we incorporate an additional step to identify the informative set, as outlined in Algorithm Theoretical
guarantees for these algorithms are provided in Theorem and Theorem 5
We impose the following condition, where the first two parts correspond to the kernel and distributional assumptions that are standard in local linear regression.
Condition 5. (i) The kernel K is probability density function, symmetric around zero. Fur- thermore, defining Kaj J& K2 (u)w' du, [Kaal and [2ss] are both finite.
(ii) The marginal density (k) of (k) and the conditional density of ( given U(k), (k) exist and are twice continuously differentiable. Besides, it satisfies that
(o.0)) sup sup (x) (k) (2)) H3
0 2.((6)
for some H3 >
(ii)) h and noh co.
(iv) Σx o(1).
(v) (k) have a common bounded domain.
(vi) with == (Xn((X xx{(x(1)) x))), j =0, 0, 1, and oo (k)
= ug (u (1)
(k), (k)
(k)
are bounded.
28

---

Algorithm 3 Local Wasserstein Transfer Learning (LWaTL)
Input:
E parameter X, bandwidth and query point x E R.
Target and source data {(x; (0) ) U (Uigesk {(o!)) (h) ) regularization
Output: Target estimator m L,4 (z).
I: Weighted auxiliary estimator
K
(z) = no
20 (I),
TLA
k=0
(1) where (x) " E
SiL (x,, 1t) F and nA = Σ k= n
(k)
(ik)
2: Bias correction using target data
Fo)(x) = arg min 100,))
SL (z, 1) - -9l2 + Allo - (x))222
3: Projection to Wasserstein space
m (0)
L.A (z)
arg min
F (2))122
NEW
The following theorem establishes the rate of convergence for the LWaTL algorithm.
Theorem Assume Condition 5 holld and the regularization parameter satisfies X x no for some c >> O. Then, for the LWalL algorithm and fixed x E TR?,, we have
1112//1122--11 1122-6
(.) m0)) = Op h +no
11/22/222222 -112221112--111
3D 4h +h 1/2 ((
1/2
K
+ I 11)
Th
Do
k=
k=0
where U maxisksK 115 (0) (x) f(h (z)]
Proof. Note that
LL) (x))
s (m m 00) (x)) (x)) + (m
(0) (x),
(0) (x), (m) (x)))
2
( (x) 11 (x) (x)
For the first term, We could use Theorem 3 of 26 since we can check all its assumptions easily, which is similar to the proof of Lemmal | Here we should consider the metric space BV((0, 1). H.) for some H4 > 0 endowed with the probability measure naturally induced by the canonical embedding of the Wasserstein space.
For the second term, first note that using Theorem of (26]. we have ()) 5%0 (x)]l2 (p(o)) 1/2). Then similar to the proof of Theorem we have
1111 (x) For(2)) < fo) (x) fa( XII (z) (x)
2 (x) ( - fl0 (z)]l2.
Hence in E, (115 (x) F100 (x)1l2 - $
11 (x) For (xr 13 < 2 (0) (x) - (x) < 2\(1 + (1 - (z))22),
(A.9)
1111 (k)
(0) 1|2. For the first term of A.9 using Theorem 3 of 26 and (i),
where the = maxx
(ii) in Condition 5 we have
1/ - + O1P).
For the second term of A.9 we could follow the proof of Lemma 3 to study the asymptotic rate of convergence. The difference is that we do not assume covariates are sub- Gabssan but
29

---

Algorithm 4 Adaptive Local Wasserstein Transfer Learning (ALWaTL)
Target and source data ((x; v/ j U ( (UigagK (xx yi (k )( ) regularization
Input:
parameter X, bandwidth h, number of informative sources L, and query point x e R.
Output: Target estimator m (,,, (z).
I: Compute discrepancy scores, For each source dataset k = 1,..., K, compute the empirical discrepancy
VES = 115 (z) - 7 (1)1l2,
where (M (z) n E
(R) 4D
(z, F (a)) Construct the adaptive informative set by selecting
the L smallest discrepancy scores
( K Vi is among the smallest L values).
2: Weighted auxiliary estimator
fa(=)= Σxe ah kEAu(()
Σ
na (z).
Pha (1)
3: Bias correction using target data
no
Fo arg min (0,1( no
(0) Si (1,, 1 10
gll2 + Alll (x))22
4: Projection to Wasserstein space
m (.)
(v) = arg min
F.-1 - (z))122
NEW
we have upper bounds for moments related to covariates and the kernel. In detail, we define (k)
(0) (x."* and D. (k) () = | gale | Sk (x)
ALL hx) = (x(( x)
(a 1
Then similar to (A.6) and the proof of Theorem 4 in (26) we have for small 8 >>00 0,
ni
E sup
(
(z,h)D
Σ
og Efs! (z,
(g, (x,h))(
daa( (1t)
)0 no + n2
A=0
k=
0 vTkk) = =0(8h-1/2
1/2
110 + n
Another term we need to bound is Ek En 1 (x, h) - ((() (z, (e). Note that we can define
BM := ({ (1)))))))) > [ ( n3 (1 n
Eme |Kn(X - (x - =01 < 12 M))) for
some large M > 0. Then easy to check that
K CAL
P((B)))
<
n
for some CA1 > 0. In BM, observe that
Str (z, h) _(k) (z, 1)) < (w/(h) Kn (x,
(k)
(k)
x1)1 + (W [x - (xx(x()) 2)).
where
(k)
(k)
w-(k) x
(k)
wi(k)
un
21
(a) 12
(a (1)))²
(o 22) )2
(a 1
Note that in Byg it is easy to check
[w (k)
< CA2
Σ
h
(k)
1=0
30

---

and
Ca2
Σ h h-1-s uj 107
(1)
for some constants CA2 > Hence in Bas.
RM K
(k) Si
(x,, h))
BiL (x, 11)1 >1)
((8)
R=0
K
(Wo + h Winl) > >t)
2 CA3
0m0
1
$ 12
C44
11k
( m In addition, we can check that f((I) E (V, Hs) exists almost surely for some Hs > 0 by showing EE
Hence ΣK E 1 (z. h)
3/1) (z, h)l
Lk=1 Ee 1 (z, h)l Op()). Also utilizing the same technique in the proof of Lemma 3 It suffices to show that (1)) (x)]2 (1). is just simple generalization of Lemma 2
[26.. Eventually, we have
K
11 (x) Fon(x)]l2 = Op 1 11/2
n
1/2
+
T2k +
no
nA
k=0
k=0
and
(2))
(0)
(x),
2/ +h2+h -h-1/2 0,( h (n'+m 1/2-c
-1122
n 1)
1/2
Σ
n
no
+
n2
k=0
Remark 3. Theorem can be extended to general metric spaces and serves as parallel version 6
of Lemma 3 Besides, if is much larger than no and vz O(n=*) with A > 15-102 then
among bandwidth sequences = nn" the optimal sequence achieved at r = 2 leading to convergence rate
36-2eh42
d (m (z), (x)) 0 (n
which surpasses the convergence rate local Fréchet regression Op 4/5
26
There is also a parallel version of Theorem 3 built upon the following condition.
Condition 6. Suppose the regularization parameter satisfies À x nh -1/2 for some >> 0 and for some >> e, There exists a non-empty subset A C (1...., K} such that
h22 + (m.h)-1/2 + (moh) 1/2
maxiEA WIYI.
o(1),
O(1),
minkEAC thE,L
h2 (n.h) 1/2 ny 1/2 h-12te
where AC -==11 KHA. min S K ni and Vh.r. 15 (z) 1 (x) andd Then for the ALWaTL algorithm we have
Theorem 5. Assume Conditions
(0)
Op
11tt + -nn
-1
1/2-c
max UhR + h? AEA
+ h -
Σ
n 1/2
kEAUÇO} CKEAU(() my
kEAU(())
31

---

Proof. We first prove P(Å= = A) - 1. Note that
P(Å= A) 2 Pimax Wi,h < min vkk,h)
KEA
NEAD
21 - P(
max ( (10k, wk.h + WRn, 1),, >
max VKLLL min Ve,z
ISkS
kEA
REAC
1 - K max ISK'SK
P( 1// + liber thx.e.)
max the, REA
min 112,
READ
According Theorem 3 of (26]. |R th, (h²). Utilizing Theorem 4 of [26] |EE. GR 0p((mph)1112 (noh) (12)) According to Condition
A A L. m L.R.L
00. h22 + (m.h) 1/2 + (nngl))
Hence
P( A)= 1- o()) where we substitute A for {l...., K3 we have
Then utilizing Theorem
dy m (0)
L. (,)) m
(x:)
0p
11a :2o
11/2 1112--
max LR,L + li2
h-1/2
Σ kE)u(()
4
EKeAURo) nk 1)
kEAU(a)
Remark 4. n., is much larger than no. then the simplified rate Op(h2 + no Po h --1c''-)). Among bandwidth sequences h = nī. the optimal choice is achieved at r* 5-2c" leading to the convergence rate
-1/2 1//2////-
(m
(0) L. (.))), m(x)) = Op
which is faster than the convergence rate of local Fréchet regression (n 4/5
26
E
Transfer Learning for Wasserstein Regression with Empirical Measures
Fréchet regression assumes that each distribution is fully observed. However, this assumption is often impractical, since in real-world settings one rarely encounters datasets where each observation itself constitutes full distribution. To address, Wasserstein regression with empirical distributions has been proposed [43] where distributions are replaced by their empirical versions, reaching a convergence rate of (n 1/22 + min where Nmin representing the minimum number of observations. In this section we do the same modification to handle this obstacle, providing algorithms for global and local Wasserstein transfer learning with empirical measures (WaTI.-EM)) respectively. The only difference between them and those with full distributions is that we substitute empirical distributions 2.(()
1 (1))) defined as N;
Σ 1 8 Vi
for the true but unobservable distributions V; (k) S S mk. 1 S k S K,
N, where we recall the settings in Subsection 3.2 and Appendix D additionally assume {ut 3 are
independently sampled from uy and define ó= as the Dirac measure at z. In addition, we denote Nmin mini, (,(A).
Theorem 6. Assume Conditions 7 and 2 hold and the regularization parameter satisfies À X 0. Then, for the WaTL-EM algorithm and fixed RP, it holds
(n 1/2 N --144/1- for some that
Σk Thk + 1/22 N-11/1 -1/24
m
(z)) (z))
Or 0m(mm11/ 1o--1/+---1// 1 "
min
no + nA
32

---

Algorithm 5 Wasserstein Transfer Learning with Empirical measures (WaTL-EM)
(k) ((
Input:
Target and source data {(=; {vij J==1 )) U (UISASK ((x) {uig Jz=1 ) regu-
larization parameter A, and query point & E IR..
Output: Target estimator m MG,EM(3)) (x.).
1: Empirical measures: 1/1
-==1 6 (g). (EM(3)
2: Weighted auxiliary estimator:
notRa kkz M
[() EM (x), where JEM JEM (x)
(K) " E. SiG (z)
3: Bias correction using target data: Fo,FM()) arg min (EL2(0.11 ginggige no
1 E (0)
(x)l -
A|og - TEM
(0) 4: Projection to Wasserstein space: MG arg new || - Fo
Algorithm 6 Local Wasserstein Transfer Learning with Empirical measures (LWaTI.-EM)
Input:
(k) Target and source data {(=;0) (Viy 1== = U ( (UISAsK ((/!") (uiy j==1 | regu- larization parameter A, bandwidth h and query point æ E R.
(0) (o
Output: Target estimator LL,h(2) (z).
Empirical measures: D!!) N. E5=1 8 (N)
2: Weighted auxiliary estimator FL,EM())=
K
nk nAJn.EM
H.M (z),
2(k)
110 + TA
k=0
where 7(0) (x) n E SiL (x, h) p(())
and nA
Σk1 kmml Tkg-
3: Bias correction using target data
No.
Ea (x) arg min
00) Si
(z. h)ll
-9l2 + Al1g Fn. 20
De (0,1) n2
4: Projection to Wasserstein space
(.) = arg min
FT11
fun.Es
NEW
where vb = maxigAsk (x) -2)) (x))l2 and source.
quantifies the maximum discrepancy between the target
Proof. First note the following result has been established 443 (n 1/2 + N-1/1) 1/1
720 (x) - (x) =
Also, an analogy of Theorem T can established in a similar way, just note that analysis in proof of
3 we have established that
K Tk
S (k)
(0))=
i=
33

---

Then using Theorem 7.9 in [3] we have
B
(z)l
F o {(=;",
{utg) 1=ر J U (UiSAsK {(a, (K),
(0) (o0)
{utj ) مرر )12))
(k) N/th
(z)1= Op
2N 2N
1/4 ni min
k=1
Hence have 11f fEM ll2 Op(NNmi ) Then the augments in the proof of Theorem still
hold F-11 (o)r> f(0), are replaced by their empirical-ment measure version. And the rate turns to be
Op (16 1/2 N
+ Σt=OVTE + (no + aa) N Nm notnA
Theorem 7. Assume Condition Sholds and the regularization parameter satisfies X x (no 1/2 11/2 N 1/41 -e for some > O. Then, for the LWaTL-EM algorithm and a fixed x E IRP, it holds that min
1/2
d (m (m)
Op (16= -1/2 1/11///11//- 1/2
min N 1 ( (N-1/4
K
+ 1r1 + 122 h-1/2
7
)1)
71
110
+
n1
A=0
k=0
where dh = m 11 (x) )1h)) (z))1-
Proof. Using the same arguments in the proof of Theorem in 43 we can show that
| (x) fN (z)l22 Op(m h 1/2
N min
1/4).
Analysis in the proof of Theorem 4
shows
bi = Op())
Then using Theorem 7.9 in 3
have
(k) (z. h)ll - FFbbll ((x!0)
(0)
3 U (UiSksK (tatt) Luis ا )
(k)
(Vig
1
S (2, Op
1/4 n min
Hence we have I fl. = Op (N
/i Then the augments in the proof of Theorem still
((U) fr. replaced by their empirical- version. And the rate turns to be hold if y000
Op (m) 1/2 n 112 + N
(no
min ( (N + th. + 12 + h-112 EK 1 ++ EK ))) I1
min
34


Reviews:
[{"review_id": "RyPwSiGzTY", "reviewer": "Reviewer_BkLd", "summary": "This paper proposes WaTL, a novel transfer learning framework designed for regression models where the outputs are probability distributions residing in the Wasserstein space. The core of the method builds on Fréchet regression: it first constructs a weighted estimator using both source and target domain data, then corrects for distribution shift bias using the target domain data, and finally projects the estimator back into the Wasserstein space. To mitigate negative transfer, the authors also introduce an extended variant of WaTL that performs a pre-selection of source domain datasets based on a discrepancy score, selecting only those most similar to the target domain. The paper provides a theoretical convergence rate analysis of WaTL and validates its performance on both synthetic and real-world datasets, demonstrating promising empirical results.", "questions": "1. Instead of training a new estimator for each source domain from scratch, could WaTL be adapted to use existing pre-trained models to reduce computational overhead?\n2. Have the authors considered using RKHS-based metrics such as MMD in place of the Wasserstein distance? It would be helpful to discuss the trade-offs and motivations for choosing Wasserstein geometry over kernel-based alternatives.\n3. The experimental comparison is limited to models trained only on the source or only on the target domain. A more informative baseline might include a model trained on combined (source + target) data, to better investigate WaTL’s performance improvements.\n4. The projection back into the Wasserstein space is an essential step in WaTL. Could the authors elaborate on whether this step is easy to compute?\n5. A brief clarification or context would be helpful for the notation E in line 80 and later.", "limitations": "yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "- Strengths\n1. The paper tackles an important and underexplored area in transfer learning, i.e., regression models where the outputs are probability distributions in Wasserstein space. This non-Euclidean setting naturally arises in many modern applications. The motivation for addressing distribution shift in such geometric spaces is well-justified, and the paper fills a noticeable gap in the literature.\n2. The paper is well-written, with a clear structure and precise technical presentation. Notation is generally well-defined, and the flow of ideas is easy to follow. The balance between theoretical depth and practical applicability is well-maintained, making the paper accessible to both theoretical and applied audiences.\n\n- Weaknesses\n1. WaTL requires training a separate estimator for each source domain. When multiple source domains are needed, this could become computationally intensive, especially if training each estimator is expensive.\n2. The method assumes access to data that sufficiently captures full probability distributions, which may not always be feasible in real-world applications.", "was_revised": true, "final_justification": "After reading the rebuttal and other reviews, all my concerns have been satisfactorily resolved. I believe this is a good paper, in particular, it tackles and proposes an effective solution to the challenging non-Euclidean setting. So I remain inclined to recommend acceptance.", "rebuttal": "We thank the reviewer for the thorough evaluation and constructive feedback. We appreciate your recognition of our work's importance in addressing the underexplored area of transfer learning for **probability distributions in Wasserstein space**, as well as your positive comments on the paper's **clarity and technical presentation**. Below, we provide detailed responses to address each of your concerns and questions.\n\n---\n\n> WaTL requires training a separate estimator for each source domain, which could become computationally intensive when multiple source domains are needed.\n\nThank you for raising this important practical concern. We conducted computational cost experiments comparing separate vs. combined estimation approaches across various settings. The results demonstrate that separate estimation is actually more computationally efficient:\n\n| Size of Target Site | K (Number of Source Sites) | Size of Source Sites | Method | Mean Time (ms) |\n|:---------:|:---:|:-------------------:|:--------:|:--------------:| \n| 50 | 5 | (10,10,10,10,10) | separate | **0.475** |\n| 50 | 5 | (10,10,10,10,10) | combined | 0.527 |\n| 2500 | 5 | (500,500,500,500,500) | separate | **7.593** |\n| 2500 | 5 | (500,500,500,500,500) | combined | 8.410 |\n| 5000 | 5 | (1000,1000,1000,1000,1000) | separate | **14.707** |\n| 5000 | 5 | (1000,1000,1000,1000,1000) | combined | 16.078 |\n\nThis efficiency gain occurs because each domain-specific estimator operates on smaller datasets and can be parallelized across domains. However, we acknowledge that leveraging pre-trained models could further reduce computational overhead, which we will discuss as promising future work in the revised version.\n\n> The method assumes access to data that sufficiently captures full probability distributions, which may not always be feasible in real-world applications.\n\nThank you for highlighting this important practical limitation. We acknowledge that the assumption of fully observed distributions is indeed restrictive in many real-world scenarios. However, our framework naturally accommodates empirical measures, and we address this limitation in several ways:\n\n**1. Theoretical foundation for empirical extensions**: While our main theoretical analysis focuses on fully observed distributions to establish fundamental transfer learning principles, the framework seamlessly extends to empirical measures. As discussed in Appendix E, the convergence rates can incorporate additional terms reflecting finite sample effects. The key theoretical insight is that empirical Wasserstein distances converge to true Wasserstein distances at a rate of $O_p(m^{-1/2})$ for $m$ samples per distribution.\n\n**2. Practical implementation with finite samples**: Our real-world applications (Section 6) already demonstrate this extension by using empirical quantile functions constructed from finite samples. For instance, in the NHANES physical activity study, we work with empirical distributions derived from accelerometer readings, and in the mortality analysis, we use empirical age-at-death distributions from country-level data.\n\n**3. Formal convergence guarantees**: The extended convergence rate becomes:\n$$d^2\\_W(\\hat{m}^{(0)}\\_G(x), m^{(0)}\\_G(x)) = O_p\\left(n_0^{-1/2+\\epsilon}\\left(\\psi + \\sum\\_{k=0}^K\\frac{\\sqrt{n_k}}{n_0+n_A} + (n_0+n_A)^{-1/2}\\right) + m^{-1/2}\\right)$$\nwhere $m$ represents the minimum number of samples per distribution. This shows that the finite sample effect ($m^{-1/2}$ term) is additive and does not compromise the transfer learning benefits.\n\n**4. Computational advantages**: Working with empirical measures actually offers computational benefits, as empirical quantile functions can be computed efficiently and stored compactly, making the approach more scalable than methods requiring continuous distribution estimation.\n\nWe will incorporate this formal extension and provide clearer guidance on sample size requirements in the revised manuscript.\n\n> Could WaTL be adapted to use existing pre-trained models to reduce computational overhead?\n\nThis is an insightful suggestion for enhancing computational efficiency. While our current framework requires domain-specific estimation to capture the geometric structure of the Wasserstein space, incorporating pre-trained models as initialization or feature extractors could indeed reduce training time. This represents a promising avenue for future work that could combine the benefits of transfer learning in parameter space with our geometric approach in distribution space. We will add this discussion to the future work section.\n\n> Have the authors considered using RKHS-based metrics such as MMD instead of Wasserstein distance?\n\nThank you for this insightful methodological question. While MMD and other RKHS-based metrics offer computational advantages in high-dimensional settings, we chose Wasserstein distance for several principled reasons:\n\n**1. Geometric foundations and theoretical guarantees**: The Wasserstein space provides a natural Riemannian manifold structure that aligns perfectly with Fréchet regression theory (Petersen & Müller, 2019). This geometric framework enables our rigorous convergence rate analysis (Theorems 1-3) and ensures that our transfer learning operates within the intrinsic geometry of probability distributions, rather than relying on extrinsic embeddings.\n\n**2. Structural sensitivity for transfer learning**: Wasserstein distance captures meaningful distributional shifts crucial for transfer learning—including support differences, mode shifts, and tail behavior—through its optimal transport formulation. This sensitivity to geometric structure makes it particularly suited for detecting when source domains are truly informative for the target, which is essential for our adaptive algorithm (AWaTL).\n\n**3. Computational tractability in our setting**: For univariate distributions, Wasserstein distance admits the closed-form expression $d_W^2(\\mu_1, \\mu_2) = \\int_0^1 |F_{\\mu_1}^{-1}(u) - F_{\\mu_2}^{-1}(u)|^2 du$ (Equation 1), making computation both efficient and interpretable without requiring kernel approximations.\n\n**4. End-to-end optimization**: Unlike kernel-based approaches that typically involve two-stage procedures (embedding followed by regression), WaTL operates directly in the native distributional space, enabling unified optimization without information loss from intermediate representations.\n\nWe acknowledge that MMD-based approaches have merits, particularly for high-dimensional distributions, and we will add this methodological discussion to Section 2.2 to provide readers with a clearer rationale for our design choices.\n\n> The experimental comparison is limited to models trained only on source or target domains. A more informative baseline might include a model trained on combined data.\n\nYou raise an excellent point about baseline completeness. We conducted additional experiments on the Human Mortality dataset, comparing our method against a naïve combination of source and target data. The results show that WaTL achieves an RMSPR of **0.022**, while the combined baseline yields 0.033, demonstrating that simple data pooling is insufficient due to distributional differences across domains. On the Physical Activity dataset, the pooled baseline yields RMSPRs of 41.15950 for females and 49.63236 for males, whereas WaTL improves these to **39.11525** and **48.23983**, respectively. These findings validate our bias-correction mechanism in Algorithm 1, Step 2, and underscore the necessity of transfer learning even when target labels are available. We will include this comparison in the revised experimental section.\n\n> Could you elaborate on whether the projection step back into Wasserstein space is easy to compute?\n\nThe projection in Algorithm 1, Step 3, is computationally tractable. Since the Wasserstein space $\\mathcal{W}$ is a closed and convex subset of $L^2(0,1)$, the projection exists and is unique. In practice, this optimization problem can be solved efficiently using convex optimization solvers like OSQP (Stellato et al.,2020), with computational complexity that scales well with the problem size. The quantile function representation (Equation 1) further facilitates this computation. We will add computational complexity details in the revised version.\n\n> A brief clarification would be helpful for the notation $\\mathbb{E}$ in line 80 and later.\n\nThank you for pointing this out. The notation $\\mathbb{E}$ represents the expectation operator. We will add this definition clearly in the notation section to improve clarity.\n\n**References**\n\n- Petersen, A. and Müller, H.G., 2019. Fréchet regression for random objects with Euclidean predictors. Annals of Statistics, 47(2), pp.691–719.\n\n- Zhou, Y. and Müller, H.G., 2024. Wasserstein regression with empirical measures and density estimation for sparse data. Biometrics, 80(4), p.ujae127.\n\n- Stellato, B., Banjac, G., Goulart, P., Bemporad, A. and Boyd, S., 2020. OSQP: An operator splitting solver for quadratic programs. Mathematical Programming Computation, 12(4), pp.637-672.\n\nWe sincerely appreciate the time and thought you have devoted to reviewing our work, and we are grateful for your insightful questions, which have helped enhance the quality of our paper. We would be thankful to know if our response has adequately addressed your concerns, and we are more than happy to provide any additional clarification if needed."}, {"review_id": "CDIuKRlJJe", "reviewer": "Reviewer_jbHn", "summary": "The paper studies transfer learning in regression tasks where the response is a distribution in Wasserstein space. The goal is to utilize relevant source samples to achieve a small Wasserstein error. The paper proposes two algorithms:\n\n- WaTL (Wasserstein Transfer Learning), which assumes that all source domains are informative.\n\n- AWaTL (Adaptive WaTL), which selects a subset of informative source domains in a data-driven manner.\n\nThe authors provide theoretical guarantees for the rate of convergence, corroborated by some experimental results.", "questions": "What is the main obstacle to extending the current results to the multivariate case? Would the rates change in this extended setting?", "limitations": "The paper does not have any potential negative societal impact.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\n - The paper studies a new setting where the responses are probability distributions, a setting not covered by the existing transfer learning literature.\n\n- It provides finite-sample error bounds that strictly improve upon target-only Fréchet regression when the sources are sufficiently close.\n\nWeaknesses and limitations:\n\n- The setting assumes that each response distribution is known in its entirety, whereas in practice we usually observe only finite samples. Although this issue is discussed in the appendix, there is no formal result for the case where the distributions are not observed directly, but only through samples.\n\n- The rates are compared only to vanilla Fréchet regression, and no minimax lower bound is provided.\n\n- The results are limited to univariate distributions in Wasserstein space.", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the careful evaluation and constructive feedback. We appreciate your recognition that our work addresses **a novel and important setting** in transfer learning with **distributional responses**. Below, we provide detailed responses to address your concerns and questions.\n\n---\n\n> The setting assumes that each response distribution is known in its entirety, whereas in practice, we usually observe only finite samples. No formal result is provided for the case where distributions are observed only through samples.\n\nThank you for highlighting this important practical consideration. While our main theoretical analysis focuses on the case of fully observed distributions to establish the fundamental transfer learning principles, our framework naturally extends to empirical measures:\n\n1. **Theoretical extension**: As noted in Appendix E, our convergence rates can be extended to incorporate an additional term reflecting the number of independent samples per distribution, following the approach of Zhou & Müller (2024). The key insight is that the empirical Wasserstein distance converges to the true Wasserstein distance at a rate of $O_p(m^{-1/2})$ for $m$ samples per distribution.\n\n2. **Practical implementation**: In our real-world applications (Section 6), we already use empirical quantile functions constructed from finite samples, demonstrating the practical feasibility of our approach.\n\n3. **Formal extension**: The convergence rate would become $d^2_W(\\hat{m}^{(0)}_G(x), m^{(0)}_G(x)) = O_p\\left(n_0^{-1/2+\\epsilon}\\left(\\psi + \\sum\\frac{\\sqrt{n_k}}{n_0+n_A} + (n_0+n_A)^{-1/2}\\right) + m^{-1/2}\\right)$, where $m$ is the minimum number of samples per distribution. We will add this formal result in the revised version.\n\n> The rates are compared only to vanilla Fréchet regression, and no minimax lower bound is provided.\n\nThis is a valid point that reflects the current state of the field. To our knowledge, minimax optimal rates for Fréchet regression in the Wasserstein space have not been established in the literature, making this a fundamental open problem. Our work provides the first finite-sample analysis for transfer learning in this setting, with rates that provably improve upon single-domain Fréchet regression (Petersen & Müller, 2019) when sources are sufficiently similar ($\\psi \\lesssim n_0^{-1/2-\\epsilon}$ in Theorem 2). Establishing minimax lower bounds for both standard and transfer Fréchet regression represents an important direction for future theoretical work.\n\n> The results are limited to univariate distributions in Wasserstein space.\n\nWe acknowledge this important limitation and appreciate the opportunity to clarify the challenges and our future research directions. The restriction to univariate distributions stems from fundamental computational and theoretical considerations:\n\n**1. Computational complexity**: For multivariate distributions, Wasserstein distance lacks the elegant closed-form expression $d_W^2(\\mu_1, \\mu_2) = \\int_0^1 |F_{\\mu_1}^{-1}(u) - F_{\\mu_2}^{-1}(u)|^2 du$ available in the univariate case (Equation 1). Computing multivariate Wasserstein distances requires solving computationally expensive optimal transport problems, which would significantly impact the scalability of our transfer learning framework.\n\n**2. Theoretical challenges**: The geometric structure of multivariate Wasserstein spaces is substantially more complex, involving intricate geodesic computations and more sophisticated empirical process theory. Our current theoretical analysis (Appendix C) relies heavily on the tractable structure of univariate quantile functions, which does not directly extend to higher dimensions.\n\n**3. Curse of dimensionality**: We anticipate that convergence rates would degrade with dimension $d$, likely incorporating additional factors dependent on the support dimension, consistent with known limitations in optimal transport theory.\n\n**4. Future extensions**: Despite these challenges, our framework establishes the theoretical and methodological foundation for multivariate extensions. We envision several promising directions:\n   - **Sinkhorn regularization**: Using entropy-regularized optimal transport for approximate but computationally tractable multivariate Wasserstein distances\n   - **Sliced Wasserstein approaches**: Leveraging one-dimensional projections to maintain computational efficiency while capturing multivariate structure\n   - **Hybrid methods**: Combining our approach with dimensionality reduction techniques for high-dimensional distributional data\n\n**5. Current impact**: Even in the univariate setting, our framework addresses numerous important applications, as demonstrated in Section 6 with mortality analysis and physical activity monitoring, where distributional outputs naturally arise as univariate quantities.\n\nWe will expand the discussion of these multivariate extensions and computational trade-offs in our revised manuscript.\n\n> What is the main obstacle to extending the current results to the multivariate case? Would the rates change in this extended setting?\n\nThe main obstacles are both computational and theoretical:\n\n1. **Computational obstacle**: The univariate Wasserstein distance admits the elegant quantile function representation $d^2\\_W(\\mu_1,\\mu_2) = \\int_0^1|F^{-1}\\_{\\mu_1}(u) - F^{-1}\\_{\\mu_2}(u)|^2du$ (Equation 1), enabling efficient computation and analysis. In the multivariate case, this closed form disappears, requiring expensive optimal transport computations.\n\n2. **Theoretical obstacle**: Our convergence analysis relies heavily on the bounded variation function space $BV((0,1), H)$ and associated covering number bounds (Appendix C). The multivariate extension would require more sophisticated function spaces and significantly more complex empirical process theory.\n\n3. **Rate changes**: Yes, we expect the rates to change substantially. The convergence rates would likely incorporate dimension-dependent factors, potentially following the pattern $O_p(n^{-1/2+\\epsilon}d^\\alpha)$ for some $\\alpha > 0$, reflecting the fundamental curse of dimensionality in optimal transport and high-dimensional statistics.\n\nDespite these challenges, extending to multivariate distributions using alternative metrics such as the sliced Wasserstein distance (mentioned in our conclusion) represents a promising direction that could maintain computational tractability while capturing multivariate structure.\n\nWe will incorporate these clarifications and the formal empirical measure result in the revised manuscript to better address the practical applicability of our framework.\n\n**References**\n\n- Zhou, Y. and Müller, H.G., 2024. Wasserstein regression with empirical measures and density estimation for sparse data. Biometrics, 80(4), p.ujae127.\n\n- Petersen, A. and Müller, H.G., 2019. Fréchet regression for random objects with Euclidean predictors. Annals of Statistics, 47(2), pp.691–719.\n\nWe deeply appreciate the time and effort you have dedicated to reviewing our work, as well as the insightful questions that have contributed to improving our paper. We would be grateful to know if our response has fully addressed your concerns, and we are more than willing to offer any further clarification should you require it."}, {"review_id": "PmilBXF1iO", "reviewer": "Reviewer_2tfp", "summary": "In this paper, the authors propose a framework, called Wasserstein transfer learning (WaTL), for transfer learning in regression models, with complex data structures. Specifically, WaTL can well hand the cases where the informative subset of transferable source domains is known or unknown.  Experiments on numerical simulations and real-world applications are done to verify the effectiveness of the proposed method.", "questions": "1.The authors motivate the method by stating the growing interest in extending transfer learning to more complex data structure but fail to give concrete examples. What are the examples of complex data structure, how the data structure complexity presents new challenges to existing transfer learning methods? Moreover, in second paragraph, the authors specifically use samples of univariate probability distributions, which, to this reviewer, is not considered as “more complex data structure” than multimedia data, e.g., images or videos. \n\n2.The line 28-29, it is not clear how this gap affects the effectiveness of the corresponding transfer learning methods. Moreover, it seems that it is restricted to methods using Wasserstein space, while the comparison with other transfer regression methods, e.g., transfer gaussian process, is not well discussed.\n\n3.Due to negative transfer, AwaTL is a more practical algorithm. The current way of measuring ‘relatedness’ or ‘discrepancy’ through AwaTl is a deterministic way of calculating $||f_0(x) -f_k(x)||$, the reviewer wonders if it can be done by using some learnable parameters? \n\n4.How the number of target labelled instances affect the performance of AwaTL?\n\n5.Time series extrapolation is a typical transfer regression setting. It is encouraged to include empirical results on such tasks.", "limitations": "Yes", "rating": 2, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strength:\n\n1.The paper introduces a Wasserstein transfer learning for transfer regression setting. \n\n2.WaTL algorithms for cases where the informative subset of transferable source domains is known, or unknown are introduced.\n\n3.Asymptotic convergence rates for the WaTL algorithm in the above two cases are presented.\n\n4.Empirical studies are done, and well support the effectiveness of the proposed method.\n\nWeakness:\n\n1.The paper is not well written and hard to follow. There are quite a lot of equations written in lines, which easily confused the reader. Moreover, a clear notation table is needed for better understanding of the algorithms.\n\n2.The motivation is not clear. See detailed questions below.\n\n3.The related works are insufficient. Wasserstein distance is a widely used technique in Transfer learning, but related works are not included. Moreover, transfer regression is a well-define transfer setting, and there are many methods, e.g., transfer Gaussian process, proposed for this setting, but they are not well discussed in this paper.\n\n4.The comparison baselines can be further improved by including more transfer regression methods, e.g. transfer Gaussian process.", "was_revised": true, "final_justification": "After the rebuttal, some of my concerns are addressed. However, I am still not very convinced on the empirical evaluation, which to me  is not solid enough. The main content only inlcudes one real-world dataset and it is artificially consctructed to fit the transfer setting. In fact, NHANES dataset is used as the test dataset in exisitng graph mining papers, e.g., [ref 1 2]. Moreover, the improvement compared with target-only baseline is quite marginal. In sum, I will keep my negative score. \n\n[ref1] Mining health knowledge graph for health risk prediction \n\n[ref2] Mining Heterogeneous Information Graph for Health Status Classification", "rebuttal": "We thank the reviewer for the detailed feedback and constructive suggestions. We appreciate your attention to the presentation quality and methodological concerns. While we respectfully disagree with some assessments, we acknowledge areas for improvement and provide detailed responses below to address each of your concerns.\n\n---\n\n> Clarity and presentation\n\nWe acknowledge this presentation concern and will significantly improve the manuscript's readability in the revised version, including better equation formatting, a comprehensive notation table, and improved section organization. We appreciate this feedback as clarity is essential for communicating our technical contributions.\n\n| Variable | Definition |\n|---------|------------|\n| $X$ | Predictor / covariate vector |\n| $\\nu$ | Distributional response |\n| $W$ | Wasserstein space |\n| $d_W$ | 2-Wasserstein metric |\n| $F_\\nu^{-1}$ | Quantile function of distribution $\\nu$ |\n| $m(x)$ | Fréchet regression function |\n| $m_G(x)$ | Global Fréchet regression |\n| $m_L(x)$ | Local Fréchet regression |\n| $s_G(x)$ | Weight function (global) |\n| $s_L(x,h)$ | Weight function (local) |\n| $\\theta$ | Mean of covariate $X$ |\n| $\\Sigma$ | Covariance matrix of $X$ |\n| $n_0$ | Target sample size |\n| $n_k$ | Source sample size for source $k$ |\n| $K$ | Number of source domains |\n| $f^{(k)}(x)$ | Auxiliary estimator from source $k$ |\n| $\\hat{f}(x)$ | Weighted auxiliary estimator |\n| $\\hat{f}_0(x)$ | Bias-corrected estimator |\n| $\\lambda$ | Regularization parameter |\n| $\\psi$ | Similarity measure |\n| $A_\\psi$ | Informative source set |\n| $\\hat{A}$ | Estimated informative set |\n| WaTL | Wasserstein Transfer Learning algorithm |\n| AWaTL | Adaptive Wasserstein Transfer Learning |\n\n\n> Motivation and problem complexity\n\nThank you for this important clarification request. Complex data structures in our context refer to data that do not naturally reside in Euclidean spaces and lack basic algebraic operations (addition, subtraction, scalar multiplication). Key examples include:\n\n**Concrete examples**: (1) Probability distributions (our focus) - cannot be meaningfully \"averaged\" as $p(x) + q(x)$ is not a valid density, (2) Networks/graphs - lack vector space structure, (3) Phylogenetic trees - require specialized metrics, (4) Symmetric positive-definite matrices - live on Riemannian manifolds.\n\n**Challenges for traditional transfer learning**: Conventional methods assume linear combinations of source and target data are meaningful, but this fails for non-Euclidean data. Our framework addresses this by using geometry-respecting metrics (Wasserstein distance) rather than assuming a linear structure.\n\n> The transfer Gaussian process is not well discussed.\n\nWe will expand Section 1.2 to provide more comprehensive coverage. However, we respectfully note a key distinction: Most existing work using Wasserstein distance in transfer learning (e.g., Courty et al., 2017; Redko et al., 2017) focuses on **domain adaptation** where inputs are transported between domains. In contrast, our work addresses **regression with distributional outputs** where responses themselves are probability distributions. This represents a fundamentally different problem setting analogous to how Li et al. (2022) developed specialized methods for high-dimensional linear regression transfer learning. We are willing to cite and discuss some papers on transfer Gaussian processes, such as Wei et al. (2022). Still, it's worth noting that transfer Gaussian processes operate in Euclidean output spaces and cannot directly handle our Wasserstein space setting without significant modifications.\n\n> More transfer regression methods.\n\nWhile we understand this suggestion, the comparison with methods like transfer Gaussian processes would not be methodologically sound because: (1) These methods assume Euclidean output spaces where linear operations are valid, (2) They cannot directly handle distributional outputs without fundamental modifications, (3) Our baselines (Only Target, Only Source) provide the most relevant comparison for demonstrating transfer learning effectiveness in the Wasserstein space. Adding incompatible methods would require extensive adaptations that would obscure the core contributions of our geometric approach.\n\n> Comparison with multimedia task\n\nRegarding the comparison with multimedia data: While multimedia data is high-dimensional, it typically can be embedded in finite-dimensional Euclidean spaces where linear operations remain valid. Probability distributions, however, represent a fundamentally different challenge—they reside in **infinite-dimensional curved manifolds** where the constraint that densities integrate to 1 creates non-linear geometric structure.\n\nThis distinction is crucial: multimedia data retains vector space properties despite high dimensionality, whereas probability distributions form **infinite-dimensional curved spaces** where traditional linear combinations are meaningless. This **geometric complexity**—the infinite-dimensional curved nature of the distribution space—not mere dimensional complexity, drives our methodological innovations in Algorithm 1 (bias correction) and theoretical analysis in Theorems 1-3. Our framework explicitly respects this **curved manifold structure** through Wasserstein geometry, enabling meaningful transfer learning where traditional linear methods fundamentally fail.\n\n> How lines 28–29 gaps affect transfer learning, and why Wasserstein methods?\n\nThe gap refers to the absence of transfer learning frameworks that respect non-Euclidean geometry. Traditional methods fail because they assume linear combinations of outputs are meaningful, which is false for distributions. Our Wasserstein space focus is motivated by: (1) It provides the natural metric for probability distributions with well-established theory (Villani, 2009), (2) The quantile function representation (Equation 1) enables computational tractability while preserving geometric structure, (3) It supports the Fréchet regression framework essential for our theoretical guarantees.\n\n> Learnable parameters to measure discrepancy\n\nThis is an intriguing suggestion for future work. Our current approach in Algorithm 2, Step 1 uses the empirical discrepancy $\\hat{\\psi}_k = \\|\\hat{f}^{(0)}(x) - \\hat{f}^{(k)}(x)\\|_2$, which is theoretically grounded—Lemma 1 bounds the deviation between empirical and true discrepancy. earnable parameters could potentially improve selection, but this would require additional theoretical analysis to ensure consistency of the adaptive selection procedure (Theorem 3), and may also introduce additional computational overhead. We will note this as a promising future research direction.\n\n> How does the target label size affect AWaTL performance?\n\nThank you for this practical question. The effect of target sample size on AWaTL performance has both theoretical and empirical dimensions:\n\n**1. Theoretical convergence**: Theorem 3 shows that AWaTL's convergence rate scales as $O_p(n_0^{-1/2+\\varepsilon})$ with target sample size $n_0$, indicating polynomial improvement as more target data becomes available.\n\n**2. Dual benefits of target data**: Target samples serve two critical functions in AWaTL:\n   - **Source selection**: More target data enables reliable estimation of discrepancy scores $\\hat{\\psi}_k$ for identifying informative sources\n   - **Bias correction**: Larger target samples improve the final estimation step\n\n**3. Empirical validation**: We are conducting experiments to validate these. The results will be included in the revised manuscript.\n\n> Time series extrapolation setting\n\nWe thank the reviewer for this suggestion. However, we must respectfully clarify that time series extrapolation is a fundamentally **different task** from the one we address. Time series extrapolation aims to predict future **scalar or vector values** based on past observations, with a focus on modeling temporal dependencies. In contrast, our work focuses on learning a regression function that maps a set of covariates to a **full probability distribution**. Because the problem settings, prediction targets, and core challenges are entirely different, applying our method to time series extrapolation would not be a meaningful evaluation of its contributions. Extending our geometric framework to handle time-dependent distributional data is an interesting but separate research direction for the future.\n\n**References**\n\n- Courty, N., Flamary, R., Habrard, A. and Rakotomamonjy, A., 2017. Joint distribution optimal transportation for domain adaptation. Advances in neural information processing systems, 30.\n- Li, S., Cai, T.T. and Li, H., 2022. Transfer learning for high-dimensional linear regression: Prediction, estimation and minimax optimality. Journal of the Royal Statistical Society Series B: Statistical Methodology, 84(1), pp.149-173.\n- Redko, I., Habrard, A. and Sebban, M., 2017, September. Theoretical analysis of domain adaptation with optimal transport. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases (pp. 737-753). Cham: Springer International Publishing.\n- Wei, P., Ke, Y., Ong, Y. S., & Ma, Z. (2022). Adaptive transfer kernel learning for transfer Gaussian process regression. IEEE transactions on pattern analysis and machine intelligence, 45(6), pp.7142-7156.\n- Cédric Villani.，2009. Optimal Transport: Old and New, volume 338. Springer.\n\nThank you again for your detailed feedback. We hope our clarifications, particularly regarding the unique problem setting of regression with distributional outputs, have helped address your main concerns. We believe our work offers a novel and theoretically-grounded solution to a challenging problem, and we would be grateful if you would consider these clarifications in your final assessment. We look forward to the discussion."}, {"review_id": "NqgxQVfBKX", "reviewer": "Reviewer_yT2Y", "summary": "The paper introduces an approach to transfer learning called Wasserstein Transfer Learning, designed for regression models where the outputs are probability distributions. \nThis paper introduces two algorithms: one for scenarios where the informative subset of source domains (the available source domains that are similar enough to the target one to yield relevant information for the learning process) is known, and another adaptive algorithm for cases where this subset is unknown which includes a way of identifying this subset. \nA rigorous theoretical analysis is given, establishing asymptotic convergence rates of the estimators. \nThe methodology is validated through simulations and real-world applications. \nThe results show that WaTL outperforms methods using only target or only source data.\nOverall, the paper presents a new approach on transfer learning by addressing the challenges of regression models with distributional outputs.", "questions": "Questions:\n\na. Is the transfer learning learnt estimator what would happen if an estimator was learnt using directly every available data, source and target at the same time? How worse would it be compared to the transfer learning one ?\n\nb. In the experiments what is the reason not to use the second algorithm with informative subset selection was used ? Which groups would be selected/rejected and why?\n\nc. Could the authors give more details on how their work connects with existing literature on transfer learning and the use of Wasserstein?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\n1- The paper tackles an interesting issue: transfer learning for probability distributions output. \n\n2-The theoretical contributions are thorough : convergence rates for the learnt estimators and guidelines for the choice of the regularization parameter are given in section 4.\n\n3- In section 3.3 and Algorithm 2 an adaptative selection procedure to identify informative subset of the source data is given and its ability to find those subset is illustrated in the numerical experiments of section 5.\n\nWeaknesses:\n\n1- The paper deserves more pedagogical clarification. Due to the focus on mathematical content some part of the methodology lack motivation/explanation. For example in section 3 in the algorithms the step that connects the source and target data via regularization (step 2 of Algorithm 1, step 3 of Algorithm 3) are only explained in two sentences while that are at the core of the transfer learning objective.\n\n2- While the problem tackled is explained, it is not clear how this paper fits into the existing literature on Wasserstein (and optimal transport) methods in transfer learning (and domain adaptation).\n\n3- Some key aspects are not defined early for example the term : \"informative subset of transferable source domains\" is used multiple times before being explained.\n\n4- The assumption \"all source data are informative enough.\" is not enough detailed.\nEven if I agree that this kind of assumption is common un transfer learning/domain adaptation, but it can be formalized (and sometimes estimated from samples) by explicitly defined a measure of distance between the distributions (see for example, On the analysis of adaptability in multi-source domain adaptation, Redko et et. 2019) \n\n5- In the empirical study there is no comparison with a baseline that would be computed on both the source and target data at the same time. It would be useful to compare it with the learnt estimator since the transfer learning estimator uses labeled target datas anyway.", "was_revised": true, "final_justification": "After discussing with the authors I have updated my rating to 4 and and improved my grading on the clarity and quality from 2 to 3. I believe the authors answered adequately my concerns regarding clarity and the lack of formalization of some key aspects by suggesting revisions. Therefore if the revisions are incorporated into the final paper I consider the final rating of 4 to be adequate. \n\nOne of my main concern about the paper quality was on the experimental side where I considered that in the real data setting a baseline learner on every distribution was lacking. The authors suggested adding it to the final revisions and have provided in their comments the numerical results. These performance of the method is closer to the suggested baseline than it is to the initial baseline but still show improvement over it.", "rebuttal": "We thank the reviewer for the thorough evaluation and constructive feedback. We appreciate your recognition of our **theoretical contributions** and the **novel problem setting**. Your suggestions for improving clarity and pedagogical explanation are valuable, and we provide detailed responses below to address each of your concerns.\n\n---\n\n> The paper deserves more pedagogical clarification. The regularization step that connects source and target data (Step 2 of Algorithm 1) lacks motivation/explanation while being at the core of the transfer learning objective.\n\nThank you for highlighting this crucial point. You are absolutely right that this step deserves a more detailed explanation. The bias correction step \n\n$$\n\\hat{f}\\_0(x) = \\underset{g \\in L\\^2(0,1)}{\\text{arg min}} \\left[ \\frac{1}{n_0} \\sum\\_{i=1}\\^{n_0} s\\_{iG}\\^{(0)}(x) \\| F\\^{-1}\\_{\\nu\\_i\\^{(0)}} - g \\|\\_2\\^2 + \\lambda \\| g - \\hat{f}(x) \\|\\_2\\^2 \\right]\n$$\n\nserves two essential purposes:\n1. **Bias correction**: The weighted auxiliary estimator $\\hat{f}(x)$ in Step 1 may be biased due to distributional differences between source and target domains, even when sources are informative.\n2. **Information transfer**: The regularization term $\\lambda \\left\\|g - \\hat{f}(x)\\right\\|_2^2$ pulls the solution toward the auxiliary information while the fidelity term ensures consistency with target data.\n\nThis follows the established paradigm in transfer learning (Li et al., 2022; Lin & Reimherr, 2024) where source information serves as a regularization prior that is corrected using target data. We will expand this explanation significantly in the revised manuscript and provide intuitive justification for this critical step.\n\n> It is not clear how this paper fits into the existing literature on Wasserstein methods in transfer learning and domain adaptation.\n\nWe will substantially expand Section 1.2 to clarify this positioning. The key distinction is that existing Wasserstein-based transfer learning work (e.g., Courty et al., 2017; Redko et al., 2017) focuses on **domain adaptation where Wasserstein distance measures distributional shift between input domains**. In contrast, our work addresses **regression where the outputs themselves are probability distributions residing in Wasserstein space**. This represents a fundamentally different mathematical setting:\n\n- **Existing work**: Transport input distributions $P_{\\text{source}}(X) \\to P_{\\text{target}}(X)$ while maintaining Euclidean outputs. For instance, Courty et al. (2017) considered wifi localization regression dataset with integer-valued outputs and other classification datasets in the experimental study in section 5\n\n- **Our work**: Regression with distributional outputs $Y \\in \\mathcal{W}$ where responses live in Wasserstein space\n\nThe output space geometry requires fundamentally different algorithmic and theoretical treatments.\n\n> Key terms like \"informative subset of transferable source domains\" are used before being explained.\n\nYou are absolutely correct. We will restructure the introduction to define this concept immediately after introducing the problem setting. Specifically, we will define the informative set $A_\\psi = \\lbrace 1 \\le k \\le K : \\| f^{(0)}(x) - f^{(k)}(x) \\|_2 \\le \\psi \\rbrace$ early in Section 3.1, along with intuitive explanation that these are source domains sufficiently similar to the target to provide useful information rather than introducing negative transfer.\n\n> The assumption \"all source data are informative enough\" needs formalization with explicit distance measures.\n\nThank you for this important suggestion. We will formalize this assumption more rigorously by defining the parameter $\\psi = \\max_{1 \\leq k \\leq K} \\left\\|f^{(0)}(x) - f^{(k)}(x)\\right\\|_2$ which quantifies the maximum discrepancy between target and source domains. The assumption \"all sources are informative\" translates to the formal condition $\\psi \\lesssim n_0^{-1/2-\\varepsilon}$ in Theorem 2, ensuring that domain discrepancies are small enough that transfer learning improves upon target-only approaches.\n\n> No comparison with a baseline computed on both source and target data simultaneously.\n\nThis is an excellent suggestion. We conducted additional experiments comparing WaTL with a naïve pooling of all available data under the same experimental settings described in the paper. On the Human Mortality dataset, the pooled baseline yields an RMSPR of 0.033, whereas WaTL achieves **0.022**. On the Physical Activity dataset, the pooled baseline yields RMSPRs of 41.15950 for females and 49.63236 for males, while WaTL improves these to **39.11525** and **48.23983**, respectively. These results demonstrate that simple data pooling underperforms due to distributional differences, validating our bias-correction mechanism and underscoring the necessity of transfer learning even when target labels are available. We will incorporate this important comparison into the revised experimental section.\n\n> Why wasn't the adaptive subset selection algorithm used in experiments? Which groups would be selected/rejected?\n\nWe used AWaTL in the simulation study (Section 5, Figure 1(b)) to demonstrate its source selection capabilities. In the real-world applications, we used WaTL because: (1) Domain expertise suggested all demographic groups could provide relevant information, (2) The relatively small number of source domains (3-4) made manual assessment feasible. However, you raise an excellent point—we will add AWaTL results to the real-world experiments to show which demographic groups are selected/rejected and provide insights into learned domain relationships.\n\n> Could you give more details on how your work connects with existing transfer learning and Wasserstein literature?\n\nWe will expand this discussion significantly. The key connections are:\n\n**Transfer learning literature**: Our work extends the \"source + target\" paradigm (Li et al., 2022) to non-Euclidean output spaces, requiring new algorithmic innovations (projection to Wasserstein space) and theoretical analysis (empirical process theory on function spaces).\n\n**Wasserstein literature**: While existing work uses Wasserstein distance for measuring input domain shift, we leverage Wasserstein geometry as the native space for distributional outputs, requiring fundamentally different mathematical treatment of the regression problem itself.\n\n**References**\n\n- Courty, N., Flamary, R., Habrard, A. and Rakotomamonjy, A., 2017. Joint distribution optimal transportation for domain adaptation. Advances in neural information processing systems, 30.\n- Li, S., Cai, T.T. and Li, H., 2022. Transfer learning for high-dimensional linear regression: Prediction, estimation and minimax optimality. Journal of the Royal Statistical Society Series B: Statistical Methodology, 84(1), pp.149-173.\n- Lin, H. and Reimherr, M., 2024. Smoothness adaptive hypothesis transfer learning. arXiv preprint arXiv:2402.14966\n- Redko, I., Habrard, A. and Sebban, M., 2019. On the analysis of adaptability in multi-source domain adaptation. Machine Learning, 108(8), pp.1635-1652.\n\nThank you again for your detailed and constructive feedback. We hope our clarifications and planned revisions have fully addressed your concerns. We believe our work offers a novel and theoretically-grounded solution to a challenging problem, and we would be grateful if you would consider these clarifications in your final assessment. We look forward to the discussion."}]

Author rebuttal:

Meta-review: This paper proposes a new method for transfer learning with 1D distributional output
using optimal transport and Frechet regression. They propose a method to transfer information from multiple source using bias correction on target data. they also propose a more adaptive method that select the most relevant sources domains to better handle domain variability. They study both  methods theoretical ad provide generalization bound that prove the interest of their transfer learning strategy. Numerical experiments on simulated and real world show that the proposed approach works better than classical baselines.

The reviewers found the paper interesting since it brings the first approach for transfer learning for distributional output with theoretical guarantees.  But they also had some
concerns about the paper being hard to read and follow, the missing theory with finite distributions,
limited experiments (distrib. in 1D, baselines missing), and missing positioning in the
literature. 

The authors did a detailed response to all those points with clarification
and additional experiments. Most of the reviewers were satisfied with the
responses and the paper and recommended an accept. One reviewer still had
concerns about the limited numerical experiments but was OK with the paper being
accepted thanks to the other contributions.  

I think the paper introduce a novel and interesting perspective on transfer learning so I recommend an accept
But I expect the final version to include the
discussed improvements including the full numerical experiments (with the other
baseline, Wasserstein regression and varying target sample size), new theoretical
results (finite distributions), and the additional details in the paper to make
it more clear.
Decision: Accept (poster)