Diversified Flow Matching with Translation Identifiability
Authors: Sagar Shrestha
Xiao Fu
Venue: ICML 2025 poster (2025)
Keywords: unsupervised domain translation
flow matching
GAN
identifiability
Primary area: general_machine_learning->unsupervised_and_semisupervised_learning

Abstract: Diversified distribution matching (DDM) finds a unified translation function mapping a diverse collection of conditional source distributions to their target counterparts. DDM was proposed to resolve content misalignment issues in unpaired domain translation, achieving translation identifiability. However, DDM has only been implemented using GANs due to its constraints on the translation function. GANs are often unstable to train and do not provide the transport trajectory information---yet such trajectories are useful in applications such as single-cell evolution analysis and robot route planning. This work introduces *diversified flow matching* (DFM), an ODE-based framework for DDM. Adapting flow matching (FM) to enforce a unified translation function as in DDM is challenging, as FM learns the translation function's velocity rather than the translation function itself. A custom bilevel optimization-based training loss, a nonlinear interpolant, and a structural reformulation are proposed to address these challenges, offering a tangible implementation. To our knowledge, DFM is the first ODE-based approach guaranteeing translation identifiability. Experiments on synthetic and real-world datasets validate the proposed method.

Diversified Flow Matching with Translation Identifiability
Sagar Shrestha Xiao Fu'
Abstract
et al. 2023). ) medical imaging (Kong al. 2021 Song
Diversified distribution matching (DDM) finds unified translation function mapping a diverse collection of conditional source distributions to their target counterparts. DDM was proposed to resolve content misalignment issues in unpaired domain translation, achieving translation identi- fiability. However, DDM has only been imple- mented using GANs due to its constraints on the translation function. GANs are often unstable to train and do not provide the transport trajectory information-yet such trajectories are useful in applications such as single-cell evolution analysis and robot route planning. This work introduces diversified flow matching (DFM), an ODE-based framework for DDM. Adapting flow matching (FM) to enforce a unified translation function as in DDM is challenging, as FM learns the translation function's 's velocity rather than the translation func- tion itself. custom bilevel optimization-bart training loss, a nonlinear interpolant, and struc- tural reformulation are proposed address these challenges, offering tangible implementation. To our knowledge, DFM is the first ODE-based approach guaranteeing translation identifiability. Experiments on synthetic and real-world datasets validate the proposed method.
al. 2024). and single-cell data analysis (Tong et al. 2023 Kapušniak et al. (024).
UDT is commonly realized by transporting the distribu- tion of the source domain to that of the target domain (see, e.g., (Zhu et al., 2017 Liu et al.) 2023b)). Distribution transport core task in modern machine learning. It is heavily studied in the context of domain adaptation, transfer learning, and generative models. Distribution transport can be realized by popular tools such as MMD (Long et al., 2016), GANs (Goodfellow et al. (2114), Shrödinger Bridge (De Bortoli et al. 2021), and continuous normalizing flows (CNF) (Lipman et al. 2023). However, it has been noted that UDT often loses control of the content be produced in the target domain. For example, in writing style transla- tion, a handwritten digit "7" could be translate to printed "3" under perfect distribution transport (Moriakov et al. 2020 Shrestha & Fu 2024). Similar issues are seen in Fig. where source pictures are translated to cartoons of unre- lated persons. This "content misaligmment" issue arises as UDT does not have identifiability of the intended translation function (i.e., the function that translates handwritten "7" to printed "7" and the function that converts profile pictures to cartoon faces without losing identities). There could be an infinite number of translation functions that can attain perfect distribution transport among the domains (Moriakov ct al. 2020 Shrestha & Fu 2024).
Many attempts have been made to address the content mis- alignment issue; see, e.g., (de Bézenac al. 2019 Zhu
1. Introduction
et al. 2017 Taigman et al. 2017 Liu et al. 2017 Xu et al.
Unpaired domain translation (UDT) aims to translate sam- ples from one domain to another (c.g., photographs to car- toons) while keeping the high-level semantic meaning (or "content")). Here, "unpaired" means that the translation is done without using paired cross-domain samples. UDT has achieved significant empirical successes in various appli- cations, such unpaired mage-to-imagee translation (Zhu
2022 for various regularization strategies to empirically
enforce translation identifiability. Notably, the recent work (Shrestha & Fu 2024) proposed the so-called diversified distribution matching (DDM) approach, which showed that UDT can be made identifiable if the translation function is learned from simultaneously transporting a number of diverse conditional distribution pairs.
let al. 2017: Choi et al. 2018 Huang et al. 2018 Yang
Challenges. The DDM criterion provides a theory-driven approach to avoid translation non-identifiability. However, similar as many works in this domain (e.g.. (Zhu et al. 2017
'School of Electrical Engineering and Computer Science, Ore- gon State University, Corvallis, Oregon, USA. Correspondence to: Xiao Fu <xiaao fullo..f@@eegeeeee www.oostatt.cccoooom
Xu et al. 2022 de Bézenac et al. 2019 Xie et al. (023 the DDM approach imposes structural regularization on the translation function realized by GANs. GaN-based
Proceedings of the 4nnd International Conference Machine
Learning. Vancouver, Canada. PMLK 267, 2025. Copyright 2025
by the author(s).

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Diversified Flow Matching with Translation Identifiability
methods sometimes suffer from numerical instability due to its adversarial training nature. More importantly, GANs learn the translation functions but not the continuous inter- mediate states between the source and target domains, i.e., the transport trajectories, yet the latter is critical in many ap- plications such as robot navigation and single-cell evolution inference (Tong et al. 2023 Liu et al. 2023a 2018).
Source
CycleGAN
FM
RM-OT
DOM-GAN (Q=4)
To overcome these challenges, natural thought is use
Figure 1: [Columns 2-4] Content misalignment issues in both GAN and FM based UDT (CycleGAN(Zhu et al. 2017), FMQLipman et al., 2023), FM-OT (Tong et al. (023) [Column 5] Result by DDM-GAN (Shrestha & Fu. 2024
flow matching (FM) (Lipman et al., 2023 Albergo al. 2023) for UDT. FM learns the velocity of the translation function and thus easily recovers the trajectories, In ad- dition, FM methods are friendly to train, using nonlin- ear least squares instead of min-max adversarial criteria. Nonetheless, using FM identífiabllit-ddiieen UDT turns out be quite nontrivial, as most existing works (e.g.. (Zhu jet al.
2. Background
Unsupervised Domain Translation. Consider two data domains (e.g., photos and sketches), denoted by X' S R and y S IRd, respectively. Assume that there exists a deter- ministic continuous mapping that translates every æ E X' into its content-aligned counterpart y E v...e.
2017; Xu et a 2022 de Bézenac et al.. 2019 Xie jet al.
2023 Shrestha & Fu 2024)) enforce identifiability
via imposing constraint/regullaiizttion on the translation function-yet, unlike GANs, FM does not have an explicit expression of the function. Shifting such constraints onto the function veloeity/traeecorry requires completely different designs, which have been elusive in the literature.
x ~ Pæy = gx(),,
(1)
where Pa: is the distribution of x and g" X + y is a differentiable bijective map. We stress that there might exist many g is such that g(x) E ) for all x E X, but our interest lies in content-preserving g (e.g., the one that changes the style of handwritten "7" to its printed version but keeps its identity). The goal of UDT is to estimate g" from the unpaired samples of Pa and py-
Contributions. In this work, our interest lies in an FM- based learning framework for DDM-based distribution whiph we call diversified flow matching (DFM). Our detailed contributions are as follows:
DFM with Transaltion Identifiability, We custom design the loss function and interpolant (i.e., the function to guide the trajectory of the flow transport) for saanttifaailiitgguaaattttt DFM. Conventional FM uses nonlinear least squares losses
Translation via Distribution Transport. In the literature (Zuu et al. 2017 Park et al. 2020)). the UDT problem is generally addressed by finding invertible translation func- tion g such that the distribution of the translated samples g (x) matches that of y, i.e.
and linear interpolants (Liu et al., 2023b Tong et al., 2023), which fail to realize DDM as they generate conflicting tra- jectories among different conditional distribution pairs. We propose to use a nonlinear, private interpolant function for each conditional distribution pair, and show that a bilevel optimization loss with this design provably retains the trans- lation identifiability of DDM.
find invertible g
(2)
subject to 9#p= = Py
Tangible Implementation. We propose an implementation that exploits the non-overlapping property of conditional distributions. This way, we show that the computationally demanding bilevel optimization loss can be recast into a more manageable two-stage approach, consisting of an in- terpolant learning stage and a flow training stage-both of which admit differentiable unconstrained losses and can be solved using simple pro-ppropagttion.
where 9 9wP. represents the distribution of g(z). The criterion can be realized by many computational tools, e.g., GANs (Goodfellow et al. 2014), and more recently, diffusion-based tools such as Shrödinger bridge (De Bortoli et al. 2021). and FM (Lipman et al.) 2023). It was noticed in the literature (Galanti et al.) 2018 Moriakov et al.) 2020) that solving 2 sometimes produces content-misalignn translations- mansnanions-mmnning that the desired g* in the ground- truth translation model (I) is not identified. Fig. (columns 2-4) shows the content misalignment issue that exists in both GANs and FM based UDT.
We test our method over synthetic data and real-world appli- cations (i.c., robot crowd route planning and unpaired image translation). The results corroborate with our theoretical analyses and algorithm design.
Many works showed that imposing more structural informa- tion on g in (2) could establish content alignment (or, the identifiability of the translation function g )) see, e.g., Be
Notation. We largely adhere to established conventions in machine learning; see also Appendix A
naim & Wolf (007)): |enaim et al. (2018) Xu et al. 2022);
2

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Diversified Flow Matching with Translation Identifiability
U1 black hair
In addition, it was also shown that the DDM criterion is robust to small violations of the SDC (see Appendix B.
U2 female
Pylur
Par u
DDM-GAN and Challenges. In (Shrestha & Fu 2024 Problem was solved using GAN-based framework:
Unified
g())
min max g (d Σ Pr(u((t) (Fy-)) [log [Nogrd (1)
P
Paluu w0
y M2
(1- + E Empp (al(t))
U1 Black Hair
where dd) is discriminator for the gth pair of conditional distributions. Cycle-consistency and backward translation were also used to enforce invertibility of which is omitted here for conciseness. The DDM-GAN formulation showed promising results (see Fig. D but two main challenges exist: First, GAN-based training is sometimes numerically unsta- ble due the min- min-max nature. Second, more importantly, the learned deterministic g does not contain the trajectory reflecting how Pa was changed to Py but such trajectories are critical information for number of domain translation problems, c.g., robot navigation and single-cell evolution inference [Tong et al. 2023 Liu et al. 2023a 2018).
U2: Female
The idea of DDM. The variable v9 can often be defined as attributes that are not supposed to change across domains. In (Shrestha & Ful 2024). it was shown Q 2 suffices underpin the translation identifiability.
Figure 2:
Moriakov et al. 2020). Among them, Shrestha & Fu (2024) showed an interesting theoretical result. To elaborate, the method in (Shrestha & Fu 2024) first defines correspond- ing conditional distributions Palu=a(u) and Pylu=u(o) for E [Q]. where ( is auxiliary information. For example, for face to cartoon translation in Fig. 2 n.(0) could be de- signed to represent some face attributes, e-g.. gender, that not supposed to change during the translation. Then, they proposed to learn a unified g for matching Q pairs of such conditional distributions using the so-called diversified distribution matching (DDM) criterion:
3. Proposed Approach
As mentioned, besides GANs, diffusion and flow based methods can also be used for distribution transport. The latter genre is known for their relatively simple training processes and the ability to reveal the transport trajectories. Hence, we are motivated to design an FM (Lipman et al.) 2023) based UDT method for carrying out the DDM prin- ciple 3). This turns out be nontrivial task. To see this, we start with some preliminaries of FM.
(DDM) find invertible g subject P Pyluto, Vig E [Q),
3.1. Preliminaries on Flow Matching
FM is an instance of class of generative models called continuous normalizing flow (CNF) (Lipman et al. 2023). CNFs learn time- varying differentiable map f. Rd
where we used Patu(() Pala==t()) Shrestha & Fu 2024
introduced the following condition:
Rd VI E [0, 1]. called the flow, such that fx) = 2,.
Definition 2.1 (Sufficiently Diverse Condition (SDC)). For any two disjoint sets A, B c X, where A and B are connected, open, and non-empty, there exists a u(A.B) {ui,..., ( such that S& Palu,a,s) (æ))xx + Ss Palu (w)dx. Then, the set of conditional distribu- tions (Paluto) 12 is called sufficiently diverse.
where 20 = x, 21 y. The flow St has a velocity field
V, : Rd + Rd:
d
ve v.( di ((()).
(5)
Using the vector field U2. one can easily translate given sample x to any intermediate state in the trajectory from æ to its corresponding y:
Under the SDC, there are at least two PDFs among (Pejuce) 1 that are sufficiently different over any A and B. was also shown that
f((x) = x =x+ I V. (2(2))ds, Le [0,1]
Theorem 2.2 (Translation Identifiability). (Shrestha & Fu 2024 Suppose that (Pajusso 1 satisfies the SDC. Let 9 be any optimal solution of the DDM criterion Then, we
(6)
and we have f((x) = g(z) = y. In practice, v, for trans- porting Pa to Py is parameterized by a neural network and
have 9- = g", a.c.
3

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Diversified Flow Matching with Translation Identifiability
learned via the following nonlinear least squares loss (AT
bergo et al.
2023 Lipman al. 2023 Liu et al. 2023b):
min Ve
(7)
pympp(..))
(0.1])
(...
(ve.dr.d..))
(a) Left:
Puiuto)-
Right: pliner (ar, y, t) as inter- polant trajectories.
where p(x, y) is any joint distribution of x, such that Ix p(a,, y)dæ= = Py)( and Jy p(z, y)dy = Pæ (z), and I: Fd x R [0, 1] + Fd is the so-called interpolant function which is differentiable with respect to t and satis- fies [a, y. 0) = x and [22, y. 1) y. The independent coupling p( y) = Pa (2)) (7) and the linear interpolant
Pai ul)
Ilnnear (xe, y, t) (1 t) + ty
Palei m
are most commonly used in the FM literature.
(c) Actual trajectories com- (d) 00.2 + 0.5y). puted from
3.2. Challenges of FM-based DDM
Figure (a) Samples of two pairs of conditional distribu-
While FM appears to circumvent some well-known chal- lenges of GANs-such as the instability of min-max optimization introduces its own distinet difficulties when used to realize the DDM criterion in (3). To explain, let us start with the following definitions:
tions. (b) Linear interpolant at t E [o, 1] for interpolant
trajectories. (c) Actural trajectories learned by solving (d) ūj points towards the - -E(x).
Definition 3.1 (Transport of Measures). A vector field v. is said to transport distribution w to r7 if the corresponding flow 9u = f, (cf. Eq. satisfies: [gu] #w = 17.
et al. 2023b)) This implies that the samples from Palus gets "reflected" back to Pylaz and that from Pælus u2 to Pylur i.e., "x" to "e" and "x" to "e" as shown in Fig. (())
Definition 3.2 (DDM Satisfaction). A vector field u, is said to satisfy DDM in (3) if it transports (0)) to Pylu(0) for all q E [Q). i.e., [9 (P Pyju(), Vig E [Q].
Using Private and Learnable Interpolants. The previ- ous example shows that the commonly used interpolant rlinesr y, 1) (1 t)a: + ty does not work for DDM. Note that the interpolant "guides" the velocity, and it is well known that the velocity is the conditional mean of the time- derivative of the interpolant (Albergo et al. 2023). Hence, to learn a legitimate i2 that transports Palu( to Pyluto for all q that satisfies the DDM criterion (3). a suitable interpolant needs to be selected.
Next, we show that the classical interpolant in (8) fails to attain DDM satisfaction.
Classic Linear Interpolant Fails. A naive way to imple-
ment DDM using FM is follows:
To this end, propose the following strategy: First, we let each pair Paju(s) and Pyju(e) use their own private inter- polant, denoted by [() Second, we design the f to be nonlinear, learnable interpolants. To proceed, we define a set of learnable I follows:
minimize 12
(FM luear Pg))
9=1
wtt where Pg = p(x, 3) )l = Pajutei Py) win (y). The
goal of (9) is to enforce unified vector field to transport Pojutt Pyjuls) for each u(9). Unfortunately, the loss ( cannot attain this goal in general. Fig. 3 shows typical failure case. The source and target distributions are both Gaussian mixtures with two modes. There, the red "x" are supposed to be translated to red "o" (same for the blue modes). However, the linear interpolants for two pairs of conditional distributions intersect at = 1/2.This intersec- tion results in 3 (f + }y) = 2(H + ty E[æ]], where o, is the optimal solution of (9) (see Fig. 3(d) and Appendix 3.4 for derivation; also see similar visualizations in (Liu
I = (Lo (x, v, Rd Rd x [0,1| Rd Lo(æ 0) x, Io y. 1) y. To differentiable w.r.t. t}
Using the private interpolant, a natural formulation to realize the DDM 33 appears to be the following:
Σ CPM (vt, (4)) p(x, y ))l()))
minimize 2
(10)
12 9=1
4

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Diversified Flow Matching with Translation Identifiability
xis Ma als Mi:
criterion:
minimize
|1 (0)
vel1² + 1 1r
(12a)
subject to
0 = arg min LFM (w) I(9), wl))
w 99)
(12b)
where p p(x, y ) for short and 1 is the indica- tor function of set I. Problem [2) is bilevel optimization problem, where we ensure DDM satisfaction by using two constraints (12b). The lower level optimization (12b) in- troduces private vector field v(() for each q and the upper level optimization forces consensus among all v/ Conse- quently, (T) holds for all q E [Q] when (12) is optimally selveedrecovering the DDM criterion in (3) This leads to the following proposition:
Figure 4: [Left] Success case of solving (10) [Right] Fail- ure case of solving I0P.
where I(9) E I and has learnable parameters 0g- The hope is that I(0) will be learned in a way such that at simultaneously minimizes each c(g) for q q=1l 1, . in (10) in order to attain DDM satisification.
Proposition 3.4. Suppose that there exists flow S Rd + IR, continuously differentiable in time and space, from Pa to Py such that fi = =g" Suppose there exist a diffeomorphism from the standard Gaussian N Ia) to Palu(u); Vg. Let Dt denote a solution of Problem [1) and
Pitfall of Loss I0 However, turns out that 10 is not a
correct criterion. Fig. shows the result of solving 10) on
the Gaussian mixture example (see details in Appendix B.4 One can see that it sometimes works [Left] yet sometimes fails [Right]-no matter how hard one tunes the optimiza- tion algorithm for solving (IO). This drives us to discover the following fact:
denote go gg(x)) x + 1 (zz)).. Then, under the same model in ( when {Pajuto 19=1 satisfies the SDC, we have 9s = g" holds a.e.
Fact 3.3. The problem formulation in 10 is not equivalent
to the DDM criterion in 3
Proposition 3.4 shows that is viable to use an FM-based loss attain the same conclusion of Theorem 2.2 Note that Theorem 2 was established under the premise that DDM is attained, yet Proposition 3.4 specificclly needs the distribution matching part to be realized by FM. This gap is filled by the bilevel loss design in (2)..
To explain, note that we hope using the loss in (10 to find
6,(2) = R[O, (2r, 1) (4)) (ar, 1) = z], (11)
for all q E [Q). where the expectation is taken over p(a, 0)) Under the model in ( such i, and ((0) for all q could exist (which we will discuss later in more details). However, when minimizing the (0),
3.4. Implementation: Exploiting Structural Constraint
Unlike (T)), which suffers from theoretical flaws, 12 is theoretically sound in establishing translation identifiability. However, it requires solving a bilevel optimization problem. While this can be tackled using off-the- techniques such as implicit gradients and gradient unrolling (see (Zhang) et al. 2024 ), computational efficiency remains concern.. In addition, for problems where the lower level optimization part is nonconvex and intractable, convergence of bilevel optimization is hard to guarantee.
it is possible that one finds (ör, (ū,,T()) (()) for certain g's ( I
such that (g)
((g) LFM (0,, 7(9) yet e +
E[0,i(9)
y. z). This pathological case could happen because 7(4) is a learnable term whose scale can be changed to attain an overall small FM How- ever, such small loss values do not reflect the real goal of distribution transport. In other words, due to the change- able ((9) and the setting Q > 1, having a small value of PM (Vr,7 (00) does not mean [1] is met for q, which makes (10) problematic for enforcing DDM.
((9) (z. v.
((g)
To avoid these computational barriers, we propose to sim- plify the Problem (12) by exploiting the structural property of conditional distributions that naturally arises in many cases. Specifically, the auxiliary information u often corre- spond to semantic attributes/labels (e.g., gender for image translation), which induce roughly pon-overlapping clusters. To utilize this structure, let us assume the following:
3.3. Proposed Criterion: A Bilevel Learning Loss
The above discussion shows that we need IT to hold for each q. Hence, instead of using a sum of LSSStypi loss,
Assumption 3.5 (Non-overlapping Supports). For
we propose the following diversified flow matching (DFM)
(Pajuto) 0 1p we have supp(Pæ)u.) O supp(pa)u,) d
5

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Diversified Flow Matching with Translation Identifiability
and supp(pyla.) n supp(pylu,) Φ. Vi + j. where supp(p) {= p(z) > 0).
the values of [(x(.) y the two values are close in time and space.
(1) and [(x((), y (22) apart, if
Simplifying the Bilevel Loss. Let I denote the learned
Under the above assumption, it is possible to design proba- bility paths Putes: Vq, that satisfy the boundary conditions Pl(e) = Paju(o) and Pkgg) = Pylu(e), such that P/cc 's sup- ports are non-overlapping among all E [Q]. Learning unified vector field that simultaneously follow the P( for all q E [Q] will then ensure that the vector field satisfies DDM.
non- -nnersttrgg interpolant. Then [2 can be reformulated
as follows:
minimize ve.0!"
Σ
11 vell
(15a)
(9)
9=1
subject to v/()) = arg min CFM (wy, p(o))
(15b)
Designing such -nt--eettrsectngggtoo P can be achieved by designing the interpolants themselves. This is due to the well-known fact:
wi
In contrast to (12). I is not learnable in Problem (1).. This
allows to further re-express the formulation as:
Fact 3.6. (Albergo et al. 2023) The distribution Pr is the same as the probability distribution of the random variable I(9) (x, y, t) where æ, y ~ p(z, y ())).
minimize
LeM((ur p(a, ylu
(16)
Therefore, it suffices to select interpolant I (99) for all g that do not "intersect" with each other in the following sense: Definition 3.7 Non-intersecting Interpolants). The inter- are non-intersecting Vi,j E [Q). j. 1=1
Note that we have eliminated the slack variable representing the private w(() and the consensus loss in (15a), as no private r is needed in our loss function.
polants (I(9) ( Q
The proposed algorithm is referred to as DFM and is detailed in Algorithm in Appendix
1 (x(0, y()) (6) .t) + 16) (z()), (0),
for all (x/1), g()) E supp(p(æ,, y () E (i.1}.
4. Related Works
As we will see, non-intersecting interpolants will allow to greatly simplify the bilevel loss so that the unified vector field can be learnt efficiently.
UDT via Distribution Transport. Distribution transport is arguably the most widely used approach in UDT (as well as closely related techniques such as domain adaptation (Long et al. 2016)). In the literature, distribution transport is realized using various methods such as minimization of maximum mean discrepancy (MMD) (Long et al., 2016))
Learning Non-intersecting Interpolants. A major benefit of exploiting the non-overlapping support structure in As- sumption 3.5 as follows: Under Assumption [3.5] assume that ( for all q E [Q] are learnable universal path repre-
GANs (Zhu et al. 2017 Huang et al.
2018 Choi al. 2021
senters. Then, there exists set of f(9) for q [Q] that are non-intersecting. This is simply because the starting and ending points mapped by 7(() and ((5 are completely dif- ferent. Further, using Assumption 3.5 one can use a unified = (() to express the non-intersecting interpolants for transporting Pelu(o) to Pyluco for all q E [Q]; that is,
2020 bridge matching (De Bortoli al.
|
Liu et al.
2023a De Bortoli et al. 2024). and FM (iu et al. 2023b
Eyring et al. 2024).
Translation Identifiability. The non-uniqueness of the distribution transport maps in UDT is a well-known issue MMorikov al.) 2020; (Shrestha & Fu, 2024 Galanti et al., 2018). Many hypothesized that the desired map g is likely to be the optimal transport (OT); see (Liu et al.)
2023b
T(x, 1) = Σ a. vesupp(oto)) f(0)
(x. ( 1)
(13)
and (De Bortoli et al. 2021 2024 While OT maps have been shown to be effective in some applications such as single cell analysis (Bunne et al., 2024). it is in general not clear if g is the OT map. There also exist many empirical ways to constrain the feasible set of the transport maps, making the found ğ with intended behaviors (Amodio & Krishnaswamy 2019 Xu et al.) (022). Recently, (Shrestha & Fu 2024) showed that it is possible to identify general (non-OT) translation function under the SDC.
To find non-intersecting unified interpolant, we use the following learning criterion:
Q-
Q
i = arg min
Σ
Σ
E, to
(111-112111 - xx (14)
1
1= 1=1-1
()) Yo, (x) ()) ,41) - I (1) ,y (12)112)],
Bridge and Flow Matching for UDT. Following the rise of diffusion and FM based generative models, many works have emerged to apply the continuous density transport
where Yo (a) = max(exp(-dd -aap /2/20²²) ,,)) and y > 0 is a
small constant. Simply speaking, Problem (1) tries to push
6

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Diversified Flow Matching with Translation Identifiability
perspective onto UDT. Two major classes are notable. The first class is the FM-hased approaches that use deterministic continuous time process characterized by an ODE (Liu et al.
5.1. Synthetic Data Validation
Setting. We use two layer MLP with 64 hidden units and SeLU activations to represent (e(: o) well as I Ig. We use Adam optimizer with an initial learning rate of 0.001 for and 0.0001 for Io. We use batch size of 512. Details of the proposed algorithm is presented in Algorithm | We run both phases of Algorithm for 2000 iterations.
2023b Eyring et al. 2024 Kapušniak et al., 2024 Kornilov et al. 2024, Gazdieva et al.
2023). The second is diffusion and schrödinger bridges (Liu et al. 2023a De Bortoli et al. 2024, Sasaki et al. 2021). characterized by SDEs. These methods do not consider the translation identifiability and thus issues in Fig. [ arise. In addition, they are designed to transport among only one pair of distributions, which is hard to use for DDM as in this work.
Metrics. We use the Earth Mover's Distance (EMD) to assess the distribution matching between Pylu(e) and Poju(e); Vy. Similarly, we use translation error to assess the identifiaiility with respect to the true translation function g", which is defined as follows
Using Auxiliary Information in FM. Auxiliary informa- tion has been incorporated into FM in the context of learn- ing conditional generative models (Atanackovic et al., 2024 Zhu & Lin 2024). These approaches typically learn sepa- rate vector fields (y (n())) for each conditioning variable n(9), treating the auxiliary information as a condition. As a result, they avoid the need to construct unified across all conditional pairs and do not require designing nonlin- ear, learnable interpolants. This makes them fundamentally different from the setting considered in this work.
N Σ
1 N
Translation Error
1
ynl12
where yn g" (2x,) target translation and Dn = I go()n))
In + S De(zz)dt is the predicted translation.
3D Gaussian blobs. Fig. 5 shows the results of a setting where we generate Gaussian mixtures p(æ) with d = 3 and use y = g"(x) - to generate samples from p(y), where each Gaussian component is with unit vari- ance. Trajectories obtained by the baselines FM-cond and FM-cond-OT show the "reflection" effect discussed in Fig. 3 which results in p())u(f) being very different from p(y) ().. However, the proposed method success- fully finds vector fields that circumvent the reflection issue associated with this setting, correctly transporting Pajats) to Pyluc)) Vgg E {1,
5. Experiments
Interpolant Construction. In order to construct the learn- able I, we parametrize Ie using the following form:
Io () = (1 - 1)ar ly + ((1 (a0 U, (17)
where Yo: : Idd x Rd x [0, 1] + Idd is a neural net- work. Note that Ig constructed this way satisfies the boundary conditions by design, i.e., (a y, = and Lo y, 1) == y. Similar constructions have been used in the literature (Kapušniak et al.) 2024).
2D Gaussian blobs. Fig. 6 shows the performance of DFM and baselines when = 2. Note that the = 2 case is arguably more challenging than the d = case, as the transport trajectories have less space to explore for collision avoidance. Nonetheless, one can see that DFM learns Ut such that the two Gaussian blobs travel at dif- ferent speed (see the color bar of / in Fig. 6 to avoid col- lision and reflection, successfully transporting Paju(() to Pyju(e) 4q E [1,2]. This interesting behavior is not ac- quired by FM-cond or FM-cond-OT, showing the impor- tance of designing T.
Baselines. The major baselines used throughout this sec- tion are plain-vanila flow matching (FM) (Albergo et al.]
2023 Liu et al., 2023b Lipman et al. 2023) and FM with minim-bated optimal transport coupling (FM-OT) (Poola- dian al. 2023 Tong et al. 2023). We also modify the plain-vanilaa FM and FM-OT to incorporate auxiliary vari- ables following Eq. tto show that without custom de- signed interpolants, simply using auxiliary variables does not attain DDM satisfaction or translation identifiability. The two modified algorithms are referred to as FM- cond and FM-cond-On respectively. For Sec. [5.3] and the nav- igation experiment there, we further use metric flow match- ing (MFM-OT), and its auxiliary-van varrialee modified version MFM-cond-OT (also following Eq. (.). For Sec. 5.2 and the image translation task there, we also include GAN- CycleCAt (Zhu et al.) 2017) and DDM-CAN (Shrestha & Ful 2024) and diffusion- -assed baselines SDEdit (Meng et al. 202T and EGSDE (Zhao et al. 2022).
Quantitative Results Table 1 shows the mean and stan- dard deviations of EMD and TE attained by various methods averaged over 10 trials. It shows that the proposed method successfully transports Pa) to Pyju(n) Vq and identifies g more accurately relative to the baselines.
5.2. Image Translation
In this subsection, we demonstrate the efficacy of the pro- posed method on an important UDT task, namely, unpaired image to image translation. We use images of human faces from the CelebAHQ dataset (Karras et al.) 2017) with
7

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Diversified Flow Matching with Translation Identifiability
Tou di1 We
y04
9%
59
Input
FM-cond
FM-cond-OT
Praposed (front view)
Proposed (top view)
Figure 5: Trajectories returned by all methods for the 3D synthetic data.
Input
FMcond-O0 OT
xtt
0.5
gtn xit
0.4
53
50
Figure 6: Trajectories returned by all methods for the 2D synthetic data. Colorbar indicates time t.
Table 1: EMD and TE attained by the proposed method and baselines for the synthetic data.
FID vs Iterations
biall
trial 2
trial trial bials
200
EMD
TE
Method
2D Gaussiuns
3D Ganssians
Gaussians
3D Gaussians
150
FM-cand
9.64 0.51
0.45
0.43
10.115 0.388
FM-cond OT
9.56 1: 0.41
9.44 :: 0.49
9.86 4: 0.37
9.994 :: 0.428
100
DFM (proposed)
0.06
0.05
2.42 0.04
3.196 0.034
50
Iteration
30, 000 images as the source data P= and Bitmoji faces with
10k
20k
30k
40k
50k
4084 images (Mozafari 2020) the target data Py- To avoid data imbalance, only randomly selected 5000 images from CelebAHQ. Both image domains are resized to have a size of 256 x 256. The Bitmoji faces are first center-cropped to 80% of the original image before resizing. All FM-based methods are trained on the latent space of the VAE from Stable Diffusion v1 (Rombach et al. 2022). We FID measure the distribution matching performance, while translation identifiability is visually checked from con- tent alignment. The auxiliary information used for this task is the gender, i.e., #1 = "male" and = "female". More details and hyperparameter settings are in Appendix [D.1
Figure 7: Convergence issues of DDM-GAN
DDM-GAN the sequel.
We observe such issue in the considered experiment (see Appendix B.5 for results). As mentioned, such hardness in adversarial optimization is reason motivating DFM approach. Therefore, we present the best FID attained by DDM-GAN in the sequel.
Result. Fig. g shows the qualitative results of translation obtained by all methods. One can see that CycleGAN, FM, and FM-OT, which does not use the auxiliary information of gender, suffer from content misalignment issue. DDM-GM does not show satisfiable content alignment either, probably due to the numerical instability as demonstrated in Fig. 7 FM- cond, the naive FM based implementation of DDM us- ing linear interpolants in ( show better content alignment than the other baselines (c.g., see first row). Nonetheless, DFM shows the best alignment, supporting the translation
DDM-GAN's Convergence Issues. As discussed in Sec. 2. GANs could encounter convergence challenges in some scenarios. Fig. shows case where the dataset size is up to 5000 and
that FID exhibits quite erratic behaviors when the number of iterations increases. Note that FID > 70 is unacceptable quality (see Appendix for an example). As we mentioned, such hardness in adversarial optimization a reason motivating our DFM approach. Therefore, we present the best FID attained by
2. One
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Diversified Flow Matching with Translation Identifiability
Source
DMM
DOM-AN
CydeeGN
FM
FM-OT
FM-cond
SOE-e
FOSDF
30
T
+ 3
Figure 8: Human faces to Bitmoji translation by all methods.
Table 2: Quantitative results on the Face to Bitmoji task.
information. This motivated us to design a FM-based DDM approach. We first revealed the unique challenges of impos- ing DDM-induced constraints on flows. Then, we designed bilevel formulation with private learnable nonlinear inter- polants, provably recovering the DDM criterion using FM. We also provided an efficient two-stage implementation, avoiding computational barriers. Experiments demonstrated that DFM effectively computes the DDM criterion, serving as the first translation adletifiability-guaaattee flow model.
Method
FID DreamSim Train Time (hrs) Infer. Time (s)
GAV-based
DOM-GAN [Shrestha & Pu 2024
26.20 0.58 (0.05)
13:60
0.911
CycleGAN
Diffrrion bared
SDEdit (k-3)
63.37 (0.06)
27.22
BOSDE (k=5)
69.93 0.56 (0.06)
31.77
65.82
FM-hhzed
HM (Albergo et al 2023
43.22 (1.66 (1.06)
3,08
5.28
FM-comu 5
22.65 0.60 (0.06)
3.15
5.30
FM-OT Tang
(0.06)
DHM (Chirs)
22.20 0.59 (0.05)
6.17
5.27
Limitations. First, our UDT framework is restricted to one-to-one translations, whereas many applications, such as image translation, can benefit from one-to-many ttmm map- pings. Extending FM-based methods to enable identifiable one-to-many translations is promising yet challenging di- rection. Second, our efficient computing scheme relies on non-overlapping supports of the conditional distributions. For overlapped cases, how to efficiently realize the bilevel loss is worth considering in the future.
identifiability claims of DDM.
Table 2 presents the quantitative results of all methods. FID assesses the distributional similarity between the translated and target images, while DreamSim evaluates content align- ment between the source and translated images. The pro- posed method achieves the best balance between FID and DreamSim, indicating that it preserves high-quality domain translation without sacrificing content consistency.
Acknowledgment
5.3. Swarm Navigation
This work was supported in part by the National Science Foundation CAREER Award ECCS-2144889.
We also test our algorithm on an interesting robot swarm navigation problem (Liu et al. 2018 2023a); please refer to Appendix C.I for details.
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. There many potential societal consequences of our work, none which we feel must be specifically highlighted here.
6. Conclusion
In this work, we introduced DFM, a computational frame- work that integrates FM into the DDM criterion. DDM is a powerful criterion for UDT, which provably attains trans- lation identifiability in UDT, solving content misalignment issues. DDM had only been realized by GANs, encoun- tering numerical instability and losing transport trajectory
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Diversified Flow Matching with Translation Identifiability
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Diversified Flow Matching with Translation Identifiability
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Diversified Flow Matching with Translation Identifiability
Supplementary Material of "Diversified Flow Matching with Translation Identifiability"
A. Notation
1. (t,(t, represents the partial derivative of f with respect to t.
2. [fl#p and #p represents the purw-forward of the density p by the map f & y, i..., satisfies f (p((A) (A)) for a measruable set S V, where f is the pre-image of f and p(B) denotes the measure set under the distribution specified by density p.
3. Ls E IRdxd represents the identity matrix.
4. 0,f, represents the partial derivative of fe with respect to I.
5. V e represents the divergence of vector field v.
B. Details on Proposed method and Challenges
B.1. Robust Identifiability (Shrestha & Fu 2024)
Theorem B.1. (Shrestha & Fu 2024 Let g be any optimal solution of the DDM criterion Let dia(A) = su w
z|12 measure the size of set. Let V {AA, B) SDC is violated on (A, B)}. Assume that g" is an L-Licchitz continuous function and that (aX(e. max(dia (B)( Then,
(Ø(x) g" (æ) S 2rL, Vo E X.
That is, g* is identified reasonable accuracy by the DDM criterion if the SDC holds approximately. Morevoer, the translation error increases only linearly with the size of sets in which the SDC condition is violated.
B.2. Algorithm DFM
Algorithm DFM
Require: 11 02, initialized parameters o,
while Stopping criterion for is not met do
2: Sample (x:( y())) p(x, y (l))) and by (0, 1), 44 [Q]
3:
"t,,
(q)
- (1 (z) + ttyy( + +ty(1 - (z ((1)), Vg E [Q]
4: Linterp (0) + Σ
1 Σj=1+1 Yop (1t; ty Yo, (11= 21122
(i) (3)
5:
0 ++0 - V Cinterp
6: end while
7: while Stopping criterion for u Φ) is not met do
8: Sample (x, y) ~ p(æ, 9) and ~ U(0,1) 1)
9:
2t t (1 t)x ty + tto y)
10:
(()) [v, 0,,,11
11:
o à a2 (0)
12: end while
13: return v)(:;))
In Algorithm represents gradient- gradient-based update direction. The specific optimizers used in the experiments are described
in their corresponding experiment sections.
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Diversified Flow Matching with Translation Identifiability
FID vs Iterations
100
80
60
40
Training iteration
20
10k
20k
30k
40k
50k
Figure 9: Image quality during the GAN training when convergence issues were encountered.
B.3. Derivation 0g (2 + ( in Fig.
If the vector field v, (z) is solution to
then can be expressed as follows:
(z(2)=
0₂1 linear (x, y.t)
linear
(x, y.
(1) pu a (tu youp ((())
x (1 - f)x ty
w -p pal(()) yrup ul4))
i (z) (x Pelll
12z 2mc
Eg-pa [2x - 2x;
2(z B[xx])
Hence By points towards the mean of the two clusters, Paju(u) 1,2, resulting in reflection.
B.4. Pitfalls of Problem 23
23 works in practice, we run simulations on 2D and 3D synthetic data as presented in Fig. 4 We observed that directly using Problem (23) is challenging from optimization perspective, since the unified vector field need to minimize LFM simultaneously with respect to all conditional distributions. Instead the following variable splitting based implementation that decouples LFM loss minimization and vector field unification was found to be optimization friendly:
In order evaluate how well Problem
LFM (0(9) (1)) p(x, )(l + AE, ( (v((z)) -500 ((0)()))
minimize v. 0 1=1
(18)
where Pt (4)
implies 2, 1 (x, y, t). (x. y) p(x, )))
Experiment Details. The 2D Gaussian blobs for Parus and Palus for Fig. 4 are generated with unit variance at locations
(1, and (1, 1)). = x is used to generate Pylu: and Pylua- For 3D plot, the location of the blobs are (1, 0) and 11)
B.5. Convergence issue of GAN-based UDT
Fig. 9 shows an example training trail of DDM-GAN for the setting considered Sec. 5.2 along with the sample translated images. It shows that once the convergence issues are encountered, the translation quality is unacceptable and can be considered an optimization failure.
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Diversified Flow Matching with Translation Identifiability
C. Additional Experiments
C.1. Swarm Navigation
Navigating a large swarm of robots, on land or in air, requires computing the control policies (i.e., the velocities) for individual robots that meet the specifications for their collective behavior (Liu et al. 2018). CNF-based methods can be used recover individual robot control policy based on the robot's location and time such that the entire swarm navigates
from source location to the destination (Liu al. 2023a Kapušniak al. 2024 Liu al. 2018).
Problem Description. We consider swarm navigation a complex land surface. The surface is specified by LiDAR measurements of Mt. Rainier (Legg & Anderson, 2013) containing 34,183 points. We consider a challenging scenario where there are two different swarms of robots with their own source and destination locations on the surface as shown in Fig. [0 Columm IJ. We consider the source locations of the first and second swarm distributed as Pælu, and Paluu and the destinations distributed as Pylmy and Pyluz. respectivly. Then, the goal to transport samples from Paju(s) to Pylu(e), Vq, which aligns with the objective of DFM. This problem can be understood clusters of robots so that they move from source destination clusters, while avoiding collision among the clusters.
DDM problem. We want to control the
Neural Networks and Hyperparameter settings. We use a 3-layer MLP with 64 hidden units and SeLU activation to represent both Lg and vr(:;g) We use Adam optimizer for both interpolant and the vector field with an initial learning rate of 10 and 10 respectively. We use a weight decay of 10-5 for both networks. We use 01 = 0.1, 02 = 1.5.
Regularization for surface adherence In order to encourage the swarm to stay close to the land surface, we add a regularization from (Kapušniak al., 2024) in the first phase of our method, i.e., interpolant training in Algorithm Since Problem (12) itself does not encourage trajectories to stay close to the land surface represented by LiDAR measurements, we use additional regularization from (Kapušniak et al. 2024) that encourages trajectories to stay close to the surface. To explain, let D {m,} ) be the set of LiDAR measurements. Then, Line 4 and 5 in Algorithm | is modified as follows:
(() λ1 Cittepp(0 (0) A2 (0).
where
(0.21 ) Gf D) ())(((())))) (0.2(
Lmim (0)
where G( D) is a data dependent metric introduced in Kapušniak et al. 2024 to "pull" the interpolant paths closer to the manifold represented by D.
For our experiments Sec. 5.3 we use "LAND" originally introduced in Arvanittss et al. (0016) (see details in Kapušniak et al. 2024 with hyperparameter a = 0.125. We set \ 5000 and X₂ 1.
we use Gaussians to represent Paju( and Pylu( We use a variance of 0.02 for Palu(n) and that of 0.03 for Py)u()) 4q E 21, We use K = 4000 samples for each of the conditional distributions.
Dataset. In the experiment in Fig. T0
Metric, We use surface adherence (SA) metric to measure how close the trajectory is on average to the surface specified by
LiDAR measurements. SA is defined as follows:
1 N SA
Σ
Σ | - [NN([=>/1:;; D1:2)ssl
(19)
N
where (N( D1:2) represents the nearest neighbor of x, in the set D while only considering the first and the second coordinates .i.e., x. location without the height.
Result. Fig. shows the trajectories obtained by the proposed method and the baselines. One can see that the trajectories returned by the baselines almost overlap for different swarms, and poorly adhere to the land surface (c.g.. crossing underneath the mountain) However, the proposed DFM method returns different trajectories for different swarms while staying closely to the surface.
Table 3 shows the SA obtained by all methods averaged over 5 trials. Combined observation from Table 3 and Fig. I0 shows that the proposed method shows better swarm navigation, in terms of transport and surface adherence, compared to the baselines. Note that the color in Fig. [00 indicatts the time t. This experiment also shows that DEM is useful in tasks requiring simultaneous trajectory estimation between multiple pairs of distributions.
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Diversified Flow Matching with Translation Identifiability
Top View
Front View
Input
FM-cond-OT
MFM-OT
MFM-cond-OT
DFM (Proposed)
Figure 10: Visualization of the trajectory obtained by different methods.[Colum] I Top] source locations of two swarms (red and blue) on the top and corresponding destinations on the bottom of the plot. [Column 2 5] swarm trajectories obtained by different methods, color specify the time. From red to blue, = 0 to 1.
Table 3: Average SA with standard deviation for different methods.
Method
SA
FM-OT
Tong et al.
2023
1.09 + 0.027
FM-cond-OT
Tong et al.
2023 w/ (9)
1.08 + 0.019
MFM-OT
Kapušniak et
2024)
0.47 = 0.015
MFM-cond-OT
Kapuániak al.
2024 w/ (9)
0.43 + 0.055
DFM (Proposed)
0.32 + 0.080
D. Experiment Details
D.1. Unpaired Image to Image Translation
Neural Networks and Hyperparameter settings. We use the UNet architecture (Ronneberger et al. 2015 to represent both neural networks. We adopt similar hyperparameter configuration based on the UNet architecture (Dhariwal & Nichol 2021). For the vector field, we use the AdamW optimizer (Loshchilov) 2017) with an initial learning rate of 10 4 and parameters 31 0.9, 32 = 0.999, the exponential moving average (EMA) (Tarvainen & Valpola| (017)) of the weights with decay parameter 0.9999. We use the same hyperparameter settings for the interpolant, except that the learning rate set to 10- head channels is 32 and attention resolution is 8. We use 0.1 and 02 = 10. We train the models for 100k iterations. All baselines are also trained for the same number of iterations.
le 8, and no weight decay. We use a batch size of 64, and dropout 0.1. take
For the image translation experiment, we observed improved training stability and efficiency when modifying Algorithm I so that 0-updates and ø-updates are interleaved, resulting in Algorithm 2 The primary reason that learning the vector field in the second phase of Algorithm [ with respect to a complex nonlinear interpolant appears challenging, as evidenced by oscillations in the objective function. However, training the vector field and interpolant tandem alleviates this issue. as the randomly initialized Io at the beginning of training close linear interpolant. This results in gradual training of vector field from almost linear to increasingly nonlinear paths.
E. Proofs
F.1. Proposition 3.4
To prove Lemma 3.4 first consider the following definition:
16

---

Diversified Flow Matching with Translation Identifiability
Table 4: Hyperparameters of the UNets for unpaired image translation task.
Hyperparameter
Value
Attention resolution
16
Heads channels
64
Heads
1
Channels multiple
2,
ResNet blocks
4
Channels
128
Algorithm 2 DFM with Interleaved Training Require: p. 11 d2, initialized parameters Φ)
1: while Stopping criterion is not met do
2: Sample (x(1), (1) y ( ~ p(z, y (1))) and to ~ (0,1))) Va E [Q]
# (1 - tg)z(1) + tgyl1 + +tg(1 (20 gl Vg [Q]
(1)
4: Linterp (0) + QQ-1 1--1 De == No. (14 - 1y 1,17%,, (1 2 (,0112)
(1)
5:
0 0 VoLinterp (
6: LFM(Ø) t क Eq==l |11(2
(9)
- -, r,
7: $ 6- Φ V (p)
8: end while
9: return v+(:i)
Definition E.1 (Interpolable density, Albergo et al. (2023) Def. D.I). A probability path with t E [0, 1] is interpolable there exists a time dependent invertible map /,, R. + Idd with e [0, 1]. such that ps the pushforward of 2b, of the standard normal density, i.e., [+bl = Pt-
time and space,
This definition will be helpful in proving Proposition 3.4
Proposition 3.4 Suppose that there exists flow fi Rd + Rd, differentiable in time and space, from Px to Py such that fi g". Suppose there exist a diffeomorphism from a (0,, Ia) to Palu(s) Vq. Let Tr denote a solution of Problem 12) and denote Ia (2) = æ + f @r(z,)dl. Then, under the same model in when (Paju(n) ) satisfies the SDC, we have 9s = g", holds a.c.
Proof. Let v/(0) denote a deffeomorphism from WN Ia) to Paju(() Further, let w/( = fi o vo This implies that (v6 #1(0,)1) = Po) and (9)
#N((0.1) = Pylu Hence, w/()) is time and space contimoousyy-diffeemtablle invertible
map from u(o) to Pylu(s).
Moreover, let pt = [v! #o = Then Pt Vg are interpolable densitics in the sense of Definition E.
Now, consider the following result from (Albergo et al.
2023
Proposition E.2 ([Albergo et al. with corresponding map v,, i.e.,
2023) Proposition D.I). Let pe be an interpolable density in the sense of Definition #1(001 = Pt. Then
E.1
I( = 1. 2b(155 (2/5'(2)) cos (t) + +े ' sin( 7t)
(20)
such that I I for independent coupling, i.e., x po and y ~ P1.
Invoking the above proposition, we have that for each 9 E [Q], there exists an interpolant function that can follow the
probability path p3 (a)
17

---

Diversified Flow Matching with Translation Identifiability
We similar reformulation technique as
Albergo et al.
2023
Appendix D) to re-express our constraints as follows:
minimize
(1) vel2 100
(21a)
q=1
subject to
pipp (4)
+ + (v p())) = =0,
(21b)
pb" Pæj u(e) pl Pylu(e)
(21c)
Pt interpolable,
(21d)
where u represents the divergence of vector field u, i.e. V u(x) Σ
aug U2.
Here, the optimization is over the
probability path p\ instead of the interpolants. Note that the constraints (1c
21d restricts the search over probability
paths that can be represented by interpolants in Z. other words, ( in Problem (21) is reparametriiztion of I in
Problem (2). Finally, the constraint (21b) is the continuity equation that ensures that the private vector fields vi follow the
probability path (.4) (Villani et al. 2009 Lipman al. 2023 Albergo al. 2023 Therefore, (21b) is equivalent (12b).
Let (f) = (x)) Then, v", Pt Vy constitutes a feasible solution to Problem 21 that can attain zero objective.
This implies that Problem (12) can also attain an objective of zero at the minimum.
Note that when the E I denote the private vector fields and interpolants returned by solving Problem 12 constraint 12bb is satisfied, g(0) transports Paju(e) to Py u(e) (Albergo et al. 2023).
Let ()) and I
Since there exists at least one solution to Problem
12 which attains zero objective value, ú, must also satisfy
(11) 00c
11
1|2
v. 0| Vgg E [Q]
Hence G, is a unified vector field that transports Pa)jla) to Pyjuce) Vg E [Q].
Therefore, 98 satisfies
Igal#pe Paju
Hence 9o is solution Problem Therefore, invoking Theorem 2.2 96 =g", a.e.
E.2. Proof of Fact 3.3
Let us define the solution sets ((4) follows:
v(4) {ve Ur transports Pmju(e) to Pyj Pp)
{ vz = arg min Ly (w,, I), VI E
where Cg = LFM(s.sP p(x, y)u(())) for brevity. Note that vla is the set of all vector fields that can transport Paju(s) to Pylu(o). is clear that (+1) (i) for * j in general. Our goal is find unified vector field that simultaneously transport Paju(n) pyl Vg. To that end, let
V = {vel | transports Palulo) to Pyjuco), Vg}
= fvel ve E v(1) & ... & v. E ((Q)
n p(1).
(22)
18

---

Diversified Flow Matching with Translation Identifiability
For the ease of exposition, let 2. Our argument will generalize arbitrary Q. Let
Vi
(1) arg min v.
(vr, I) and
0/
(1) 7 arg
min V.E,, 1
Σ (v, I(0),
(23)
Note that in Problem 23 02 E V is explicitly enforced. Whereas in Problem 10 hope for.
Uz E V is not enforced but is what we
Nonetheless the definition of o (1) implies that
C, (1 & (,, (7)).
However, if of $ V and (101!" ) < L1 (0 (00) then there may exist 7(2) I such that
C, 1 (1)) L, (t)") 1 > Cz 11 ,70) -2,
= = (T) (1) ) - C, 2 < (o, 1²)) + L2(0,, 700)
Since (T( 1) 11) ((2)) is also feasible solution to Problem (T).. we can conclude that Problem (00 is not equivalent to
Problem 23 The vector field returned by Problem (10 does not guarantee DDM satisfaction, since its solution may not be from V.
19


Reviews:
[{"review_id": "GJcWfLYNfP", "reviewer": "Reviewer_AAms", "summary": "This paper aims to address the unpaired domain translation problem with conditional information. The previous method is based on GAN. However, GAN training may not be stable; this paper proposes a Flow Matching-based method. However, the naive FM method may not work because conditional distributions may be mismatched, as shown in the synthetic experiments in the paper. The authors proposed a novel bilevel optimization loss to address this problem. One level of the loss optimizes the flow for each condition, and the other level optimizes the global flow and the interpolant between source and target distributions. The authors conducted experiments on synthetic data, Human face data, and Swarm Navigation data.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**[Intersecting trajectories in figures]**\n\nNote that Fig. 3(b) does **not** show the velocity field but the interpolant $I^{\\rm linear}(x, y, t)$ where $x, y \\sim \\rho(x, y | u^{(q)})$. The wording was meant to imply that interpolants guide the learning of the vector field. Our term “intended trajectories” could be confusing; we will change it to “interpolant trajectories.”\n\nIn Fig. 5(a) and Fig. 10, the trajectories do not cross each other (**please note the time colorbar**). This is because the two conditional populations travel at different speeds. Hence the intermediate transported distributions are not at the same location at the same time, although they pass via the same location, avoiding collision. This was briefly discussed in Sec. 5.1 (2D Gaussian blobs). We will make this clearer in the revised version.\n\n&nbsp;\n\n**[About Eq (10) and (12)]**\n\nThis is an important but nuanced point worth clarifying. We briefly discussed why Eq. (10) fails in Lines 244–268, with a more rigorous explanation in the proof of Fact 3.3. Essentially, when $I^{(q)}$ is learnable and $Q > 1$, Eq. (11) may not hold for all $q$, even though it is needed for $\\widehat{v}$ to transport $p_{x|u^{(q)}}$ to $p_{y|u^{(q)}}$. Changing $I^{(q)}$ alters the minimum of the flow matching loss $L_{\\rm FM}^{(q)}$, and since $v_t$ is coupled with all $L_{\\rm FM}^{(q)}$, minimizing Eq. (10) may not achieve the minimum for every loss. However, since $v_t$ is coupled with all $L_{\\rm FM}^{(q)}$’s, it is not guaranteed that minimum of all $L_{\\rm FM}^{(q)}$ is reached by minimizing Eq. (10).\n\n$I^{(q)}$ is also parameterized as in Eq. (17). \n\nNote that Eq. (12) does not “fail” like Eq. (10); unlike Eq. (10), which is theoretically flawed (therefore fails sometimes) as shown in Fact 3.3, Eq. (12) is theoretically sound per Prop. 3.4. However, as the reviewer correctly noted, Eq. (12) is hard to optimize due to its bilevel nature. To address this, we proposed to exploit the structural constraint in Section 3.4. \nWe will add a remark to summarize these nuances and clarify this discussion.\n\n&nbsp;\n\n**[More Experiments]**\n\n**Anonymized URL**: (https://drive.google.com/file/d/1U4gdB5qy1d98AJ1YWxR2v3fzDQp167nt/view?usp=sharing)\n\nWe should mention that the baselines FM-OT and MFM are strong, recent methods. Nonetheless, to address the reviewer’s concerns, we experimented with two additional diffusion-based baselines—EGSDE (provided by the reviewer) and SDEdit [Meng et al., 2022]—both using the same diffusion model trained on the Bitmoji domain based on DDIM. \n\nIn URL, we include these baselines and test them on the CelebAHQ-to-Bitmoji experiment (note that standard datasets, e.g., CelebAHQ Male-to-Female and AFHQ, are actually easier cases since the domain gap is smaller; hence, content misalignment rarely appears even for methods (such as MFM, EGSDE) without identifiability guarantees). \n\nR-Fig. 3 shows that translations by the new baselines differ significantly from Bitmoji images, leading to high FID despite setting the reverse diffusion repeats (K) to 3. Thus, although the ^DreamSim score [Fu et al., 2023] is relatively low, the translations are not meaningful due to high FID. In comparison, our method yields superior translations, as shown by qualitative results, FID, and DreamSim scores.\n\n*^Note that DreamSim appears to be insensitive to small but perceptible differences in translation fidelity (c.f. DFM and DDM-GAN qualitative results)* \n\n&nbsp;\n\n**[References]**\n\nThank you for the relevant references. We will include them in the revised version.\n\n&nbsp;\n\n**[Overlapping Cases]**\n\nR-Fig. 3 shows the case where the source and target conditional distributions have some overlap. The overlap is created by increasing the variance of the Gaussian components from Fig. 3 in the manuscript. One can see that small overlaps have a marginal effect on the performance of the proposed method. However, since the proposed implementation exploits the property of disjoint supports, larger overlaps can be detrimental to its performance.\n\nWe should remake that, as the partitioning of the conditional distribution is **controlled by the system designer,** the available data can always be split in a support-disjoint way. Hence, the harm brought by overlapping support is avoidable in many cases.\n\n&nbsp;\n\n**[About FID]**\n\n**In the context of identifiability-guaranteed translation, a good method retains source content while maintaining a reasonable FID (image quality).** Thus, minimizing FID is not our primary goal; we use it to show that our method does not compromise quality. Note that low FID alone does not ensure transport identifiability. Even if some baselines have similar FID scores, it does not mean they could attain content alignment.\n\nR-Table 1 shows that our method does not compromise image quality (measured by FID) while ensuring content alignment (see R-Fig. 2 and Dreamsim score), unlike baselines that often lose content despite similar FID scores.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Not sufficient. The toy examples should include the case that different conditions have overlaps. The authors need to compare with stronger baselines on more datasets.", "theoretical_claims": "I roughly checked the proof of Proposition 3.4, and didn't find issues.", "experimental_designs_or_analyses": "Yes, important baselines are missing, and experiments on more datasets on which domain translation methods are commonly tested should be included.", "relation_to_broader_scientific_literature": "The key contribution of this paper is to propose a Flow Matching method to address the domain translation problem with additional conditional information. The previous method proposed by Shrestha and Fu 2024 was based on GAN.", "essential_references_not_discussed": "Several existing image-to-image translation methods are missing:\n- Zhao et al, EGSDE: Unpaired Image-to-Image Translation via Energy-Guided Stochastic Differential Equations, NeurIPS 2022. \n- Gazdieva et al., Extremal Domain Translation with Neural Optimal Transport, NeurIPS 2023.\n- Kornilov et al., Optimal Flow Matching: Learning Straight Trajectories in Just One Step. NeurIPS 2024.", "other_strengths_and_weaknesses": "### Strength:\nThe authors proposed a Flow Matching method to address the unpaired domain translation problem with conditional information. The authors proposed a bilevel optimization loss. One level of the loss optimizes the flow for each condition, and the other level optimizes the global flow and the Interpolant between source and target distributions. \n\n### Weakness:\n\nThe experiments in this paper are not strong. \n1. The authors should compare with stronger baselines: \n- Zhao et al, EGSDE: Unpaired Image-to-Image Translation via Energy-Guided Stochastic Differential Equations, NeurIPS 2022. \n- Gazdieva et al., Extremal Domain Translation with Neural Optimal Transport, NeurIPS 2023.\n- Kornilov et al., Optimal Flow Matching: Learning Straight Trajectories in Just One Step. NeurIPS 2024.\n\n2. The authors should do experiments on more datasets, such as CelebA/CelebA-HQ, AFHQ, on which domain translation methods are commonly tested.  \n\n3. In the synthetic experiments, the authors should include data with overlapping condition distributions, such that we can evaluate how effective Eq. 12 is. \n\n4. The improvement of the proposed method is minor compared to using the naive baseline FM-cond: 22.21 vs 22.65 in terms of FID.", "other_comments_or_suggestions": "No.", "questions_for_authors": "In Fig. 3 (b), the authors drew intended trajectories. However, I am not sure whether the blue and green trajectories are really achievable or not. The blue and red lines cross. At each cross point of one blue line and one red line, the final velocity v will be the average of the two velocities of the red and green lines. So, the blue and red lines cannot cross on the 2-d plane. I have similar concerns for Fig. 5 (a) and Fig. 10 DFM. The authors need to clarify this in case my understanding is wrong. \n\nThe authors did synthetic experiments showing Eq. 10 sometimes works and sometimes doesn't work, to motivate their own formulation Eq. 12. However, the authors didn't explain why Eq. 10 doesn't work. Is the network complexity not enough, or is the initialization not good, or other reasons? How do you parametrize $I^{(q)}$ in Eq. 10 and 12? Eq. 12 is also difficult to optimize, and Eq. 12 could also sometimes fail. The authors need to justify that the optimal solution of Eq. 12 can be achieved in the experiments.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "msC2768ePp", "reviewer": "Reviewer_wzYK", "summary": "The paper introduce Diversified Flow Matching (DFM), an FM-based framework for DDM. They design a custom loss function and nonlinear interpolant to ensure translation identifiability, addressing the limitations of conventional FM methods that use linear interpolants. By leveraging the non-overlapping property of conditional distributions, they reformulate the bilevel optimization problem into a two-stage approach, simplifying computation. The method is tested on synthetic data and real-world applications, demonstrating its effectiveness in maintaining translation identifiability and improving trajectory information.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "&nbsp;\n\n**[Quantitative Metrics for Image to Image Translation]**\n\nImage similarity metrics were not presented because we observed that metrics such as LPIPS do not make sense when the images have large domain gaps (i.e., when the geometric representations of the feature spaces are largely different like in photos and bitmoji). For example, DFM and CycleGAN achieved the same LPIPS score, despite DFM being clearly better than CycleGAN in terms of alignment.\n\nNonetheless, to alleviate the reviewer’s concern, we evaluate the DreamSim score [Fu et al, 2023]. This score serves for similar purposes of LPIPS but was shown to better align with human judgement [Fu et al., 2023]. Although the metric can provide some rough idea, it still cannot capture some obvious issues (e.g., DDM-GAN has diversity issues as seen in Fig. 7, last two rows, in the manuscript):\n\n------------------------------\n\nMethod    $\\quad\\quad$      Mean (Std)  \n\n------------------------------\n\nDFM       $\\quad\\quad\\quad~~$        0.59 (0.05)\n\nDDM-GAN   $\\quad$   0.58 (0.05)\n\nCycleGAN  $\\quad~$    0.63 (0.06)\n\nFM   $\\quad\\quad\\quad~~~~$  0.66 (0.06)\n\nFM-OT  $\\quad\\quad~~$       0.66 (0.06)\n\nFM-cond    $\\quad~~$    0.60 (0.06)\n\nSDEdit\t$\\quad\\quad~~$       0.62 (0.06)\n\n-------------------------------\n\n&nbsp;\n\n**[Swarm Navigation Issue]**\n\n\nWe should mention that our learned paths are time-space nonoverlapping. This means that the swarms can use the overlapped space as long as they do not reach the same spatial point at the same time. This was actually reflected in Fig. 10 (**please note the colorbar**). We believe that such learned path plans are reasonable as they avoid collision and allow the swarms to use the same space at different times. We realize that we did not articulate this point in the simulations. We will clarify in the revision.\n\n&nbsp;\n\n**[Additional Baselines]**\n\n**Anonymized URL**: (https://drive.google.com/file/d/1U4gdB5qy1d98AJ1YWxR2v3fzDQp167nt/view?usp=sharing)\n\nTo alleviate the reviewer’s concern, we have conducted experiments with additional baseline SDEdit which is based on diffusion. The new table and figure can be found in the anonymized pdf link. One can see that SDEdit suffers from misalignment as well as has higher FID. \n\n&nbsp;\n\n**[References]**\n\nThank you for the relevant reference. We will discuss this work in our related work section.\n\n&nbsp;\n\n**[Non-overlapping support assumption]**\n\nIn fact, we found that the non-overlapping support assumption is not very hard to be met in practice, as the partitioning according to $u^{(q)}$ is controlled by the system designers. For example, in photo to cartoon translation, one can pick black/non-black hairs as $u^1$ and $u^2$, which naturally lead to non-overlapping clusters. Of course, partitioning data would reduce the amount of data in each cluster, which might make learning its corresponding $v_t$ harder. This is a tradeoff that system designers should pay attention to when splitting the data.\n\n&nbsp;\n\n**[Experimental Scope]**\n\nThe reviewer’s comment is related to an open challenge for diffusion and FM-type methods; that is, scalability with large dimension data has not been solved in general. Note that diffusion and flow matching in latent space of VAE rather than raw pixels is common practice [R1, R2] . Further, the latent space is of dimension 32 x 32 x 4, which is still quite high dimensional. \n\nWe agree that diverse modalities like text-image is a very interesting scenario. However, applying any diffusion/FM models in the raw data space remains an open problem, and solving this challenge is out of scope of this work. We will add some remarks in “limitations” to draw attention to this open challenge. \n\n*[R1] Kapusniak, Kacper, et al. \"Metric flow matching for smooth interpolations on the data manifold.\" NeurIPS 2024.*\n\n*[R2] Tong, Alexander, et al. \"Improving and generalizing flow-based generative models with minibatch optimal transport.*", "claims_and_evidence": "YES", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria in the paper are appropriate for the problem of diversified distribution matching (DDM).", "theoretical_claims": "The paper presents two main theoretical claims: Proposition 3.4 (translation identifiability of DFM) and Fact 3.3 (failure of the sum-of-LS formulation). \nProposition 3.4 relies on idealized assumptions (e.g., existence of diffeomorphisms) but is logically sound within the theoretical framework.\nFact 3.3 is rigorously supported by both synthetic examples and formal reasoning.", "experimental_designs_or_analyses": "The paper evaluates DFM through three experiments: synthetic data, image translation, and swarm navigation. While the experiments demonstrate DFM’s effectiveness, several design and analysis choices warrant discussion:\n1. Synthetic Data Experiment\nDesign: \nTests on 2D/3D Gaussian mixtures with ground-truth translation y=-x.\n  Metrics: Earth Mover’s Distance (EMD) and Translation Error (TE).\nValidity:\n  Strengths: EMD and TE are appropriate for distribution matching and identifiability. The use of multiple trials (10 runs) adds statistical robustness.\n2. Image Translation Experiment\nDesign:\n  Translates CelebAHQ faces to Bitmoji using FID for distribution matching and visual checks for content alignment.\n  Trains on Stable Diffusion’s latent space (not raw pixels).\nValidity:\n  Strengths: FID is standard for generative models. Including CycleGAN and DDM-GAN baselines contextualizes performance.\n  Issues:\n    Content Alignment: Relies on qualitative visual checks without quantitative metrics (e.g., SSIM, LPIPS, or user studies). This risks overlooking subtle misalignments.\n3. Swarm Navigation Experiment\nDesign:\n  Tests on LiDAR data of Mt. Rainier with Gaussian swarm sources/destinations.\n  Metric: Surface Adherence (SA) to measure trajectory proximity to terrain.\nValidity:\n  Strengths: SA is a meaningful metric for terrain adherence. \n  Issues:\n    Oversimplified Scenario: Swarms have non-overlapping paths (per Assumption 3.5), ignoring real-world complexities like intersecting trajectories or dynamic obstacles.\nGeneral Issues:\nBaseline Scope:\nOmission of recent flow-based UDT methods (e.g., Rectified Flow, Schr̈odinger Bridges).", "relation_to_broader_scientific_literature": "The key contributions of the paper are deeply rooted in addressing gaps and building upon advancements in unsupervised domain translation (UDT), distribution matching, and flow-based generative models. Here’s how they relate to prior work:\n1. Bridging Flow Matching (FM) and Diversified Distribution Matching (DDM)\nPrior Work:  \n  DDM (Shrestha & Fu, 2024): Introduced translation identifiability via GANs by matching multiple conditional distributions.  \n  Flow Matching (Lipman et al., 2022): Provided stable training and trajectory modeling via ODEs but focused on single distribution pairs.  \nContribution:  \n  The paper unifies these frameworks by adapting FM to DDM, enabling translation identifiability (a GAN-based DDM strength) with trajectory information (a flow-based advantage). This resolves GANs’ instability and trajectory limitations while extending FM to multi-conditional settings.\n2. Addressing Translation Non-Identifiability in Flow Models\nPrior Work:  \n  CycleGAN (Zhu et al., 2017): Highlighted content misalignment due to non-unique transport maps.  \n  Optimal Transport (OT) in UDT (Liu et al., 2022; De Bortoli et al., 2021): Assumed OT maps as solutions but lacked guarantees for general translations.  \nContribution:  \n  By enforcing DDM’s sufficiently diverse condition (SDC) through FM, the paper guarantees identifiability for non-OT maps. This directly addresses the non-uniqueness issue in prior UDT methods, aligning with theoretical insights from Shrestha & Fu (2024) but in a flow-based framework.\n3. Introducing Bilevel Optimization and Nonlinear Interpolants\nPrior Work:  \n  Linear Interpolants in FM (Albergo et al., 2023): Used linear paths (e.g., \\(z_t=(1-t)x+ty\\)) but failed for multi-conditional DDM due to conflicting trajectories.  \n  Conditional FM (Atanackovic et al., 2024): Learned separate flows per condition but lacked unified transport functions.  \nContribution:  \n  The paper proposes nonlinear interpolants (Eq. 17) and bilevel optimization (Eq. 12) to harmonize trajectories across diverse conditional pairs. This extends FM’s applicability to DDM, resolving conflicts inherent to linear interpolants and enabling a unified velocity field.\n4. Exploiting Structural Constraints for Efficiency\nPrior Work:  \n  Non-Overlapping Supports in OT (Villani et al., 2009): Leveraged distribution separation for tractable transport.  \nContribution:  \n  By assuming non-overlapping supports (Assumption 3.5) and designing non-intersecting interpolants (Eq. 14), the paper simplifies bilevel optimization into a tractable two-stage process. This aligns with structural insights from OT but tailors them to FM-based DDM.", "essential_references_not_discussed": "Deep Momentum Multi-Marginal Schrödinger Bridge:\nThe proposed DFM framework aims to unify multiple conditional distribution pairs under a single flow, which inherently aligns with multi-marginal SB formulations. Including this reference or related paper is critical to contextualize DFM’s bilevel optimization strategy against state-of-the-art SB advancements, especially since DFM implicitly addresses multi-marginal coupling.", "other_strengths_and_weaknesses": "Strengths:  \n1. Originality:  \n  The integration of FM with DDM is a novel contribution. While DDM was previously restricted to GANs, this work creatively adapts FM to enforce identifiability via bilevel optimization and learnable nonlinear interpolants.  \n  The design of private interpolants for each conditional distribution pair and the reformulation under non-overlapping structural assumptions demonstrate innovative problem-solving.\n2. Significance:  \n  Translation identifiability is a critical issue in UDT, and DFM provides a principled solution with theoretical guarantees.  \n   The ability to recover transport trajectories has practical implications for applications like robot navigation and single-cell analysis, where path information is essential.\n3. Technical Soundness:  \n   Theoretical results (e.g., Proposition 3.4) rigorously connect DFM to the DDM criterion, ensuring identifiability under sufficiently diverse conditional distributions.  \n   Experiments on synthetic data, image translation, and swarm navigation validate the method’s effectiveness. The improved FID scores (Table 1) and avoidance of GAN instability (Fig. 6) highlight empirical advantages.\n4. Clarity:  \n  The paper is well-structured, with clear explanations of challenges in adapting FM for DDM (e.g., pitfalls of linear interpolants in Fig. 3). Visualizations (e.g., trajectories in Fig. 5, 9–11) enhance understanding.\nWeaknesses:\n1. Restrictive Assumptions:  \n  The efficient implementation relies on non-overlapping supports of conditional distributions (Assumption 3.5), which may not hold in many real-world scenarios. While acknowledged as a limitation, potential workarounds for overlapping cases are not explored.\n2. Experimental Scope:  \n  Image translation experiments operate in a latent space rather than raw pixels, which may understate challenges in high-dimensional settings. Testing on more diverse modalities (e.g., text-to-image) could strengthen claims.", "other_comments_or_suggestions": "No.", "questions_for_authors": "No.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "CAUQqRIOu9", "reviewer": "Reviewer_Y3HP", "summary": "The paper introduces Diversified Flow Matching (DFM), a novel unsupervised domain translation (UDT) framework that extends ODE-based Flow Matching (FM) from linear to nonlinear interpolants, addressing the critical limitation of translation identifiability.\n\nKey Contributions: \n\nOvercoming Linear Flow Matching Limitations\n- Prior FM approaches fail when distribution modes overlap, leading to incorrect mappings due to linear interpolants.\n- The authors explicitly demonstrate this failure case with synthetic Gaussian mixtures where mode overlaps disrupt proper translation trajectories.\n\nAdvancing Nonlinear Flow Matching for Identifiability\n- Builds on prior learnable flow models that incorporate nonlinear interpolants to improve flow-based translation.\n- While previous works in metric-based flow modeling and Schrödinger bridge methods introduced learnable transport functions, they did not explicitly guarantee translation identifiability.\n- The authors extend this concept by introducing higher-order non-intersecting nonlinear interpolants to explicitly enforce identifiability in domain translation.\n\nBilevel Optimization Loss for Enforcing Identifiability\n- A novel bilevel learning loss is proposed, ensuring a unified, non-intersecting translation trajectory across different domain conditions.\n- This approach is a unique theoretical and practical advancement, correcting mode overlap issues seen in prior FM methods.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "&nbsp;\n\n**Anonymized URL**:  (https://drive.google.com/file/d/1U4gdB5qy1d98AJ1YWxR2v3fzDQp167nt/view?usp=sharing)\n\n&nbsp;\n\n**[Limitations of Claim 2: Bilevel Loss implies Identifiability]**\n\nTo clarify, the proposed DFM and the baselines FM, FM-OT, and FM-cond do use the same architecture, and are trained using the same hyperparameter settings. Therefore, the only difference between these methods is in the loss design. Hence, the qualitative and quantitative performance gain can be attributed to the translation identifiability rather than architectural differences. We will add a remark for clarification.\n\nRegarding real-word scenario: in this work, our interest lies in a theoretically sound FM approach for realizing DDM and retaining translation identifiability. Translation identifiability’s effectiveness in real-world applications is indeed important, but a bit beyond the scope. Nonetheless, we noticed that there have been recent works using DDM and translation identifiability for medical imaging [R-1]. They did not use FM, but clearly showed the usefulness of translation identifiability. We will point readers to such references.\n\nWe will make the above points clearer in the revised version.\n\n[R-1] Song, Jiahui, et al. \"Translation Identifiability-Guided Unsupervised Cross-Platform Super-Resolution for OCT Images.\" IEEE SAM, 2024.\n\n&nbsp;\n\n**[Limitations of Claim 3: DFM achieves superior content alignment ]**\n\nWe should mention that Table 1 is not meant to show performance improvement. Table 1 only shows that FID of the proposed method being comparable to the best baseline. This ensures that the image quality of the translated images is not compromised. Whereas, the qualitative results accompanying Table 1 in Fig. 7 is used to show that the proposed method has better content alignment than all other baselines. We have now quantified the content alignment result using DreamSim scores. This can be used together with Table 1 and the figure to assess the performance of the methods. Overall, a good method in our context should have low DreamSim scores and reasonable FID scores. Only having FID scores does not mean good performance.\n\nNote that we provided two real-world experiments with image translation and swarm navigation to show generalization beyond synthetic tasks. \n\n&nbsp;\n\n**[Table 1: Translation identifiability gains]**\n\nAs mentioned, Table 1 is not used to show performance gain. Table 1 shows the image quality (measured by FID) is not compromised when we establish translation identifiability. The translation identifiability is shown in Fig. 7.\n\nThe DFM does not retain identifiability if the nonlinear interpolants are not used. We have included linear interpolant in this experiment. One can see that the performance is not promising.\n\n&nbsp;\n\n**[Ablations]**\n\n**[Alternative non-linear Interpolants]** In fact, we did not use any specific design for the nonlinear interpolant. Our understanding is that as long as it satisfies the regularity condition and is a learnable interpolant, it can serve our purpose. To address the reviewer’s concern, we have conducted experiments on different nonlinear interpolant parametrization, which is shown in R-Fig. 1. In R-Fig. 1, we use two other parametrization of the nonlinear interpolant (i,e., $(1-t) x + ty + f(t) \\gamma_{\\theta}(x, y, t)$ with different $f(t)$, namely, (i) $ (1-t) x + t y + {\\rm sin}(\\pi t) \\gamma_{\\theta}(x, y, t)$ (ii) $ (1-t) x + t y + \\sqrt{t (1-t) } \\gamma_{\\theta}(x, y, t)$. One can see that DFM is robust to different parametrization of the nonlinear interpolant.\n\n\n**[Computational efficiency]** R-Table 1 in the URL shows the training and inference time of all methods. The proposed method does not incur significant training time overhead compared to other FM based methods. Whereas, there is no overhead at all during inference. \n\n\n**[Real-world validation]** Note that our implementation did not directly optimize the bilevel loss. We exploited the non-overlapping structure of conditional distribution to obtain a simplified loss. We have provided two real-world validations as discussed previously. \n\n&nbsp;\n\n**[Overlapping supports]**\n\nR-Fig. 3 in the anonymized PDF link shows the result of overlapping supports of the conditional distribution. The figure shows that small overlaps do not significantly harm the translation. Further, note that non-overlapping supports naturally occur in applications as demonstrated in our experiments. That said, very large overlaps can indeed be detrimental to the proposed method. \n\n&nbsp;\n\n**[Code Submission]**\n\nNote that we have already provided demo code in the supplemental materials along with our manuscript. We will release the complete code along with pre-trained models before the camera-ready submission.", "claims_and_evidence": "(1) Claim 1: Linear Flow Matching Fails for Translation Identifiability\n\nEvidence:\n\n- Synthetic Gaussian mixture experiments clearly demonstrate incorrect trajectory mappings when using linear interpolants.\n- Prior FM methods fail in mode-overlapping settings, leading to poor translation accuracy.\n\nLimitation:\n\n- Needs real-world validation in high-dimensional datasets. \n\n\n\n(2) Claim 2: Bilevel Learning Loss Ensures Identifiability\n\nEvidence:\n\n- Proposition 3.4: Provides formal proof that the bilevel loss enforces translation identifiability by learning non-intersecting interpolants.\n\n- Empirical results:\n\n(a) Synthetic data: DFM avoids trajectory conflicts where FM-cond fails.\n(b) Image translation: FID scores demonstrate improvements, but more qualitative analysis is needed to attribute gains specifically to translation identifiability.\n(c) Swarm navigation: DFM preserves transport trajectories where standard FM methods fail.\n\nLimitation:\n\n- Table 1 lacks clarity → Needs explicit quantitative and qualitative validation that the observed FID improvements are due to translation identifiability rather than other architectural enhancements.\n\n- Real-world experimental results lack qualitative and quantitative ablation studies on the effect of the bilevel optimization loss.\n(a) No direct comparison is made between DFM with and without the bilevel loss in a real-world scenario.\n(b) Translation identifiability should be evaluated explicitly in real-world tasks, particularly in structured medical or multimodal datasets where domain shifts are crucial.\n\n\n\n(3) Claim 3: DFM Achieves Superior Content Alignment Over FM-cond and GAN-based DDM\n\nEvidence:\n- Face-to-Bitmoji translation experiments show better content alignment than FM-cond and GAN-based DDM.\n- Table 2 (synthetic results) shows clear performance improvements, suggesting the effectiveness of DFM’s nonlinear flow matching.\n\nLimitation:\n- Table 1’s performance improvements require stronger justification.\n- Need further ablation studies → Testing whether DFM’s gains persist without nonlinear interpolants.\n- More real-world experiments needed to confirm generalization beyond synthetic tasks.", "methods_and_evaluation_criteria": "Technical Approach's Contribution : \n\n- Introduces a custom bilevel optimization loss with nonlinear, non-intersecting interpolants to explicitly enforce translation identifiability.\n- Unlike prior FM-based methods, the bilevel optimization ensures that each interpolant follows a unified translation trajectory, avoiding trajectory conflicts.\n- Simplifies the bilevel optimization problem by leveraging non-overlapping conditional distributions, making it computationally efficient.\n\nEvaluation Metrics\n\n- FID Scores → Measure image translation quality.\n- Earth Mover’s Distance (EMD) → Assesses distribution transport accuracy.\n- Translation Error (TE) → Evaluates content alignment precision.\n\nStrengths:\n\n- Clear synthetic data evaluations that visualize trajectory correctness.\n- Shows that the bilevel loss improves identifiability beyond prior GAN/FM-based methods.\n\nWeaknesses:\n\n-  Table 1’s performance gains need further explanation → Whether the gains stem from translation identifiability or other factors is unclear.\n- Table 2 (synthetic) provides stronger validation than Table 1 → More real-world experiments should match this clarity.\n- Ablation studies are missing for:\n(a) Alternative nonlinear interpolants.\n(b) Computational efficiency trade-offs.\n(c) Real-world experimental validation of the bilevel optimization loss's effect on translation identifiability.", "theoretical_claims": "I have no comments on that.", "experimental_designs_or_analyses": "I recommend that the author review and supplement the following parts:\n\n(1) Ablation Study: Effect of the Bilevel Optimization Loss on Translation Identifiability\n- Ablation experiments should test DFM with and without the bilevel loss to show how much it contributes to translation identifiability.\n- Qualitative trajectory comparisons should illustrate the effect of the bilevel loss on mode separation and content preservation.\n- Quantitative experiments should assess domain translation accuracy and alignment differences with and without the bilevel loss.\n\n(2) Real-World Verification of Bilevel Learning Loss Effectiveness\n- Demonstrate a real-world use case where translation identifiability affects model performance.\n- Quantify performance improvement using classification accuracy, error reduction in domain adaptation tasks.\n\n(3) Addressing Ambiguity Between Table 1 and Table 2 Results\n- Table 2 (synthetic experiments) shows clearer performance improvements than Table 1, strongly supporting DFM’s effectiveness.\n- Table 1’s real-world results, however, remain ambiguous, particularly regarding: \n(a) Whether the observed performance difference between DFM and FM-cond is due to translation identifiability or other confounding factors. (b) Lack of qualitative trajectory visualizations and structured ablation to confirm that the bilevel optimization loss directly improves translation identifiability.\n\n(4) Other suggestions\n- Additional experiments aligning real-world results (Table 1) with synthetic evaluations (Table 2).\n- Qualitative trajectory comparisons in real-world settings to explicitly visualize translation improvements.\n- Ablation study breaking down the exact contributions of translation identifiability in Table 1 results.\n- If real-world dataset expansion is infeasible, at least include additional qualitative and quantitative results in the appendix.", "relation_to_broader_scientific_literature": "Additional Clarifications Needed:\n\n- Clarify how DFM differs from prior work in nonlinear flow matching, particularly in handling mode overlaps and enforcing identifiability.\n- More discussion on real-world applications where identifiability is a critical bottleneck.", "essential_references_not_discussed": "I have no clear reference to mention, but please refer to my comment on Relation to Broader Scientific Literature.", "other_strengths_and_weaknesses": "There are some weaknesses that could limit the paper’s broader impact. \n\n- Empirical validation remains narrow, as the evaluation focuses on controlled synthetic settings and simple image translation tasks, without testing on real-world, high-dimensional structured datasets. \n\n- Real-world validation of the bilevel loss is missing → Needs qualitative & quantitative ablation studies.\n\n- Table 1 results require stronger interpretation to confirm that improvements arise from translation identifiability rather than other confounding factors.\n\n- Computational trade-offs remain unexplored → Scalability to larger datasets needs further analysis.\n\n- The lack of open-source implementation raises reproducibility concerns, which should be addressed before the camera-ready submission.\n\n\n====\n\nI will assess the authors' incorporation of feedback, along with other overall considerations, and determine whether to adjust or maintain the score accordingly.", "other_comments_or_suggestions": "Some mathematical derivations (e.g., Proposition 3.4) could be more clearly structured for better readability, and a summary table comparing DFM to existing methods would help contextualize its contributions.", "questions_for_authors": "- What is the computational overhead of DFM compared to these baselines? \n\n- Does the bilevel optimization framework introduce significant training inefficiencies?\n\n- How sensitive is DFM’s performance to the specific choice of nonlinear interpolant? \n\n- Have alternative nonlinear functions been tested, and if so, how do they affect stability and identifiability?\n\n- Have you tested runtime performance across different dataset sizes or on high-dimensional data?\n\n- Would the method perform well in cases where conditional distributions have overlapping supports, which may occur in real-world applications?\n\n- Do you plan to release code before the camera-ready submission?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ifkZz1mULB", "reviewer": "Reviewer_dKp3", "summary": "This paper proposed a flow matching model for diversified distribution matching, called DFM. The proposed method formulates a bilevel optimization problem to learn an interpolant and train a flow model on this interpolant. This paper demonstrates that the standard flow matching models fail in the DDM task. To address this, DFM proposes a method for learning a non-intersecting interpolant (Eq. 14) by leveraging the class label information. Intuitively, the objective encourages the trajectories of different classes to be well-separated. DFM model is evaluated on synthetic data, image-to-image translation, and swarm navigation (appendix).", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**[Novelty of Proposition 3.4]**\n\nThe novelty lies in how to use FM-based losses to attain the same conclusion of Thm 2.2. Note that Thm 2.2 assumes that distribution matching is already attained. But Prop 3.4 specifically needs the distribution matching part to be realized by FM. How to achieve this while not violating the constraints in Thm 2.2 was very unclear before we came to the loss design of the bilevel objective. Coming to our design was entirely nontrivial and this constitutes the major novelty. We will articulate this point by adding a remark after Prop 3.4.\n\n&nbsp;\n\n**[“$ \\times 1_{\\mathcal{I}}$”  instead of “$+1\\_{\\mathcal{I}}$” ]**\n\nThank you for your careful reading. We seemed to have missed the definition of the identity function $1_{\\mathcal{I}}$ leading to confusion. The operator should be $+$ when \n$1_{\\mathcal{I}}(x) = 0$ for $x \\in \\mathcal{I} $ and infinity otherwise. Whereas the operator should be $*$ when $1_{\\mathcal{I}}(x) = 1$ for $x \\in \\mathcal{I} $ and infinity otherwise.\nWe will include the definition to avoid this confusion.\n\n&nbsp;\n\n**[Presentation ]**\n\nSome experiments were moved to the appendix due to space constraints. To alleviate the reviewer’s concerns, we will restructure the paper to include partial results from all experiments along with baseline comparisons in the main paper and move the rest to the appendix.\n\n&nbsp;\n\n**[ LPIPS score ]**\n\nImage similarity metrics were not presented because we observed that metrics such as LPIPS do not make sense when the images have large domain gaps, which is the case in the considered experiment. For example, DFM and CycleGAN achieved the same LPIPS score, despite DFM being clearly better than CycleGAN in terms of alignment. Also note that LPIPS score was still not presented for the CelebAHQ to Bitmoji translation task in DDM-GAN .\n\nNonetheless, to alleviate the reviewer’s concern, we evaluate the DreamSim [R1] score in the table below, which was shown to better align with human judgement than LPIPS score. Although the metric can provide some rough idea, it still cannot capture some obvious issues (e.g., DDM-GAN has diversity issues as seen in Fig. 7, last two rows, in the manuscript).\n\n*[R1] Fu, Stephanie, et al. \"Dreamsim: Learning new dimensions of human visual similarity using synthetic data.\"*\n\n\n------------------------------\n\nMethod    $\\quad\\quad$      Mean (Std)  \n\n------------------------------\n\nDFM       $\\quad\\quad\\quad~~$        0.59 (0.05)\n\nDDM-GAN   $\\quad$   0.58 (0.05)\n\nCycleGAN  $\\quad~$    0.63 (0.06)\n\nFM   $\\quad\\quad\\quad~~~~$  0.66 (0.06)\n\nFM-OT  $\\quad\\quad~~$       0.66 (0.06)\n\nFM-cond    $\\quad~~$    0.60 (0.06)\n\nSDEdit\t$\\quad\\quad~~$       0.62 (0.06)\n\n------------------------------\n\nWe will add the above result in the revised version.\n\n&nbsp;\n\n**[Standard Benchmark Dataset]**\n\nNote that standard benchmark datasets (CelebAHQ Male to Female and AFHQ) can actually be considered as easy cases because the domain gap is relatively small. Hence, the issue of content misalignment was not observed by the existing methods [1,2]. For this reason, we intentionally did not include them, as they did not help make our point.\n\nHowever, in the case of CelebAHQ-to-Bitmoji, the content misalignment issue is clear, and therefore helps to prove our point. \n\n&nbsp;\n\n**[Necessity of Thm 2.3]**\n\nIndeed, we presented Thm 2.3 for showing the robustness of Thm 2.2. As Theorem 2.3 is not the contribution of this work, we only presented the theorem to provide sufficient context. However, we understand the reviewer’s concerns that Theorem 2.3 was not used explicitly in any of our own theoretical analysis, so might not be necessary to present it. \n\nWe will move the theorem to the appendix to save space for other important revisions to the paper.\n\n&nbsp;\n\n\n[**Comparison with (Liu et al., 2023)**]\n\nNote that the example in the motivation is not claimed to be the contribution of our work. It is merely to illustrate the issue with using linear interpolants to the readers. Hence, we believe that a similar example showing reflecting trajectories does not undermine our work. Nonetheless, we understand that the example might be obvious for some readers familiar with the work. Hence we will cite [1] before elaborating on the example.  \n\n&nbsp;\n\n**[ UDT terminology ]**\n\nThe work can be considered unsupervised in that no paired data is required at all. However, we understand the reviewer’s concern and will consistently use “unpaired domain translation” for UDT instead of “unsupervised domain translation” in the revised version.", "claims_and_evidence": "- DFM model is supported by the experimental results.\n- However, there is concern regarding the novelty of Prop 3.4, as it is mainly a restatement of the assumption that $f_{1}^{\\star}=g^{\\star}$ and Thm 2.2.", "methods_and_evaluation_criteria": "- DFM aims to address the unsupervised domain traslation (UDT) problem. However, this model requires class labels for input-output data, raising concerns about whether this setting is also considered as UDT.", "theoretical_claims": "- There is concern regarding the novelty of Prop 3.4 (See **Claims And Evidence**).\n- \"+ $1_{\\mathcal{I}}$\" in Equation 12 may need to be replaced with \"* $1_{\\mathcal{I}}$\".", "experimental_designs_or_analyses": "- DFM model is evaluated on synthetic data, image-to-image translation, and swarm navigation (appendix).\n\n- The image translation experiment is conducted on a relatively non-standard benchmark of CelebAHQ-to-Bitmoji. For reference, see [1, 2].\n\n- There are concerns regarding the presentation. Most of the experimental results comparing DFM with baseline methods are placed in the appendix, including Fig 9, 10, 11, and Tab2. Additionally, all swarm navigation results are placed in the appendix.\n\n- While the main target task is diversified distribution matching, the image translation experiments only evaluate marginal distributions, not conditional distribution matching. For reference, DDM-GAN measured the LPIPS score on a similar experiment [3].\n\n\n[1] De Bortoli, Valentin, et al. \"Schrodinger Bridge Flow for Unpaired Data Translation.\" NeurIPS 2024.    \n[2] Liu, Guan-Horng, et al. \"I $^ 2$ SB: Image-to-Image Schr\\\" odinger Bridge.\" ICML 2023.    \n[3] Shrestha, Sagar, and Xiao Fu. \"Towards identifiable unsupervised domain translation: A diversified distribution matching approach.\" ICLR 2024.", "relation_to_broader_scientific_literature": "This paper proposed a flow matching model for the diversified distribution matching (DDM) task.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "**Strengths**\n- The paper is well-written and easy to follow.\n- The motivation is well-supported by the experiments in Section 3.2.\n\n**Weakness**\n- There are concerns regarding the necessity of Thm 2.3.\n- The motivation section presents results that are somewhat expected, as observed in [1].\n- Additional concerns are included in other sections.\n\n\n[1] Liu, Xingchao, Chengyue Gong, and Qiang Liu. \"Flow straight and fast: Learning to generate and transfer data with rectified flow.\"", "other_comments_or_suggestions": "No", "questions_for_authors": "No", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The reviewers generally appreciated the theoretic development and the integrating of prior research. Particularly, the integration of translation identifiability via Diversified Distribution Matching (DDM) and flow matching (FM) was interesting. The theoretic development seems to be the main strength of the paper.

However, the original goal for merging DDM and FM is to provide an empirically better UDT method that preserves content better. But, as multiple reviewers noted, the experiments do not seem to clearly outperform previous methods (particularly when the compared baselines are stronger). Some noted that the original experiments did not include strong baselines. 

Additionally, the main claim of the paper relies on qualitative assessment of content/semantic preservation. But the qualitative results (especially with the stronger baselines) are not particularly convincing. By looking at the new results with the added baselines, it is not clear that DFM outperforms them empirically (e.g., see rows 3 and 4 in R-Fig. 2 where EGSDE arguably performs better, DFM seems to perform better for rows 1 and 2). Thus, the key benefit of the proposed method is not clearly demonstrated.

While theoretic identifiability is interesting in its own right, this paper builds upon the prior identifiability results. And thus, the significance of the proposed approach hinges on better empirical performance.

Given this, I think this is an interesting work but the experiments need strengthening and clarifying to more clearly demonstrate the benefits of the theoretically developed methods.
Decision: Accept (poster)