Projected Coupled Diffusion for Test-Time Constrained Joint Generation
Authors: Hao Luan
Yi Xian Goh
See-Kiong Ng
Chun Kai Ling
Venue: ICLR 2026 Poster (2026)
Keywords: diffusion model
constrained diffusion
Primary area: generative models
TL;DR: A test-time framework enabling multiple pretrained diffusion models to generate joint samples enforcing task-specific constraints.

Abstract: Modifications to test-time sampling have emerged as an important extension to diffusion algorithms, with the goal of biasing the generative process to achieve a given objective without having to retrain the entire diffusion model. However, generating jointly correlated samples from multiple pre-trained diffusion models while simultaneously enforcing task-specific constraints without costly retraining has remained challenging. To this end, we propose Projected Coupled Diffusion (PCD), a novel test-time framework for constrained joint generation. PCD introduces a coupled guidance term into the generative dynamics to encourage coordination between diffusion models and incorporates a projection step at each diffusion step to enforce hard constraints. Empirically, we demonstrate the effectiveness of PCD in application scenarios of image-pair generation, object manipulation, and multi-robot motion planning. Our results show improved coupling effects and guaranteed constraint satisfaction without incurring excessive computational costs.

Published as a conference paper at ICLR 2026
PROJECTED COUPLED DIFFUSION FOR TEST-TIME CONSTRAINED JOINT GENERATION
Hao Luan' Xian 2oh See-Kiong Ng!.3
Chun Kai Ling'
School of Computing, National University of Singapore Paculty of Computer Science and Information Technology. Universiti Malaya 'Inttittee of Data Science, National University of Singapore
paoluan@comp nus edu.. sg, u21027566 60 s sisw. um, edu. (seekiong, churkail (eteaiil@@uus @nus edu sg
ABSTRACT
Modifications to test-time sampling have emerged as important extension to diffusion algorithms, with the goal of biasing the generative process to achieve given objective without having to retrain the entire diffusion model. However, generating jointly correlated samples from multiple pre-trained diffusion models while simultaneously enforcing task-specific constraints without costly retraining has remained challenging. To this end, we propose Projected Coupled Diffusion (PCD). novel test-time framework for constrained joint generation. PCD in- troduces a coupled guidance term into the generative dynamics to encourage co- ordination between diffusion models and incorporates a projection step each diffusion step to enforce hard constraints. Empirically, we demonstrate the effec- tiveness of PCD application scenarios of image-pair generation, object manipu- lation, and multi-robot motion planning. Our results show improved coupling ef- fects and guaranteed constraint satisfaction without incurring excessive computa- tional costs. Our code is available at at/htpps:// ///t:thhbhttt/ com/ BdwundLuan/pen /pcd
INTRODUCTION
Diffusion models have achieved remarkable success in generative modeling, with a plethora of ap-
plications ranging from image (Rombach et al. (2022). video (Ho et al. 2022). language (Li et al.)
2022]. graph generation (Niu al. 2000 Madeira
M et al. 2024 Luan et al. 2025). ), as well as roboties (Janner et al. 2022, Chi et al. [023 Carvalho et al.) 2023). One of the crucial factors underlying
these achievements the use of test-time conditional sampling techniques such as classifier guid-
ance (Dhariwal & Nichol 2021 Song et al. 2021b), inpainting (Lugmayr et al., 2022 Liu et al.
2023
reward aligmment (Uehara et al. 2025 Kim et al. 2025), and projection (Christopher et al,
2024
Sharma al. 2024).
While these methods are primarily designed for sampling from univariate distributions, numerous real-world tasks require sampling highly correlated variables from joint distributions, e.g. image
pairs (Zeng et al., 2024). multimedia (Ruan et al. 2023 Tang et al. 2023 Hayakawa et al. 2025).
traffic prediction JJang et al.] [2023 Wang al. 2024b). and multi-robot motion planning (Shaoul
et al. 2025; Liang et al., (0255.. Directly training a diffusion model to capture one single joint
distribution costly and inefficient. First, high-quality annotated datasets of joint behaviors are
scarce, expensive, and often proprietary. One example is real-world traffic trajectory data, which is essential for prediction and planning in autonomous driving (Li al. 2023). Second, training joint distributions becomes increasingly computationally demanding as the number of variables grows (Gu al., 2024), and relearning the entire joint distribution becomes necessary when marginals are changed. For instance, coordinating robot teams may require retraining the entire model even if only one robot's behavior differs from its marginal of the pretrained joint for new task.
Inspired by compositional modeling (Du et al. 2020 Liu et al. 2022 Du & Kaelbling 2024 Wang ct al. 2024a Cao et al., 2025), we opt for a more practical approach modeling multiple marginal
"These authors contributed equally.
This work done National University of Singapore.

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Published as a conference paper at ICLR 2026
distributions independently - each cheaper and simpler to train - and to couple them during test- time in a sensible way to obtain the required joint distribution. Unfortunately, such test-time cou- pling alone does not efficiently guarantee adherence to task-specific hard constraints such safety protocols and physical limits. To address such limitation, we propose extending standard I Langevin dynamics by combining projection methods with coupled dynamics. Our method cleanly integrates multiple pretrained diffusion models through a coupling cost while explicitly incorporating a pro- jection step to ensure strict adherence to task-specific constraints throughout joint sampling.
Contributions. We propose Projected Coupled Diffusion (PCD), a novel test-time framework uni- fying both coupled generation leveraging multiple pre-trate prained diffusion models and cnnjection-based generation to enforce hard constraints only specified at inference. PCD suitably generalizes some conditional sampling techniques including classifier guidance. We show empirically that PCD with both a coupled cost and projection is superior in joinily generating highly correlated samples with hard constraints compared to alternatives with the absence of either or both components.
2 RELATED WORK
Diffusion Models and Guidance Diffusion models conceptualize generation as a progressive de- noising process, i.e., Denoising Diffusion Probabilistic Models (DDPM) (Ho et al. 2220). or equiv- alently, as a gradient-descent-lke procedure that leverages the score of the data distribution within a Langevin dynamics framework (Welling & Teh, 2011 Song Ermon 2019 [Sng et al. 2021b). Improving DDPM, DDPg et al. (2021a) introduced DDIM to accelerate sampling, and Karras et al.) (2022) systematically clarified key design choices for practitioners. Guidance mechanisms form an important class of conditioning techniques for diffusion sampling. Dhariwal & Nichol (2021) first introduced classifter guidance (CG) to steer pretrained diffusion models at inference time without retraining, while Ho & Salimans (2022) proposed classifier-free guidance by integrating condition- ing signals directly during training. Building upon CG, subsequent work has extended guidance
beyond classifiers to include analytic functions (Guo al. 2024 Lee al., (025) and property predictors (Meng& & Fan 2024 Feng et al. 2024) in tasks beyond image generation.
Constrained Diffusion To address requirements of constraints in real-world tasks, researchers resort to constraint-guided diffusion generation (Yang al.) 2023 Kondo al., 2024 Feng al. 2024). However, such paradigm falls short of enforcing constraint satisfaction. This incentives the introduction of projection at each step of diffusion (Bubeck et al., 2015, Christopher et al. 2024
Liang et al.] 2025). A primal-dual LMC method by [hamon et al. (2024) handles both inequality and equality constraints, Zampini et al. (2025) proposed Lagrangian relaxation of projection in the latent space. Lou & Ermon (2023) proposed Reflected diffusion for hypereube-constrraned data while Metropolis sampling (Fishman et al.) 2023b), log-barrier and reflected dynamics (Fishman
et al. 2023a) were also introduced to Riemannian diffusion models.
Diffusion for Joint Generation Previous studies in joint generation using multiple diffusion mod- els primarily targeted multimodal generation. [arr-Tal et al.) (2023) and Lee et al.) (2023) demon- strated panoramic image synthesis by synchronizing several image diffusion models. X et al. (2024) and Hayakawa et al.] (2025) demonstrated synchronized audio-video generation, and Tang al. (2023) introduced a framework for generating and conditioning content across combinations of a set of modalities. Joint high- and low-level robot planning is also studied in (Hao et al. 2025).
PRELIMINARIES
Notation. Denote by - the set of all positive integers, || the Euclidean norm and ||- || the Frobenius norm. Let X E RD= and Y E RR be random variables where Dz, Dy E Z+ are their dimensionality, respectively. We denote (xr as the probability density function of random variable X, likewise for Y, and may omit the subscript indicating the random variable when it is clear from the context for notational brevity. Let (p,, Σ) be normal distribution with mean μ and covariance Σ. Denote IIx (x) R D. + D. as a projection onto a nonempty set Kxx C RP::
Txx (x) A arg min || -
Wee break ties arbitrarily if arg min.mii..... ||= x|| is unique. Uniqueness always holds for convex Kxx.
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Diffusion and Score-based Generative Models. We examine diffusion models' inference from the perspective of Langevin dynamics. Let Ex(x) 2. + be a continuously differentiable
energy function with Lipschitz-comtinuour gradients and Z Sgo. exp (-Ex (z)) dz < This
energy function defines probability density (z) 1/Z exp Ex (x)). To sample from px (x),
one may leverage Langevin Monte Carlo (LMC) Roberts & Tweedie 1996, Welling & Tch 2011).
Given initial sample from prior distribution Xo Px (z) and fixed time step size ò RT the LMC iterates are as follows:
X = X, VV (X.,) +
where 6 (0, 2 and V Volog px (x) is called the (Stein) score function of p.x (z). When d - and T + 0o, the distribution of XT converges to px (I) under some regularity conditions (Welling & Tch 2011: Song& & Ermon 2019). In practice, the analytie form of Ex(x) is not accessible. Instead, the score Volog (I:) or equivalently the gradient - -xx (ar) is approximated by a neu- ral network parameterized by and trained via denoising score matching (Song & Ermon 2019):
sxx (z,t) 2 VylogPx,(11)
Classifier guidance. Classifier guidance (CG) is a "soft" way to steer the diffusion sampling pro- cess toward desired distributions. Given a desired attribute yo as (a constant) condition, the objective of CG is to sample from target conditional distribution Pxjy (x y 3o). CG achieves this by perturbing the original learned score with a likelihood term to obtain the posterior score:
V Vglog (z y = yo) = V px (r) + V Vogogyyjx (y = Yo | xx)
where V V.logpx((1) (4)) is approximated by the trained score model 5x. and the likelihood P (y x)) can be modeled by classifier, predictor, or a differentiable function.
Projected diffusion. To enforce the hard constraint X E Kx, projected LMC performs a projec-
tion operation at every diffusion time step:
X 11 (X, + 8V 2V ex) Kx
with 6, ~ (0, 261). The convergence properties of Eq. 2 is analyzed by Bubeck et al. (2015)
when the constraint set Cx convex and the distribution px (x) is log-concave.
4 PROJECTED COUPLED DIFFUSION
We study the problem of generating correlated samples (X, Y) subject to the test-time constraints of X Kxx and Y E Ky, with two pre-trained scores diffusion models sxx t) and sy (y, t), parameterized by 0 and é. respectively, without retraining either of them.
Coupled Dynamics through Costs. To facilitate correlation between the generated X and Y, we propose using cost function couple their diffusion dynamics, i.e., Eq. (I) for X and likewise for Y. Let the cost function c(I, y) : DR x RDy + IR be continuously differentiable Then the coupled joint dynamics of (X, Y) are
X X, 78V C.V2ccXK (X,
Yi) + (X,, + Ex,t
(3a)
Y Y, - ydV c( Y)) + 6 (Y,, + EY
(3b)
where y E R+ is coupling strength parameter, <x E (0 25I D. ), cY,, E (0, D5I are i.i.d. Gaussian noise drawn at each diffusion time step. The cost function c(x, y) can either be an analytic function neural network, e.g., trained classifier a regression model, and demonstrate the use of both in our experiments.
As an extension, for each cost function instance c(x. y). we can derive its posterior sampling (PS) variant Cps (ar, y, f) with the DPS method (Chung et al. (003). Concretely, by Tweedie's 's formula (Efron) 2011 may obtain point estimate for each vaniale's s denoised version through the trained scores, and then compute the cost with those estimates:
Cps (x. V. t) = e Xx + xx (X,, o (
where ox, and or are the noise levels at time step &. associated with the score models.
II practice, this be relaxed cost functions that possess subgradients.
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21.0%
Bock
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Location of Smax Divk
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(a) Toy example selting
(b) Conditional sampling
(c) Sampling w/ coupling
(d) PCD (proposed)
Figure I:
Toy example of fitting two blocks with different sizes into a narrow corridor. (a) Setting:
the bigger block (purple) and small block (green) with lengths 6 and 2. Both score functions those of two Gaussians centered at the middle of the red corridor of length (b) Naive conditional approach: the big block occupies the center and the small block is highly possible to overlap if to stay within the corridor (orange for probability mass of overlapping positions, blue for that of non- overlapping positions). (c) The coupled dynamics can place both blocks at different sides with much lower overlapping probabilities, but still can go out of the corridor.
(d) PCD produces solutions with low overlapping probabilities and guarantees both blocks stay within the corridor.
Remark 1. The PS variant does not exactly match the description of a cost function in Eq. (3) due to the extra dependence on diffusion time step t. Yet empirically we find them performing well in our proposed framework.
Projected Coupled Diffusion. On top of promoting correlations between the generated samples, we also aim to enforce the constraints given only at test-time. As such, we propose to join the coupled dynamics and projection, resulting in the Projected Coupled Diffusion (PCD
X II (X, - -VV Yi) 85 (Xt,t) + Ex,,)
(4a)
Kx
Ye+1 11 Ky
(Yi- Vi) (Y,, + EY:t)
(4b)
where 8 E R is the LMC step size parameter, 7 E R+ is the coupling strength parameter, EX,r ~ N(0 25T D.). Cy NN (0, 25I i.i.d. noise drawn per step and X, NN, ID Yo
NO0 I. Generation algorithms of our method adopting LMC DDPM are in Appendix A
We illustrate the importance of simultaneous coupling and projection through toy example (Fig- ure Tap. Two ID blocks of different lengths must fit within a corridor and avoid overlapping as far as possible. Each of the block's center (denoted by X and Y) is generated with score model that has learned a Gaussian distribution centered at the midpoint of the corridor. A naive approach is first generate X with only its learned score, and then generate Y conditioned on X via classifier guidance. However, this can lead to samples with poor mutual correlations, e.g. the first generated X occupies the center of the corridor regardless of Y. leaving not enough room to fit both (Fig- ure lb In contrast, coupled dynamics incorporates mutual influence into the generation process of both variables via the cost function, resulting much lower overlap probability, yet could violate the hard corridor constraint (Figure Tcp. Our proposed method, PCD, combines coupled dynamics and projection (Figure ld]. avoiding overlaps while enforcing the corridor constraint.
RELATIONSHIP TO OTHER METHODS
Classifier Guidance. Interestingly, our framework can encompass the prevailing technique of (CG) special case. If we (i) set the cost function as c(x. v) x log PrJx (y x), assuming
While the assumptions in (Bubeck al. 2015) do not directly apply due the non-convex (non-concave) nature of the neural networks used for score approximation, we observe that PCD performs well in practice with reasonable parameters.
'If the small block is placed first, it is highly likely to become even infeasible to position the big block without exceeding the corridor or overlapping.
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continuously differentiable density pr exists and the (approximated) gradient VePrix (y x) is accessible, (ii) let Ky be a singleton only containing the constant condition Ky = {vo) and (iii) Kx RD=, then PCD reduces
X1+1 X, + vV,lo [pY|x (yo X.)) px (X.)) + Ec: 6r ~ N,, 261)
(5)
where 7 becomes a temperature for the likelihood and trivially Y; = yo: when 7 = 1. the gradient term fully recovers the score of the posterior distribution Pxly (z y vo). In that regard, CG can be seen as PCD with one variable fixed and projection removed in the other. We numerically demonstrate this point the toy example in Figure see Appendix C.1 for details.
Projected Diffusion. We can consider projected diffusion as PCD without coupling. In PDM Chhistopher et al.) 2024). which only concerns a single variable X, the LLC dynamics of X is projected onto nonempty but not necessarily convex set C C RR,, This work lits into PCD in the sense of (i) Kx = C. Ky = { and (ii) the cost function c(x, y) = O, preserving the projection and decoupling the dynamics (turning Y, into a dummy variable). A numerical verification of PD as a special case of PCD on the toy example in Figure I is in Appendix C.1
Compositional Diffusion. Another notable line of research is in compositional diffusion (Liu et al. 2022; Wang et al. 2024a Xu et al. 2024; Cao al., 2025). Similar to ours, this class of methods also aim to "combine"
multiple distributions modeled by (enegyybbased models (EBMs) e.g., diffusion score models. Unlike ours, however, they still foeus
univariate distribution (eff. a joint distribution in ours). These works attempt to sample from a target product distribu- tion Px
prod (ar) x (x(()) Ie (x))xx))xx)))) x exp(-Fx (x:) exp N Fx (1) - -N Fig. (x))
where Fig (1) are the corresponding energy functions for = 1,....,: N. In practice, the gra- dients of the N energy functions (x) are learned (independently) by N diffusson models PCD can conceptually cover those methods by (i) projecting X, onto RDD,, (ii) projecting Y onto a singleton {yo} and rendering it a dummy variable, and (iii) setting the cost function as: c(x, v) N. -x(x)-Ex Egg. (a),, whereas only its partial gradient V,,c(r, y) is being used.
8r -x-
Joint Diffusion. Hayakawa et al. 2025 propose to decompose the "joint scores" Valog p(x, y) V
+ Vx,vvxx (Y,)X,) and likewise for V logo,p(,,,, Yo), and train one single discriminator Dø y) approximate both conditional scores: (x,logg(Y) = Vx,log I-))(X.
and Vylog p(x, y) with Bayes rule: V Vx, log p(X,,
Do(X,,Y.) likewise for Vy,logp(Xx|\\).
PCD can cover this by setting the constraint sets Ixx = RD= Ky R Pv, and the cost to
c(z, y) = - log I-Do(2.3)
Pp(x.v)
5 EXPERIMENTS
We seek to address the following research question:
How effective is our proposed PCD method in terms of jointly generating correlated samples with test- time constraints compared to generation only with projection, coupling costs, or neither?
We also explore via ablations the tradeoffs between coupling and trained distribution adherence.
5.1 CONSTRAINED MULTI-ROBOT NAVIGATION
We demonstrate PCD in constrained multi-robot motion planning tasks (Shaoul et al. 2025 and its extension to more than two variables Given start and goal location for each robot, our objective to use pretrained diffusion models to generate 2D path trajectories that: (i) avoid collisions with static obstacles and any other robots. (ii) respect hard velocity limits specified at test time, and (iii) exhibit specific motion patterns dictated by the environment. See Appendix B.2 for task details.
$ Instead of aiming for state-of-the- performance, these experiments demonstrate that simple method like PCD can achieve strong results without intricate domain-specific extensions MMD(Shaoul et al. 2025
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(a) MMD-CBS
(b) DIFFUSER
(c) PD
(d) CD (w/o pro..
(e) PCD (Ours)
Robot trajectories in Hi ghways with = 4 robots generated by the compared methods. Red crosses mark collisions; blue stars mark velocity constraint violations. The desired motion pat- tern is to circle the central obstacle counterclockwise. (a) MMD-CBS excels in collision avoidance but cannot guarantee velocity constraint satisfaction. (b) Vanilla DIFFUSER fails generate both collision-free and velocity constraint-compiin trajectories.
Figure 2:
(c) Projection only enforces velocity
constraint but ignores collision avoidance. (d) A coupling cost facilitates inter-robot collision avoid- ance but cannot guarantee velocity constraint. (c) Our method can effectively generate collision- -ree trajectories and enforce the velocity constraint. More qualitative results are in Appendix C.3
Projection. We use projection to enforce max velocity constraints on each robot. Denote the velocity limit Umxx the (physical) time step size as At, and trajectory horizon as H. The projection can be written as an optimization of which the feasible set is our constraint set Ixx:
min X Xex
- 1X--112
(6a)
S.L. I X11l S Umax At,
(6b)
[ - X S Umax At, h= 2 H,
(6c)
where X is the Slffiusion-prediiie trajectory for one robot in matrix form, X R is the position vectors at (physical) discrete time step h and zo R² is known starting position. This convex optimization problem can efficiently solved in parallel using the Alternating Direction Method of Multipliers (Boyd et al. 2011 Parikh et al. 2014). Detailed derivations and algorithm are in Appendix B.2.2
Coupling Cost. We aim avoid both collisions among robots and collisions with known static obstacles via coupling, Prior work (Carvalho et al. [2023] Shaoul et al. (2025) achieves static obstacle avoidance by CG, which we have shown is a special case of PCD coupling. Thus, our coupling cost function is linear combination of robot-collision and obstacle-collision costs:
((,Y) Coooo (X, K Y) + AobssCota( (X,Y),
where X, Y E Hx2 trajectories of robots' positions. We experiment with two robot collision cost functions, (i) log-barrier (LB) cost
CLA CLR(X, (x,(X Σ h=1 log (IIXA Yx|l (r)
where a > 0 is parameter, and (ii) a "squared hinge distance" (SHD) cost:
CSHD (X,Y) = Σ h=1 (1 YA s sr] (r ||X, - IN)
(8)
where 1[:] is the indicator function and r > is the active range parameter. For N > robots, X ;-.., N extend both costs to co ...... XN) Σ Eigisen co X)) with
(LB, SHD}. We follow Carvalho et al. (2023) in designing the abstacle-avoidanee cost. See details in Appendix B.2
Experiment Setup. We test with 2 and 4 robots on four different tasks from Shaoul et al. (2025) For each task, we use the pretrained models from Shaoul al. (2025) and choose three Umax
Baselines. We compare our method with a vanilla diffusion model DIFFUSER (Janner et al., 2022 and MMDD-BB (Shaoul al. 2025). We evaluate each method 100 trials, each with initial

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Task Empty, Robots, Max Vel. 0.703
Task Highways, Robots, Max Vel. 0.878
METHOD Metric
SU(%)T
RST
"minCS($)1
"minDAt
SU(
RST
"minCS(S)f
"minDAf
DIFFUSER
65.0
0.616
62.3
0.990
53.0
0.208
66..7
0.979
MMD-CBS
100
1.00
11.0
0.990
100
1.00
63.0
0.960
DIFFUSER + proj.
65.0
0.615
100
0.990
54.0
0.214
100
0.978
CD-LB (w/o proj-)
100
0.993
0.0312
0.814
100
0.999
0.00
0.986
CD-SHD (w/o proj.)
100
1.00
35.3
0.990
100
1.00
34.6
0.976
PCD-LB
96.0
0.916
100
0.489
100
0.950
100
0.957
PCD-SHD
100
0.993
100
0.960
100
0.996
100
0.963
Table Performance comparison with 4 robots on two out of four tasks. Left: Empty, Umax 0. 0.703; Right: Hi ghways, Umax 0.878. "For CS and DA, which should have been Nttuples, we report here the minimum of them due to space limit. See additional results Appendix C.3
(a) PushT Task
(b) DP
(c) DP projection
(d) CD (w/o proj.)
(e) PCD (Ours)
Figure 3: (a) The PushI task: One single robot (blue circle) pushes T block (gray) target pose (green) given different initial positions of the block and robot. Robot trajectories are plotted with colormap (warmer colors indicate later time steps), with red crosses marking velocity violations. Only the first few dozen steps are shown for visual clarity. (b) Vanilla DP fails to generate trajectory pairs, each in distinct mode, and adhering to velocity limits. (c) Projection enforces velocity limits but cannot "split" trajectories apart. (d) A coupling cost encourages non non-intersecting trajectories but does not strictly enforce velocity constraints. (e) Our method generates non non-intersecting trajectories and ensures strict adherence to velocity constraints.
configuration (start and goal locations for each robot) sampled uniformly random by rejection sampling. Except for MMD-CBS, we generate 128 i.i.d. samples; for MMD-CBS we also set its diffusion sampling batch size as 128. We run 25 diffusion inference steps for all methods.
Evaluation Metrics. We evaluate performance of the methods in terms of task completion or adherence to original data distribution, constraint satisfaction, and inter-robot collision avoidance. We adopt success rate (SU) and data adherence (DA) from (Shaoul et al.) 2025) to evaluate task completion. SU is the average, over all initial configurations, of an indicator for whether at least trajectory in the batch completes the task without collision. Constraint satisfaction (CS) is an indicator of whether a trajectory satisfies the velocity constraint at all time steps. Inter-robot safety (RS) is indicator of whether trajectory tuple inter-robot collision-free. All metrics except SU are reported as empirical means over a batch of i.idd samples.
Results. Figure shows sample trajectories from compared methods. Table summarizes quan- titative results for two environments with 4 robots under one velocity constraint. All constraint- agnostic methods (without projection) achieve low CS rates. MMD-CBS unsurprisingly achieves perfect SU score by selecting and stitching single optimal trajectory tuple. PCD- and CD- methods approach this upper bound, while vanilla DIFFUSER and its projected variant lag behind. PCD-SHD shows slightly reduced data adherence, while the LB cost for collision avoidance due to its steep gradients. Similar patterns are observed in the other tasks. Overall, PCD effectively promotes inter-robot safety through coupling while enforcing hard test- time velocity constraints, with a tradeoff between coupling strength and data adherence depending on the cost function. See additional results with ablations, and runtime in Appendix C.3 and C.2
aggressively trades data adherence
"See detailed discussion of difference hetween PCD and MMD-CBS in Remark Appendix C.3

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5.2 CONSTRAINED AND DIVERSE ROBOT MANIPULATION
We evaluate our method on the PushT task (Florence et al. 2022 Chi et al. 2023). As shown in Figure Bal diffusion model is trained generate trajectories for a robot to push the gray T- shaped block from different starting positions till it matches the green target pose. Our objective to utilize such pretrained models to generate a pair of distinet trajectories strietly satisfying a maximum velocity constraint imposed at test time and do not intersect as far as possible"
Projection & Coupling Cost. As in multi-robot experiment, we enforce velocity limits via pro- jection, using the formulation in Eq. (6). We experimented with two cost functions for encouraging trajectories from pair stay away from each other. The first cost builds upon the Determinantal Point Process (DPP) guidance (Feng et al. 2025) designed for promoting trajectory diversity:
CDPP X, Y) = log (cos < (X,8)++ +
where X, 8 E R2H are the flattened vectors of the trajectories, and y > 1 is a constant. The other cost the log-barries cost Eq. (7). For both costs, we also devise their posterior sampling variants.
Setup & Evaluation Metries. We adopt DIFFUSION POLICY (DP) by Chi et al. (2023) as our base algorithm, using pretrained weights from Feng et al. 2024 Compared methods are vanilla DP, DP with only projection. DP with only coupling and PCD. We evaluate each method on 50 uniformly random initial conditions. With each method, under three different max velocity limits. We use 32 diffusion steps at inference, with other settings recommended by Chi et al. (2023). Metrics: Four quantitative metrics are evaluated: Dynamic Time Warping (DTW) (Berndt & Clifford 1994; Müller 2007), discrete Fréchet distance (DFD) (Alt & Godau 1995). J velocity constraint satisfaction rate (CS), and task completion score (TC) (Florence et al.] 2022, Chi et al. 2023). DTW and DFD quantify dissimilarity between two trajectories. CS evaluates fraction of trajectories satisfying the velocity constraint. TC measures how well the block- pushing task is accomplished, where 1.0 is the best and O the worst. We report all metrics by their empirical means over all initial staring locations and trajectory pairs. Details are in Appendix [3.
generate 100 pairs of full trajectories,
Results. We report results in Figure 3b Bel and Table 2 From Table 2 see all projection- involved methods achieve perfect velocity constraint satisfaction, outperforming both the baseline and coupling-only (CD) approaches. CD and our PCD methods consistently produce higher DTW and DFD than baseline with or without projection, suggesting that coupling effectively discourages intersecting trajectories. In terms of task completion, all methods except the baseline show degraded performance, likely due to the velocity limit enforced by projection. These results show that our framework can enforce test-time velocity constraints and spatially separate generated trajectories without significantly sacrificing data adherence. Without projection, both PS variants of the coupling costs exhibit higher DTW and DFD than their respective non-PS version, and better preserves the task completion with projection. Full results and additional ablation study are in Appendix C.4
5.3 CONSTRAINED COUPLED IMAGE PAIR GENERATION
We demonstrate a toy example of paired face generation using two latent diffusion models (LDMs) (Rombach et al. 2022 (Figure 4 Each generated pair must (i) satisfy gender and facial attribute constraints, and (ii) exhibit clear contrast between age groups. We enforce (i) via projection and promote (ii) through a classifier-driven coupling cost.
Projection. For each LDM, we generate two structurally similar exemplar images of one individ- ual at different ages (Figure 4a) using ChatGPT (OpenAl 2023), encode them via the model's VAE, and form convex hulls as feasible latent regions. At each diffusion step, we use mirror descent (Ne-) mirovsky & Yudin 1983 Beck & Teboulle| 2003) to project intermediate latents onto these hulls, enforcing strict gender and facial attribute constraints (see Appendix [or details).
Coupling Cost. To induce an age-group contrast, we use a latent classifier that classifies age groups (young, < 50) and O (old, 2 50). Our coupling cost defined as cxoR (z, y) E'E (v XOR(I,)) y), where XOR(x, y) p(a]=)(1 (10(111ppay))) + p(aly) (1 and p(al-) are the probabilities of a sample belonging to age-class a E (Y, obtained from the classifier.
7 Note that the two trajectories in pair are not pushing the block together at the same time.
'The model from (Feng et al. 2024) was trained on an augmented dataset with broader coverage than that provided byChii et al. (2023), yiclding more diverse and feasible trajectories.
Chi
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METHOD
DTW1
DFD1
CS(%)
TC1
Method XOR%
M/F%
SE-CO
SE-LA
IS-L-1
DP
3.2
.47
(65, 65)
(.93,.93)
SD
20
71/14
.40/.41
777 .75/.75
DP proj.
3.0
.43
(100,
100)
.89)
CNet
100/56
68/.75
.48/550
.4/.50
SD+P
100/100
.88/91
.15/ .06/.09
CD-DPP
3.7
.54
(63, 62)
(.92, .92)
CD-DPP-PS
4.5
.65
(57. 57)
(.91, .2)
Syne'
95/17
.62/.61
.67/.68
.70/.69
CD-LB
4.1
.60
(59, 59)
(19]
SD+C
48
51/47
.4/.466
.76/.7
.7/.75
CD-LB-PS
4.5
.65
(59, 59)
(.92,.93)
SD+C*
64
47/37
.55/660
.70/.68 .69/.69
PCD-DPP
1.6
.64
(100,
100)
(833
PCD
96
100/100 .88/192 .11/.14 .11/.13
PCD-DPP-PS
.62
(100, 100)
(.89, .89)
PCD-LB 5.1
.71
(100, 100)
.79)
Paired face-generation results. Base- lines: CNet = ControlNet-XS: Syne = SyneDiff fusion. Metrics: SE-C* = SE-CLIP; SE-L SE- LPIPS; IS-L = IS-LPIPS. Boldface indicates the best score(s) for each metric. Full results includ- ing SE-FID deferred to Table 15 in appendix.
Table
PCD-LB-PP 4.4
.62
(100, 100)
(.89, .88)
Table 2: Results of PushT task by all compared methods with velocity limit Vmmxx = 8.4. CD de- notes DP+coupling only: -PS denotes posterior sampling. See full results in appendix Table 14
(a) Exemplar Images (b) SD
(c) Sync (d) SD+C (c) SD+C1 (f) CNet' (g) SD+P
(h) PCD (Ours)
Figure 4: Paired face generation with two Stable Diffusion models (SD) (row-wise). Methods: Sync SyncDiffusion, CNet = ControlNet-XS, +C= coupling; +P= projection; T= with text prompt. (a) Exemplar images of each model, their latents form the convex hulls. (b) Vanilla SD often fails to produce faces. (c, d) SyncDiffusion and Coupling steers samples toward the target age-group con- trast but with attribute drift. (e) Adding text prompts Coupling yields faces, yet violates attribute constraints. (f. g) ControlNet-Xo. encourages gender and facial attribute alignment: Projection en- forces it; but both fail to promote the age-group contrast. (h) Our method yields pairs that satisfy the gender and facial-attributt constraints and simultaneous promote sheer age-group contrast.
Setup Metrics. We employ STABLE DIFFUSION V2. IBASE (SD-2.1) (tabiiiiyy Al 2022 as our base model and disable classifier-free guidance during sampling to achieve unconditional gen- eration unless specified otherwise. For coupling, we train a latent classifier for the model using the FFiQ-Aging Dataset (Or-El et al., 2020). Baselines: We compare
method with vanilla SD-2.1, SD-2.1 with only projection, SD-2.I with only coupling, and SD-2.1 with coupling and a generic text prompt. We also compare against SyncDiffusion (Lee et al.) 2023) and ControlNet-XS Zavadski et al. 2024 for reference To We use 100 diffusion steps and generate 25 pairs of samples using each method. See setup details Appendix B..]| Metries: To evaluate projection, we report five metrics: (i) Gender constraint satisfaction rate (M/F); (ii) Sample-Exemplar CLIP similarity (SE-CLIP) (Radford et al. (02T); I); (iii) ExemleeExemplar LPIPS (SE-LPIPS) (Zhang et al. 2018); (iv) ExeelelE FID (SE-FID) (Heusel et al. 2017); (v) Intra-Sample LPIPS (IS-LPIPS).
SE-CLIP, SE-LPIPS and SE-FID serve as proxies for adherence exemplar-specified facial at- tribute constraints (noting that satisfaction is guaranteed by design of our projection operator), while IS-LPIPS quantifies diversity across generated samples. To evaluate coupling. we measure the age-group contrast satisfaction rate (XOR) with another age-group image classifier trained on the FFHQ-Aging Dataset. We average XOR and M/F over generated pairs, SE-CLIP and SE-LPIPS
Note that ControlNet-XS requires additional training and use pretrained weights released by Zavadski al. 2024). See details Appendix [H.4
"iffuDiffusion and ControlNet-Xe are included solely for reference. We are not claiming state-of-the-art
nerformance. Instead, we demonstrating potential of PCD in image-related generation tasks.

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Published as a conference paper at ICLR 2026
exemplar pairs, IS-LPIPS over intra-model sample pairs. SE-FID is computed single scalar between the exemplar set and generated samples set of each model.
Results. We report results in Figures 4b and Table 3 Projection-based methods (SD+P, PCD) attain 100% gender satisfaction (M/F) and the strongest alignment to exemplars (higher SE-CLIP and lower SE-LPIPS) compared to vanilla SD, coupling-only variants and other baselines. Pro- jection reduces diversity (low IS-LPIPS); adding coupling partially recovers diversity. Coupling- based methods improve age-group XOR satisfaction, with PCD attaining the highest rate (96%). SyncDiffusion matches SD+C on XOR but shows slightly reduced sample variation (lower IS- LPIPS). ControlNet-XS. exhibits male-bias, producing more male samples and thus yielding 100% male gender satisfaction (M). See additional results with larger exemplar sets and ablations in Ap- pendix C.6 and runtime profile in Appendix C.2
6 LIMITATIONS AND DISCUSSIONS
First, the per-step projection operations in PCD introduce computational overhead compared to vanilla standard guided diffusion methods. In practice, however, this overhead can be mitigated through several simple strategies: (i) reduced projection frequency: projections need not be per- formed every diffusion step but can instead be applied every K diffusion steps (including the final step to enforce constraints); (ii) approximated projections: limit the number of optimization iterations within each projection operation to achieve near-convergence, and only run projections to full convergence every K diffusion steps (including the final step); (iii) warm-startin stam-starting: initialize the ADMM primal and dual variables from the previous diffusion step. These strategies can effectively reduce runtime while preserving the core benefits of PCD.
Second, PCD assumes differentiability or least the existence of subgradients for the cost func- tions as many other approaches (Dhariwal & Nichol [021] Nichol et al. [2022] Chung et al. [022] 2023 Carvalho et al., 2023] Römer et al. (025)). For various practical applications, differentiable available for reasonable yet nonfdifferentia and discontinuous costs such as
approximations
temporal logic specifications (Feng et al. 2024 Meng & Fan 2024). Moreover, alternatives that do not require gradients such Monte Carlo based importance sampling could be leveraged for
approximation (Dou & Song 2024 Phillips et al. [2024; Jung et al. 2025 Li et al. 2025).
Third, PCD assumes that the feasible region of the test-time constraints has overlap with the support of the distribution learned by the pretrained model. This issue becomes critical if the feasible solutions differ substantially from the training data. For example, restricting end-effector velocity to only 1 % of its original average in the PushT task would cause PCD to generate trajectories failing the manipulation task at most times.
Finally, PCD introduces coupling strength hyperparameter, and practitioners may need to tune for desired outcomes. Degraded data fidelity observed if the coupling strength is set too high, indicating a trade-off between data fidelity and desired correlations between variables. Recent efforts (uug al., 2025) in promoting compatibility between test-time objectives and pretrained diffusion's 's data fidelity point promising direction to addressing this issue.
7 CONCLUSION
We introduced Projected Coupled Diffusion (PCD), a test-time framework for joint generation with multiple diffusion models under hard constraints. Our method combines coupled dynamics and projection operation, generalizing existing techniques like classifier guidance and projection-based diffusion inference without requiring model retraining. Experiments on image-pair generation, ob- ject manipulation, and multi-robot motion planning show that PCD effectively facilitates mutual correlation and provides guaranteed test-time hard constraint satisfaction. Future work includes exploring more sophisticated and non con-differentiable coupled costs and non-convex constraints.
REPRODUCIBILITY STATEMENT
Implementation code is available attt:pps//ggtthbb...////dddddddd
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LARGE LANGUAGE MODEL USAGE DECLARATION
As stated in Section 5.3 the images used exemplars for the paired-faces experiments were gen- erated by ChatGPT OpenAl 2023). We also used ChatGPT (OpenAT] (023) to perform grammar check and minor language polishing for the first three paragraphs of Section with further human edition.
ACKNOWLEDGMENTS
This research/projeet is supported by the National Research Foundation, Singapore under its Na- tional Large Language Models Funding Initiative, (AISG Award No: AISG-NMLPP2024-0000 and the National University of Singapore, under the Start-Up Grant Scheme. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not reflect the views of National Research Foundation, Singapore. The authors would like to thank Ziqiao Meng, Jiaying Wu, and Sze Jue Yang for helpful discussions and feedback on early drafts.
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A METHOD DETAILS
Our proposed PCD framework can perform inference with LMC and Denoising Diffusion Proba- blistic Models (DDPM) (Ho et al. 2020). Moreover, it is also easy apply Diffusion Posterior Sampling (DPS) (Chung et al. 2023 within our framework. We present here three algorithms under PCD framework: PCD-LMC, PCD-DDPM, and PCD-DPS, detailed in Algorithm Algorithm and Algorithm 3
respectively.
Algorithm Projected Coupled Diffusion with LMC
Require: Score models a iteration T.
sy: projectors Txxx my coupling strength ?: LMC step size ò; max
Xo ~ (0, I D.). Yo (O, ID)) 2: for t = 1 to T - do
b Initialize from std. Gaussian
3:
Coupled LMC dynamics
4:
X++1 t Xi+1 ds
~
5:
Y441 V141 58l ( (Y, -2 Vi)
6:
~x~ IxNN(0,In, I,,,,I,,,,
N(0,
b i...d. noise
7:
X + X++1 + /28xx
8:
Y441 + Xx+1 + V2dey
9:
Projection step
10:
X141 + IIx (X++2)
11:
Y1+1 t IEYy (Y4+1)
12: return (Xr,Ya
> Return joint samples
Algorithm 2 Projected Coupled Diffusion with DDPM
Require: Score models x sy: projectors Ix Ix coupling strength y; DDPM noise schedule (oz )T) DDPM inference step T.
1: Xx NO, I. ). Yr
(0, ID,)
b Initialize from std. Gaussian
2: for = to do
3:
Normal diffiusion
4:
~ ~N0, D.), ey ~ NN ID,)
5:
sx + (x,,t) sy 2 (Y,,)
6:
X,-1 var
(X, + (1-a))8x) /1 - atex
7:
Yi-1 1 (Yr (1 (1-ar)8y) v1 attY
Vot
8:
Coupling step
9:
XI-1 Xt-1 - Y!)
10:
11:
Projection step X,-1 + 1Txx (X1-1)
12:
13:
Y,1 TTx (Y,-))
14:
return (Xo, Yo)
b Return joint samples
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Algorithm Projected Coupled Diffusion with DPS
Require: Score models x s projectors IIxx Hy coupling strength y; DDPM noise schedule {or)7=13 ; DDPM inference step T.
1: Pre-compute Ox II= 0t for 1 I,...,
2: Xr, YT (0.
b Initialize from std. Gaussian
3: for T do
4:
b NNormal diffiusion
5:
cx ~ ~~1N ID,), ), cy ~NO0 Ip,)
6:
8x t so(X,, t), sy + so,,,
7:
X,-1 + To. (X, + (1- (1=a)))xx) + vI 0,Ex
8:
Y- + Tar (Y, +1++1111 (1-a,)sr) + v1 - dHeY
9:
Coupling with posterior sampling
10:
Xu -- 1 (X, + (1 -
b Tweedie's formula
vāt
11:
Yot va, (Y, (1- a.))))
X,-1 + X1-1 - / x xxo, Po)
13:
f- 1-1+
- 7 c(Xo, c Yo)
14:
Projection step
15:
X,-1 + +-lx (X1-1)
16:
Y1-1 t TTKy (Y-11)
17:
return (Xo. Yo)
> Return joint samples
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B IMPLEMENTATION DETAILS
B.1 GENERAL
Computational Hardware. All experiments were run on a workstation with AMD Ryzen Threadripper PRO 5995WX 64-Core CPU, 504 GB RAM, and 2 NVIDIA RTX A6000 GPUs each with 48GB VRAM. For each experiment run, only 1 GPU was utilized.
Software and Code Bases. All experiments were run using Py ToTcch (Paszke al. 2019). Im-
age experiments were also run with Diffusers (von Platen et al. 2022).
The PushI experiment
adapted from LTLDoG (Feng al. 2024) and DIFFUSION POLICY multi-robot experiment builds upon MMD (Shaoul et al. 2025
(Chi al. 2023 The
B.2
MULTI-ROBOT EXPERIMENT
B.2.1 SIMULATION TASKS AND ENVIRONMENTS
Four simulated environments used in our experiments are from the MMD benchmark Shaoul et al. 2025), Empty, Highways. Conveyor and DropRegi onoo shown in Figure In each envi- ronment, given start and a goal position (8,g) E IR² x R² of single robot, a diffusion model is trained generate 2D trajectories X E IR #x2, where H is the trajectory length. Each environment comes with a trajectory distribution through a trajectory dataset collected under specific motion pattern:
In Empty, the motion pattern is straight-line movements from start to goal.
Hi ghways, the pattern circling around central block in counterclockwise direction when moving from start to goal, resembling a traffic roundabout.
In Conveyor, the pattern is only moving in one single direction (either left or right) when entering the corridors in the middle of the map.
In DropRegions io. the pattern is stay around one of the four "dropping points" in the map for certain amount of time steps when moving from start to goal, resembling a delivery job.
The task both environments is generate collision-fie free trajectories for N robots, given an initial configuration consisting of start and goal positions for all robots:
غ ($1.91),... ... (SN. (NN. E (Wiree x Whrel)
where 8; 9s the start and goal positions for robot i, and Wiree C IR.2 denotes the obstacle-free environment space (free workspace).
Figure 5: Multi-robot motion planning tasks with 4 robots and simulation environments (from left to right) Empty, Hi Highways, Conveyor, and DropRegion, each with their motion patterns. Each of the trajectories' colors denotes an individual robot; green and purple small dots represents start and goal positions.
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B.2.2 ADMM-BASED PROJECTION DESIGN
We repeat Eq. (6) here for readers' convenience:
min X KERRX2
||X XII
6a
S.t. [r
X1ll S Umax Al,
6b
[ Xn-1 Umax At, - HI
6c
where X is the diffusion diffusion-predietedd trajectory for one robot in matrix form, Xx, R² is the position vector at (physical) discrete time step and Zp E I² is a known starting position.
A direct approach for solving the optimization in Eq. 6 is to leverage off-the-shelf solvers to op- timize each trajectory. However, this incurs significant computation overheads upon large batch of reformulate Eq. (6) and efficiently solve batch of such problem
trajectories. Alternatively, instances in parallel using Alternating Direction Method of Multipliers (ADMM) (Boyd & Vanden- berghe 2004).
To apply ADMM, we need to introduce auxiliary variables Zh. E R22 representing the per-step positional displacements:
21 X1 20:
(10a)
Zn = XA -XA-1 for / = ........ H.
(10b)
The constraints Eq.
(bb) (6 then become:
2 S Umax At, h h= 1,..., H.
Let = [Z,, ... ZH] E x2 and define the constraint set for Z:
K2 Z R xx2 | Z S Umax At. h 1, H
(11)
together with the indicator function of K2:
0
Z E Kzz
1xc (2)
(12)
0
otherwise.
Let A E HR be a coefficient matrix
0
0
0
1
(13)
and define an offset matrix e 1x2
b
Tp 02
02
(14)
H-1
where 02 E IR? is zero column vector.
With X and Z, optimization problem Eq.
6 can be reformulated in ADMM fashion as:
min X X K2RRK22
X X1 + Txx(Z)
(15a)
s.t.
AX = b.
(15b)
The augmented Lagrangian of Eq. 15 is:
(x(X, Z,A) ||X X Ix, (Z) Tr (AT (AX b)) ||AX Z
(16)
where A IR #x2 is the dual variable, Ε is the augmented Lagrangian penalty parameter, Tr(-) denotes the trace of a matrix, and || || denotes the Frobenius norm.
The update rules of X, Z and the dual A is derived follows by ADMM:
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Xuuudtte:
X arg min (x 2* ,A*).
It has flosed-form solution given by taking the gradient w.r.t. X and setting it to zero:
2(X - X) + AT + (AX - 2t b) = 0,
which yields
x = (2I + EA A) (2X + EAT z + +EA A²)
(17)
Z-update:
zk+1 arg min Le (x (x++1,
(x*+),,,,)) ,2,A")
which is
Tr (r(-2T + _ 1 / (AXx+ - b)ll
z arg min ZEK2
The solution to the above
zk+1 1kx
Ax - b + 5 1t
(18)
where the projection operation is applied row-wise
We (Umax A!)
if | > Umax Al,
otherwise,
with w/ being the h-th row (AX b+ & 41a)
A-update:
A A++ = A + E(AX Zk+1 b).
(19)
Note that the matrix (2I + (ATAA A) in Eq. 17 is symmetric positive definite and constant across iterations, which allows for caching its inverse. A more efficient approach is to perform LU or Cholesky decomposition on (2I ı + & A) once and solve the linear system
Remark 2.
(2I + T A)X*+1 = 2X + EATZZ -
at each iteration using the cached LU or Cholesky factors.
The above derivations lead to Algorithm
B.2.3 OBSTACLE COST
For static obstacle avoidance, follow Carvalho et al. (2023) using cost based signed distance to static obstacle. Specifically, let (ox) I R be differentiable signed distance from a robot to its closest obstacle, and then the obstacle cost term reads
Cobst (x
X
[x( (x--))((x)))
(20)
where r' > 0 also parameter.
B.2.4 HYPERPARAMETERS
Let all robots share the same radius R. We set Anaoo = 1.0 and Aptis = 0.1/7. For SHD cost, we set P 6Rt and typically 7 [0.6, 3.0]. For LB cost,
1.9R and typically 7 E [0.01, 0.06). Regarding projection, typically we set the penalty & = 10, max iteration Kmax 1000, and tolerance
3 10-6
B.3 DIVERSE ROBOT MANIPULATION EXPERIMENT
B.3.1 DIFFUSION POLICY
We adopt DIFFUSION POLICY (DP) (Chi et al. 2023) as our base algorithm, using pretrained
weights from (Feng al. 2024 IT
DP is a conditional diffusion model operating receding-
1Teee model from (Feng et al. 2024 was trained augmented dataset with broader coverage than that of (Ci al. 2023) yielding diverse and feasible trajectories.
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Algorithm 4 Batched ADMM Projection for Velocity-Constrained Trajectories
Require: Predicted trajectory batch X, starting position batch Xo, Umax 0, At 0, penalty
<> O, max iteration maxx tolerance E.
1: Pre-compute A and batched b matrices
2: Pre-compute M
R A
same for all batches br same for all batches
3: Caching inverse factors of M
4: Initialize 2 +- 0, A +-
zero tensors of shape B x x 2
5: for k <- 0 to K max 1 do
6:
X-update, Eq.
7:
= EAT
Z +
1*
A broadcasts across batches
8:
X + Sodveeinearsyyttaactn 7-updale, Eg.
v)
M broadcasts across batches
9:
18
W + AX
AR
11:
for all (3, h) E EL, B} x (I..... } in parallel do
D Vectorized operation
12:
1 + min (Umax At, || W.
W Wor
Small
avoid singularity
Wa
Dual update, Eq. (79)
14:
A-1 A +
(AX++1 z - b)
Optional: Convergence check if check convergence then
16:
17:
R + x z b
D. Primal residuals
18:
S + EAT 1+++111 Z²
b Dual residuals
19:
Tiax
R | [|p][r}
$=1.... B
if maax < then
21:
break
return X k-1
horizon manner. Conditioned on an observation O IR H, (a trajectory of Ho steps where each step is 5D state vector capturing the planar position of the robot end effector and the T block's 's cen- ter and orientation), DP generates an action segment X E R representing future end-effector positions. Only the first H, < H steps executed (in simulation for our case), after which new observation O' is obtained and the process repeats, until a total of Hmax execution steps is reached.
B.3.2 DETAILED SETUP
We compare our method with the vanilla DIFFUSION POLICY (DP) as baseline, DP with only pro- jection, and DP with only coupling. We evaluate each method on 50 uniformly random initial observations. With each method, we generate 100 pairs of full trajectories A E F. " x2 (concate- nated by the executed portion of each action segments) conditioned on every initial observation. For projection, we choose three different max velocity limits, corresponding to the 80%, 90% and 95% quantiles of the velocitie. generated 32 diffusion steps at inference, and take gradient descent step for coupling. We adopt the set- ting of prediction horizon H = 16, action horizon H2 = 8, and observation horizon H, 2 as recommended in (Chi al. 2023). The maximal action steps H is set 360.
gelniitiee"eemmertteeed by the baseline across the initial observations. We use
Remark 3. We only take the executed part of the generated action segments and concatenate them along time dimension form full trajectory A R Hm for evaluation.
B.3.3 PROJECTION DETAILS
We use the same formulation and implementation of projection as in the multi-robot experiments. The parameters used are penalty Ç 6.0, max iteration Kmax 250, and tolerance 2 x 10
ICCalculated by forward difference of positions of the robot.
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B.3.4 COUPLING COSTS
For all methods at all velocity limits we use the same cost-dependent 7 value. Concretely, for DPP and DPP-PS costs set 0.2; for LB and LB-PS costs we set 7 = 0.02. These parameters are chosen based on a coarse parameter scan and selecting one among the Pareto front of the TC-DTW and TC-DFD relations.
B.3.5 DETAILS IN EVALUATION METRICS
We use four quantitative metrics for evaluation: Dynamic Time Warping (DTW) (Berndt & Clifford
1994 Müller 2007), díscrete Fréchet distance (DFD) Alt & Godau 1995). velocity constraint
satisfaction rate (CS), and task completion score (TC) (Florence et al. 2022: Chi et al. (023). DTW
and DFD quantifies dissimilarity between two trajectories and have been widely used robotics (Bucker et al. 22023 Memmel et al., (025)) and dynamical systems learning (Rana et al. 2020
Zhang et al., 2022).
each pair of full trajectories, we report the DTW and DFD between the Two corresponding segments, and average them over number of segments within each full trajectory.
The velocity constraint satisfaction rate for each full trajectory defined the fraction of action segments respecting the constraint within the full trajectory. The task completion score measures how well the manipulation task accomplished by a full trajectory given an initial observation, where 1.0 is the best, and 0 the worst. We report all metrics by their empirical mean over all initial observations and full trajectory pairs.
B.4 CONSTRAINED COUPLED IMAGE PAIR GENERATION EXPERIMENT
B.4.1 TEXT PROMPTS FOR EXEMPLAR GENERATION
exemplar images this toy example were generated using ChatGPT (OpenAl 2023). We first employ generic prompt:
"Generate simple, clean portrait of male/female in his/her early 30s, wearing a plain T-shirt, against a white studio background with soft lighting. The portrait should be a centered headshot, similar to passport photo."
to generate the seed exemplar. We then provide the seed exemplar alongside follow-up prompt instructing ChatGPT to regenerate the exact same image, but depicting the subject at an older age, or with different clothing.
B.4.2 TEXT PROMPT FOR STABLE DIFFUSION AND BASELINES
As shown in both Figure 4e and Table 3 we also run additional comparisons against SD-2.1 with coupling, SyncDiffusion and ControlNet-XS. all with the use of generic text prompt. For all text- prompted runs, we use
"High-resolution passport photo of person, facing forward with a neutral ex- pression. Wearing a plain white t-shirt, with a clean white background and even, soft lighting. The composition is centered and symmetrical, with the head at the center of the frame."
We set classifier-free guidance scale s=25 for all SD-2.1 variants, and use the authors' recommended defaults for external baselines, that is: s=7.5 for SyncDiffusion, and s-9.5 for ControlNet-XS. We also set negative prompts to null strings.
B.4.3 DETAILED SETUP FOR SYNCDIFFUSION
We choose SyncDiffusion (Lee et al. 2023 which was originally designed for panorama genera- tion with an LPIPS- based style loss to promote global synchronization and coherence, as a reference method to compare performance on promoting age-group contrast. Since SyncDiffusion allows for an arbitrary style loss, we replace the LPIPS loss with our coupling loss eXoR (x, y), parameter- ized by image classifier trained FFHQ-Aging (Or-El al. 2020). We set the latent stride
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to Slatent=64 to match SD-2.1's latent spatial dimensions, which disables any form of latent over- lapping during sampling and essentially reduces the setup to paired image generation. We employ 100 DDIM (Song et al. (021a) steps and synchronization weight Wyyn= with exponen- tial decay 0.999. We perform one synchronization step for all 100 diffusion steps. All other hyperparameters follow the defaults by Lee al. (2023) unless stated otherwise.
B.4.4 DETAILED SETUP FOR CONTROLNET-XS
ControlNet-XX XS Zavadski et al. 2024 is used for reference to compare performance in facial at- tribute preservation. We use the official ControlNet-XS implementation, We adopt the SD-2.1 Canny Edge 14M model weights provided by the authors, rather than the SD-2.1 Depth Map 14M variant, as Canny edges preserve higher-frequency facial details relevant to our attributes. The Canny thresh- olds are set hhgh 250 and flow 100 per the recommended values by Zavadski et al. (2024). We use a ControlNet control weight of Wctm==099 and set DDIM sampling steps to 100.
In this setup, note that ControlNet-XSS serves soft-constraint baseline: it encourages attribute alignment via conditioning but does not enforce feasibility, in contrast to our projection- baaed hard- constraint method.
Condition aggregation across exemplars. ControlNet-XS accepts single conditioning image per generation. To obtain single representative condition from exemplar set, compute the arithmetic mean of the exemplars in SD-2.1's latent space and decode it back to image space for conditioning. Because the exemplars are structurally and spatially aligned (see discussion in Ap- pendix B.4.7). this linear averaging preserves shared spatial attributes. Geometrically, the latent average lies near the centroid of the convex hull spanned by the exemplars, providing a neutral, exemplar-consistent condition. Conditioning images derived from M E {2, 6} exemplars per model are visualized in Figures 6b-
(a) Conditioning images for M-2 exemplars
(b) Conditioning images for M-6 exemplars
Figure 6: Conditioning images used for ControlNet-XS. (a) Conditioning images formed by taking
linear average in latent space for M-2 exemplar images from Figure 4a (b) Likewise, but for
M- exemplar images from Figure 25
B.4.5 PROJECTION VIA MIRROR DESCENT
Given two exemplar sets of sizes Mp and My, encode them via VAE encoders and obtain the latents (e) = [ -
M, E RdxM,, where is the flattened latent dimension. Define Xxx = ( A AM, the constraint set for X, where A Mr is an M, Ma-simplex, and define Ky likewise. At each diffusion step I, we project the current latent X onto xx by solving the simplex-constre constraindd problem:
.........
x(e) M, E Rs and (()) =\ (
Ax, arg min (|x( x
(21)
ΛεAx.
via Mirror Descent (MD) using the negative-entropy mirror map. which exponentiated-gradientt updates that remain AM, by construction (Nemirovsky & Yudin
yields
1983 Beck & Teboulle 2003). For the MD updates, define
Gx == x ((e) E M.X.
(22a)
bx, = x X, R.M
(22b)
and let fx, (|x(0) - 1. Its gradient is
fx,((A) = 2 bx,))
(23)
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Figure 7: Generated samples obtained by projecting intermediate DDPM latents (Eq. (29 onto the exemplar convex hulls at every step. Top: projection using the exemplar pair from the top row of Figure Aaj bottom: using the bottom-row pair. Within each row, the samples collapse to narrow mode, illustrating significantly reduced diversity induced by per-step projection onto fixed exemplar set.
Starting from (0) 1 1. each MD step performs M,
log X (k+1)
log X (k)
Xxx V/X,,(Xx..) (8)
(24a)
A X,t
(k+1) softmax( log Xx.x ((:t1)),,
(24b)
with learning rate v) > 0, which is equivalent to
(k+1) (k) Xx,1 Xx, exp exp( Vfx. (1
(K)
(25)
followed by normalization. After Kma steps, obtain Xx., and compute the final projected latent .e Y, is computed likewise. We run MD for Kmax 10, 000 steps and set its learning
X, (=) X rate 17 10 5 for convergence.
Mode Collapse. We follow the DDPM formulation of Ho et al. (2020). Let {8r)7=1 C (0, be a predefined noise-variance schedule for the forward process (( 2t-1) = =1-1)-[[i - Bg 2-1, BeI). Define
O+ = B1,
(26)
āj II axs
(cumulative product),
(27)
8=1
1 ūt- Br (posterior variance).
B =
(28)
āi
At inference, the standard DDPM sampling update with a learned noise predictor c0 t) is
21 VI-ai ae colee.1) + V F.c, Vau
1
2t-1
(~ 1, else
(29)
Recall Exx (x(e)): E AM, } (and Ky analogously for Y). When sampling with Eq.
29
observe that projecting the latents at every step ie., X, Txx (X,) and Y, myy (Y, leads to pronounced mode collapse in the generated samples as shown in Figure 7
predicted by the model timestep t can be derived from the model's 's noise prediction ca( t) using Tweedie's formula:
As shown in (Song et al.) 2021b Luo 2022), the (Stein)
1
so (2z, 1) = Vz, log
VI aj c0(21, 1).
(30)
In Figure analyze the score field along the 1-D subspace spanned by two exemplar latents from the second row of Figure 44a Specifically, we project the score V., log Pt (2x) onto the line segment (the convex hull of two joints the exemplars the direction preserved by the projection
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operator when there exactly two exemplars- - and visualize its signed magnitude. Because latent interpolatability in our setting relies on structurally and spatially aligned exemplars (see discussion on Latent Space Interpolatability in later sections), projection induces narrow feasible corridor. With projection enabled, this projected score points almost exclusively toward the same endpoint across timesteps, yielding nearly deterministic path toward the final projected latent Ep. This concentration substantially reduces sample diversity for the given exemplar set.
While this concentrates samples within a narrower neighborhood of the exemplars, it also delivers strong controllability and structural fidelity - properties that are particularly valuable for tasks such as image editing, personalization, and srrrbute-preserving transformations. We view leveraging this cociision-fiellit regime as a promising direction for such application-oriented oriented extensions.
Following Nichol & Dhariwal (2021), which highlights the impact of reverse-process variance on sampling, we additionally adopt a DDIM-style stochasticity control and scale the noise term by factor of & during sampling to increase output diversity. The updated DDPM sampling update is hence
1 (-==-=1-~~-
ea(z)1)) 1--V-=eelaan
3: VI VI-aa at
+ +. Bike,
6 e~ c~ if 1, else 0. (31)
-t-1
Vou
where k E R>1 scales the stochastic term (k- = recovers standard DDPM; k > increases the noise standard deviation by k. variance by k²)) To set k. we compute at each diffusion step the average ratio r4 = V log [ log Pe || between the model score magnitude and the magnitude of its projected component (see Figure 9.. We aggregate r, over timesteps and exemplar sets and choose k near this summary; empirically, k 20 provides good diversity- stability trade-off for experiment runs involving projection. See Appendix Cor ablation on k.
B.4.6 COUPLING GUIDANCE STRENGTH
We observe that projection dampens the gradients supplied by the coupling loss, which calls for the need of y to be scaled well beyond the values typical for classifier guidance (Dhariwal & Nichol, 202T] in order to take effect. We find that empirically, recognizable young- old contrast appears only when y 2 200. This is consistent with our earlier finding that the score component aligned with the line segment connecting the exemplar latents is roughly 13-20x weaker than the full score (as illustrated in Figure (9) the projection operator only preserves the component of gradients provided by the coupling loss that is parallel to the line segment connecting the two exemplar sets. Consequently, simple estimate of the effecttve guidance strength gives
Teff ///r
(32)
which means compensating for rr 20 reduction requires 7 increase by roughly 50x. We therefore set = 450 (yielding Toff Rv 9.0) for all experiment runs involving both coupling and projection if not otherwise specified. See Appendix Cor an ablation on 7-
B.4.7 LATENT SPACE INTERPOLATABILITY
VAEs are trained to encourage smooth, approximately Euclidean latent space by regularizing pos- teriors toward a standard Gaussian prior, making nearby latent codes decode to similar images and thereby support interpolation (Kingma & Welling 2014 Rezende al. 2014). However, find empirically that linear interpolation is reliable only when exemplars are both structurally and spa- tially aligned. Concretely, if spatial layouts differ, straight-line paths (and thus convex-hull pro- jections) tend to leave the data manifold and decode implausibly. See examples of interpolation between latents of samples from FFHQ-Aging Dataset (Or-El al., (2020) in Figure IO This ob- servation accords with geometric analyses showing that semantically consistent transitions follow curved geodesies under the decoder-induced Riemannian metric, rather than straight lines in Fu- clidean latent coordinates (Arvanitidis et al. 2018). Hence, we also experimented with spherical linear interpolations (SLERP) (Shoemake 1985) to maintain constant-norn paths under an isotropic Gaussian prior. In practice, SLERP still required close structural alignment of exemplars, and it only interpolates between two endpoints, whereas our convex- hull projection must accommodate multi- ple exemplars. Moreover, adopting SLERP consistently would call for a manifold -wwre projection, which is nontrivial and beyond the scope of this work. Consequently, we use simple linear interpo- lation but restrict convex sets to closely related exemplars to preserve visual coherence.
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Figure 8: Signed component of the model score V. 2, log projected onto the line segment necting two exemplar latents (the convex hull of two exemplars). The x-axis represents interpolation between the two exemplar latents (from left to right), and the y-axis denotes the diffusion timestep Color indicates the direction and strength of the projected score: blue values push towards the right exemplar, while red values push towards the left. This visualization, based on two exemplars from row 2 of Figure 4a, reveals that the projected score components consistently point towards one side (left), creating narrow "white" transition band that funnels every sample the left exemplar hence a nearly deterministic path and little diversity. Such behavior is observed consistently in all exemplar sets experimented and likely stems from the exemplars being relatively similar, which induces a narrow feasible region for the diffused latents. Empirically, the exemplars cannot differ too much in spatial structure. When exemplars differ significantly, interpolations between them of- ten fail to represent coherent or meaningful images (see discussion on Latent Space Interpolatability later sections).
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Mean r. over diffusion timestep t (Male Exemplars)
1200
meen h
1000
mean owtthout outieess
28.4573
800
global mean outilers) 13:1855
mean
600
400
200
0
1000
800
600
400
200
0
timestep
Mean & over diffusion timestep (Female Exemplars)
1200
1000
-*-- mcan r, outthout outters)
global mean 22.7283
m
800
global (without outliers) 12.5928
600
400
200
1000
800
600
400
200
D
diffusion timestop
Figure 9: Mean ratio r;- v,, lo (2+) [[ log ( over diffusion timesteps t, where
IT.)) projects onto the convex hull formed by exemplar latents. Top: computed with male exemplar pair (top row of Fig. 4aj: bottom: female exemplar pair (bottom row). Large outlier r; values indicate the predicted score is nearly orthogonal to the chord connecting the two exemplar latents. Outliers are defined as r μ + 1.5a, where y is the global mean and a the global standard deviation across t.
Figure 10: Linear latent interpolations between pairs of FFHQ-Aging Dataset images Or-El et al. 2020). Rows 1-3 use exemplars that differ in pose or spatial layout; midway latents leave the data manifold and decode to implausible faces. Rows 4-6 structurally aligned exemplars; the entire interpolation produces coherent and plausible images. The contrast illustrates that latent interpola- tions reliable only for closely aligned exemplars.
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Method Pair Metric
JS
TV
1.2
(PCD, CG)
3.91 X 10
5.40 x 10
1.38 x
10
(PCD, PD)
5.23 x 10
7.97
10 10-3
1.44
10
(Baseline, Baseline)
4.48 x 10-6
5.58
10
1.32
10
Table 4: Distributional metrics results for numerical verification of PCD special cases.
C ADDITIONAL RESULTS
C.I NUMERICAL VERIFICATION OF PCD SPECIAL CASES
We numerically verify that PCD recovers into two known methods, classifier guidance (CG) and projected diffusion (PD), under specific circumstances as stated in Section
Setup We experiment on the ID toy example demonstrated in Section Figure We run PCD two degenerated versions that correspond to CG and PD, respectively, as well separately im- plemented CG and PD, and then compare their empirical sample distributions over the X variable. Specifically, we discretize the range of X into 200 bins and compare the resulting histograms with sample size of 10t. For the degenerated PCD corresponding CG, fix the Y variable in PCD to the center of the corridor to match the conditioning used in CG, and adopt the same coupling or guidance strength for PCD and CG. For the degenerated PCD corresponding PD, we set the pling strength PCD as = 0. Additionally, we include a "baseline" where we draw two groups of samples from the same standard normal distribution, each with different random seed, in order to provide a numerical reference in the reported metrics to demonstrate the effects of stochasticity with the chosen sample size.
Metrics To quantify similarity between distributions, we report three common metrics: Jensen- Shannon divergence (JS), total variation distance (TV), and Lg distance (L22).
The resulting distributional discrepancies are quantified and reported in Table These divergences show that the discrepancies in sample distributions between the degenerated PCD and the corresponding special cases (CG or PD) lie within the same order of magnitude as the baseline case, wherein two groups of samples are indeed drawn from the same distribution. Visualization of the empirical distributions is in Figure [T] These results numerically verify that PCD recovers the behaviors of CG and PD in degenerated scenarios, and thus can be regarded as a generalization.
Results
C.2 RUNTIME AND MEMORY
PCD is approximately 7x slower than vanilla diffusion mainly due to the per-step projection operation. Memory overhead compared to vanilla diffusion is negligible.
Image Pair Generation Table shows the per-diffusion step runtime of each operation in sec- onds, computed across 100 diffusion steps, for the Constrained Coupled Image Pair Generation experiment, using M {2, 6) exemplars from Figure 44l and Figure 25 respecccv. In terms of no- ticeable time differences, all runs involving projection in the M-6 exemplars setup is only s 0.15 seconds slower than the M=2 exemplars setup. This small increase in runtime due our pro- jection operator being implemented as the Mirror Descent algorithm (Nemirovsky & Yudin 1983) Beck & Teboulle 2003) which scales effortlessly on GPUs; see Section [B.tion|B.4]] for details. Across both setups, we can observe that: (i) diffusion model forward passes and other miscellaneous overheads have nearly constant runtime across runs; (ii) coupling adds ~ 0.5 second per diffusion step; (iii) projection adds 3 seconds, with almost negligible dependence M. Relative to vanilla SD, per- step runtime increases by 1.5x with coupling alone, ~ 4x with projection alone, and ~ 4.5x with both.
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Histograms of Discretized Distributions
PCD-CG (ours)
Clossifier Guidance
0.220
0.0200
ap0n
0.050
0.0125
Recsvency
0.0125
0.075
0.007%
00025
0.0025
0.0
0.6
0.8
0.7
1.0
PCD vs. CG
Histograms of Discretized Distributions
PCD-Projested lours)
Projected LMC
600
0.000
0.000
3,000
0.000
0.6
0.8
0.2
1.0
(b) PCD vs. PD
Figure 11: Empirical distributions of samples yielded by degenerated PCD compared to classifier guidance (CG) and projected diffusion (PD).
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METHOD
Model
Coupling
Projection
Misc
Total
SD
0.929 10.005
0.003 _.00__.003 0.003
0.932 10.007
SD+C
0.929 10.005
0.50510.009
0.002 0.002_o.002 0.002
1.436 10.011
SD+P
0.919=0.002
2.969
0.002=0.00
3.891 +
PCD
0.919=0.003
0.498=0.010
2.984+0.030
0.002=0.00 002
00.05
(a) M exemplars.
METHOD
Model
Coupling
Projection
Misc
Total
SD
10.00
0.002_o.003
0.934 10.007
SD+C
0.929
0.507
0.002 0.002
1.438 10.012
SD+P
0.922=0.002
3.04220.030
0.003_0.000
3.967 #0.030
PCD
0.921
0.503
3.013+0.011
0.002_0.002
4.440 40.022
(b) M- exemplars.
Table 5: Per-diffusion step runtime (in seconds; mean + std over 100 diffusion steps) for each operation of the Constrained Coupled Image Pair Generation experiment. Total the per-step sum. Components: Model = SD-2.1 UNet inference; Coupling = coupling step; Projection projection step; Mise other supporting ops (e.g.. Scheduler. step ()) Results are from a single run generating 25 samples with M E {2, 6} exemplars from Figure 4a and Figure 25 respectively. A dash indicates the component is not applied.
Thsl: Empty
Task: Highways
Task: Conveyor
Task: DropRegion
50
40
40-
Santimm per Exriton mSC
Surtime ger Seteatre
30
Projett
Pris Couple
Coups
20
Couple
20
Couple
Mic
10
Number Agents
Number Agents
Numberr
Number Agents
Runtime profile per iteration (in milliseconds) for each task scaling with the number of robots in the multi-robot experiment. The inference batch size for each robot is set to 32 and the total diffusion time step configurations for each task and the standard deviations are plotted with error bars.
Figure 12:
25 for profiling. Results are averaged over 50 random initial
Multi- Navigation Figure exhibbis the runtime profiles of PCD running across different tasks and scaling with number of robots. Similar to image generation, the runtime fraction of the diffusion model's forward pass and associated overheads remains nearly constant despite the number of robots due to GPU parallelization, provided the total workload does not exceed the GPU's com- putational throughput. Runtime fraction of coupling approximately equals that of diffusion model forward pass but slightly varies across numbers of robots and tasks. The 3DMM-based projection operation takes up roughly 67% to 75% of the total runtime and scales up approximately linearly with the numbers of robots (which conforms to theoretical complexity). Note that due to the batched convergence check described in Algorithm/4 the reported projection runtime effectively the worst case within each batch per iteration. Implementing more efficient projectors such as adopting adap- tive augmented Lagrangian penalty (Boyd et al. 2011) may help accelcrate convergence.
C.3 MULTI-ROBOT NAVIGATION
C.3.1
ADDITIONAL QUALITATIVE RESULTS
16 exhibit more trajectory samples generated by the compared methods on all four tasks with N and N 4 robots. These results demonstrate the effectiveness of PCD in generating highly correlated trajectories and simultancously enforcing hard constraints compared to methods with the absence of either coupling projection, both.
Figures B
14
15
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(a) MMD-CBS (N 2) (b) DITUSER (N 2)
cc) (D
(d) CD (N 2)
PCD(Ours) (N 2)
MMD-CBS (N 4) (g)
DIPUSER (N 4)
(h) PD (N
(N
PCD (Ours)
O
(k) MMD-CBS (N- 4) ()) DIFFUSER 4) (m) (N 4)
CD
PCD(Ours) (N 4)
Figure 13: Robot trajectories in environment Hi ghways generated by the compared methods with N 2 and N robots running. Red crosses mark collisions and blue stars mark velocity constraint violations. Each row corresponds to one initial configuration (start and goal positions for each robot).
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(a) AMDDCCS(NN 2) (h) DIFFUSER
(c) (D
(d) CD (N
(c) PCD(Ours) (N 2)
(N 4) (g) DIFFUSER (N 4)
(h) (D
CD 4)
i PCD(Ours) (N 4)
(k) MMD-CBS (N 4) (I) DIRFUSER (N = 4) (m) PD (N 4) (n) CD (N = 4) (0) PCD(Ours) (N 4)
Figure 14: Robot trajectories in environment Empty generated by the compared methods with N 2 and robots running. Red crosses mark collisions and blue stars mark velocity
constraint violations. Each row corresponds to one initial configuration.
C.3.2 ADDITIONAL QUANTITATIVE RESULTS
Tables g and 9 summarize quantitative evaluation on all of the four environments with N robots. each subject to three different maximum velocity constraints. Results on all environ- robots are in Table TO constraint-agnostic methods (vanilla DIFFUSER, MMD-CBS, and all coupling-only CD- variants) achieve similar rates: as low 8-22% Empty and between around 28% and 62% in H ghways. In contrast, every projection-bast based variant (DIFFUSER with projection and our PCD-LB/SHH en- forces the constraint in all cases, confirming the effectiveness projection. Inter-agent safety scores show the similar trend. Because MMD- CBS repeatedly samples and then stitches together opti- mal trajectory (see Remark 4. it unsurprisingly attains perfect score. Among the one-shot methods, PCD- and CD- methods can almost match this upper bound, while vanilla DIFFUSER and and its projected variant performs much worse. Slightly degraded data adherence performance is again function gets affected more due its steeper gradients by
ments with N
12 and [3 In terms of constraint satisfaction,
observed in our method: the LB design. Similar trends can be observed in the other two environments. Overall, the results show that PCD effectively promotes inter-robot collision avoidance through appropriate coupling costs, while enforcing hard test-time velocity constraints. A tradeoff exists between coupling strength and data adherence, depending on the coupling cost.
Remark 4 (Comparison to MMD-CBS). direct comparison between MMD-CBS (Shaoul al., 2023) and our method is not straightforward due to fundamental differences in their sampling proce- dures. MMD-CBS a search-based approach that generates a batch of trajectories using diffusion models, selects the best one based on cost function, adopts only a partial segment of the selected trajectory to resolve collisions, and repeats this process iteratively. As a result, it produces only one trajectory per robot per forward pass, making it inefficient for generating multiple i..dd samples. In contrast, DIFFUSER and our method produce full batch i.i.d. trajectories a single pass, which preserves the benefit of massive parallel sampling from generative models. Thus, directly comparing
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(a) MMD-CBS (N 2) (h) DIFFUSER (N 2)
(c) (D
(d) CD (N
(c) PCD(Ours) (N 2)
(
(N 4)
(N
(h) PD 4)
(D
(Ours)(N
(k) MMD CBS (N 4) ()) DIRFUSER (N = 4) (m) PD (N 4) (n) CD 4) (0) PCD(Ours) (N 4)
MMD-CBS
Figure 15: Robot trajectories in environment Conveyor generated by the compared methods with robots running. Red crosses mark collisions and blue stars mark velocity constraint violations. Each row corresponds to one initial configuration.
N 2 and
1 128-s batch from DIFFUSER or ours against a single best output from MMD-CBS, or vice versa, would not be very meaningful.
C.3.3 ABLATION STUDY
We perform ablation study on the coupling strength 7 for both tasks Empty and Hi ghways with the velocity limits reported Table [ but with both and robots. Results are presented in Figure and Figure 18 In task Empty, as the coupling strength increases, SU and RS increases and saturated near .OO, with an exception in N = where RS drops bit when y too high: CS remains at 100% by projection, and DA in general gradually drops. Results of H Highwar ghways demonstrates the similar trends, with the difference where SU drops significantly for the LB cost when y is large. This is because SU also takes into accounts collisions with static obstacles. When 7 is high, the gradient of the robot-collision cost overwhelms that of the obstacle-avoidance term resulting the robots bumping into obstacles. This is supported by Figure 19 in which we report the decreasing obstacle safe rate, suggesting more trajectories are hitting static obstacles as ry increases. A potential remedy to leverage projection also enforce static obstacle avoidance as Christopher al.) (2024), but that would introduce non-convexity into the projection process and might cause convergence issues.
C.4 CONSTRAINED DIVERSE ROBOTIC MANIPULATION
C.4.1 QUALITATIVE RESULTS
We present in Figure Sheer contrast between our PCD method and others highlights the efficacy of our framework in jointly generating correlated samples while enforcing hard constraints.
comparative results the trajectories by all compared methods.
The coefficient for the gradient of the obstacle-avoidance cost is fixed in our experiments. See Ap- pendix B.2 for details.
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(a) MMD-CB5 (N 2) (b) DIFFUSER (N 2)
(c) (D 2)
(d) CD (N
PCD(Ours) (N 2)
MMD-CBS (N 4) (g) DIFFUSER (N 4)
(h) PD (N 4)
PCD (Ours)(N 4)
(k) MMD CBD (N 4) (I) DIPFUSER (N = 4) (m) PD (N 4)
(n) CD = 4) (0) PCD(Ours) (N 4)
Figure/ 16: Robot trajectories in environment DropRegi on generated by the compared methods with N and N robots running. Red crosses mark collisions and blue stars mark velocity constraint violations. Each row corresponds to one initial configuration.
Sacress Rate (S0)
InterRobot Sefety Scoee (RS) oelctt Poodiin.AM
Viitrity Condreant Salisfaction Wecckr Poediir AJNH
Data Adherence (DA)
Welecty Pooediin
Warcy Pojedier
3.1
Success Rate
0.8
1 0.5
04
044
0.7
Pig
5.7
Nici only
Saerrre
0.2
0.
Pig.
10
10"0
107
12-
10
10
16
10
10-1
10
16
10-2
10
10
(a) SU (N 2)
(b) RS (N 2)
CS) (N = 2)
(d) DA = 2)
I5U)
Safety
Velicty Safistaction Projettion AZMH
selocity A
ecticn 4M
Velerity 253; Pro eeteecion ADMM
weicely n
Mox Weerlyy Projection ATMM
2.2
3.0
2.5
Proj,
0.5
540
0.5
15 0.5
Saseline
Pesj
54
0.4
52
10
16
10
(c) SU (N 4)
(f) R$ 1 (N-
g CS (N (N=4) 4)
(h) DA 4)
Figure 17: Coupling Strength Ablation of PCD on task Empty with velocity limit Umax 0.703.
(a,b,c.d) N = 2 robots; (e.f.g.h) N robots.
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Task Empty, 2 Robots
METHOD
Metric
SU(%))
RSt
CS(9))
DA
Max Vel. = 0.703
vanilla DIFFUSER
92+27
0.92+0.27
(59 654
(0.99
1.0
MMD-CBS
10010
1.0
(8.0 12
(0.9910.073; 1.0.0.0)
DIFFUSER + projection
92
0.92
(100 100
(0.9910.085.
1.0
CD-LB (w/o proj.)
100+0
1.040.065
(7.3 5.5
(0.92+0.2:
0.
CD-SHD (w/o proj.)
100+0
1.040.0
(45+50; 52450)
(0.9940.09.
1.040.0002
PCD-LB
95
0.95
(100 100
(0.98
1.0
PCD-SHD
100+0
1.040.061
(100 100=a)
40.99 1.0=0.0092)
Max Vel. = 0.692
vanilla DIFFUSER
92+27
0.92+0.27
(40+ 44450)
(0.99
1.0+0.0)
MMD-CBS
100
1.00
(8.0 12 12132)
(0.9910.073: 1.0
DIFFUSER + projection
93126
0.92
(100 100
(0.9910.085.. 1.0
CD-LB (w/o proj.)
100+0
1. 1.040.065
(6.9+25: 4.6+21)
(0.92+0.2: 0.93
CD-SHD (w/o proj.)
100+0
1.040.0
(31+46:374 37448)
(0.99+0.09. 1.0+0.0002)
PCD-LB PCD-SHD
95
0.95
(100 100
(0.98 1.0 1.0
100+0
14.0065
(100 100
(0.98 1.0.0.012)
Max Vel. 0.675
vanilla DIFFUSER
92+27
0.
(19+ 22
(0.99 1.0
MMD-CBS
100g0
1.0
(8.0 12
(0.9949.073: 1.0
DIFFUSER projection
93
0.92
(100 100
(0.9910.085.. 1.0
CD-LB (w/o proj.)
100+0
1+0.065 06.066
2..45=25.3.6-11
(0.92+0.2. 0.93
CD-SHD (w/o proj.)
100+0
1.040.0
(18.4439, 21.3++1)
(0.99 1.0
PCD-LB
95
10.22
(100 100
(0.92 0.93
PCD-SHD
100
1.0
(100 100
(0.99 1.010.012)
Table 6: Task Empty, 2 robots, 100 random tests, sample size 128 except MMD-CBS.
C.4.2 QUANTITATIVE RESULTS AT DIFFERENT VELOCITY LIMITS
Table 14 summarizes quantitative results of all compared methods with all three velocity limits. We also report standard deviations in the tables.
C.4.3 ABLATION STUDY COUPLING STRENGTHI
We perform ablation study investigating how the coupling strength 7 affects the performance of our PCD method. Figure 21 shows the trends of evaluation metrics' (DTW, DFD, CS, TC) change as the coupling strength parameter y increase, with all three velocity limits we experimented. Above all, the general trends look expected and similar across the three velocity limits: as we increase y, the DTW and DFD monotonically increase, velocity constraint satisfaction (CS) is maintained perfect by design, and task completion score (TC) drops monotonically. This suggests that (i) the projection can guarantee velocity constraint regardless of coupling strength; (ii) as increases, the correlations between the variables get stronger, with the cost gradients gradually overwhelming the learned score and thus deviating from the original data distribution. A tradeoff exists between data adherence and correlation strength.
similar ablation also conducted on coupling-only method (CD) by removing the projection; results are in Figure 22 The same tradeoff between correlations and data adherence also exists. Unsurprisingl the velocity constraint satisfaction rates drop as increases, and the LB cost function more sensitive.
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Task Highways, Robots
METHOD \ Metric
SU(%))
RSt
CS(%))
DA
Max Vel. 0.878
vanilla DIFFUSER
23
+0.39
(72+15. 76=13)
(0.99=0.11, 0.98
MMD-CBS
100.0
1.0
(73 79 44)
(0.9910.099; 1.917
DIFFUSER + projection
92
0.81
(100 100
CD-LB (w/o proj.)
100+0
1.0.0.033
(22+42: 18
0.99=...1,0.98_.0.1
CD-SHD (w/o proj.)
100g0
1.040.020
(65 66
PCD-LB
100
0.99
(100 1004.)
(0.98 0.97
PCD-SHD
100
1.020.012
(100 100
(0.98=o.11.0..)) 0.97
Max Vel. = 0.781
vanilla DIFFUSER
93+26
0..3
(63+48. 65
MMD-CBS
10010
1.0
(67 67 147
(0.9910.099, 0.97 10.17
DIFFUSER + projection
95_22
0.81 10.39
(100 100
CD-LB (w/o proj.)
100.0
1.0.0.033
(14435- 12=32)
(0.99=0.11, 0.98
CD-SHD (w/o proj.)
100
1.0.0.0
(55 53
(0.99=0.11, 0.98
PCD-LB
100.0
0.99
(100 100
(0.98 0.96....9))
PCD-SHD
100
1.0
(100 100
(0.98_0.14, 0.97
Max Vel. = 0.647
vanilla DIFFUSER
93
4..39
(52+50, 50
(0.99 0.98
MMD-CBS
100.0
1.0
(62.49, 55)
(0.9940.099.
0.97
DIFFUSER + projection
91 91
0.84
(100 100
(0.98 0.98
CD-LB (w/o proj.)
100
0.99
229415:22=111
(0.99=0.11, 0.98
CD-SHD (w/o proj.)
100
1.0+0.029
(49+50, 46=50)
1.00==..2,..880....
PCD-LB PCD-SHD
19.9.
0.98
(100 100
(0.98 0.16
100
1.0
(100 1001o)
(0.98 0.97
Table 7: Task Highways, 2 robots, 100 random tests, sample size 128 except MMD-CBS.
C.5 ABLATION STUDY ON SAMPLING STEPS
We perform ablation study to explore the effect of the number of sample steps on the PushT task.
Setup We use pretrained DDPM model that was originally trained with T 100 de- noising diffusion steps[ and vary the number of denoising steps during sampling with T e (15,20, 20, 30, 40, 50, 75, 100} while keeping the same noise schedule. We set the velocity limit as Umax 8.4, and we use the same coupling strengths those for obtaining the results reported in T For each method, we run it on 10 uniformly random initial conditions and generate pairs of full trajectories. We report the same four metrics in Section 5.2 DTW, DFD, CS, and TC.
The performance against number of sampling steps curves are shown in Figure 23 For the baseline Diffusion Policy, diversity metrics (DTWW and DFD) appear to improve as 7 decreases. However, this trend is mainly artifact: when T < 40, the trajectory sample quality deteriorates significantly (e.g., with jerky and teleporting motions), which is also reflected in degraded task com- pletion score (TC) and rapidly dropping velocity constraint satisfaction rates (CS). Such low-quality samples often fail the manipulation task, making comparisons for T < 40 unreliable. Therefore, we shall focus on meaningful comparisons only for T 40. Surprisingly, for all methods the TC scores do not peak at the highest number of sampling steps, even for the baseline. When the number of sampling steps > 40, the improvements in diversity performance over the baseline grow steadily as T increases. This pattern can be partly explained by the coupling effect benefiting from more
Results
"This is also a pretrained model from Feng et al. 2024), but is not the one used in our main experiment. We use this model only for this ablation.
38

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Published as a conference paper at ICLR 2026
Task Conveyor, Robots
METHOD
Metric
SU(%))
RST
CS(%))
DA
Max Vel. 1.21
vanilla DIFFUSER
14435
0.64
(0.194413, 0.45+6.7)
(0.95
0.97+
MMD-CBS
861:35
1.0.0.0
(0.0 1.0
(1.010.0: 0.14
DIFFUSER projection
104:30
0.684
(100 100+0)
(0.71 0.70
CD-LB (w/o proj.)
100+0
0.96
(0.03942.0-
0.42_6..)
20.99
0.98
CD-SHD (w/o proj.)
10040
1.0.0.0
(0.007840..8. 0.2
(0.99 0.99
PCD-LB
10040
0.94
(100 1000000000000
20.85
0.85
PCD-SHD
100+0
0.99
(100 100
20.90 0.91
Max Vel. = 1.46
vanilla DIFFUSER
14435
0.64
(2.4+15, 1.7+13)
(0.95 0.97
MMD-CBS
861.35
1.0.0.0
(4.0, 5.0
(1.0.0.0: 0.98 _..4_)..4)
DIFFUSER projection
14.:35
0.67.
(100 100
(0.87 10.33, 0..31...11
CD-LB (w/o proj.)
100+0
0.9640.20
(0.1423.7, 0.75+8.6)
(0.99
0.98
CD-SHD (w/o proj.)
10040
1.0g0.0
(0.8048.9. 0.82
(0.99 0.99
PCD-LB
10040
0.95
(100 1004o)
(0.95 L..24)
PCD-SHD
10040
0.992
(100 100+o)
(0.98 0.98
Max Vel. = 1.76
vanilla DIFFUSER
14435
0.64
(36 334
(0.95 0.97+a.1s)
MMD-CBS
864335
1.040.0
(48 50
(1.0+0.0: 0.98_..14)
DIFFUSER projection
17138
0.65
(100 100
(0.94 0.96
CD-LB (w/o proj.)
100+0
0.96
(5.4+23, 4.4+21)
(0.99 0.98+0.13)
CD-SHD (w/o proj.)
100+0
1.0g0.0
(22+ 20
(0.99 0.99
PCD-LB
100
0.96
(100 100
(0.98 0..7
PCD-SHD
10040
1.0
(100 1004o)
10.9910.093, 0.9991.10)
Table 8: Task Conveyor, 2 robots, 100 random tests, sample size 128 except MMD-CBS.
accurate Tweedie's estimates used in the PS cost variants at later denoising steps, as a higher total number of sampling steps leads to greater number of steps with more accurate Tweedie's estimates.
C.6 CONSTRAINED COUPLED IMAGE PAIR GENERATION
C.6.1 ADDITIONAL QUALITATIVE RESULTS
Figure 24 presents additional pairs generated by PCD when two exemplar images (Figure 4h define the convex hull. We repeat the experiment with six exemplars per model (Figure the corre- sponding samples are shown in Figure 26
C.6.2 ADDITIONAL QUANTITATIVE RESULTS
Table 15 reports the full set of metrics including the standard deviations for the samples generated using two exemplars per model, while Table [6] for the samples generated with six exemplars per model. Quantitatively. both setups show the same trend throughout, that is: Projection- based meth- ods (SD+P, PCD) achieve 100% gender satisfaction (M/F) and the strongest alignment to exemplars (higher SE-CLIP, lower SE-LPIPS and SE-FID) compared to vanilla SD, coupling-onf) variants and other baselines. Projection reduces diversity (low IS-LPIPS); adding coupling partially recovers diversity relative to projection alone. Coupling-basee methods improve the age-group XOR satis- faction, with PCD attaining the highest rate (96% for the setup using two exemplars and 76% for the setup using six exemplars). In both experiments, SyncDiffusion matches SD+C XOR (48%) but shows slightly reduced sample variation (lower IS- IS-LPIPS). We also observe that even with the use of generic text prompt outlined in Appendix B.4 ControlNet-XS exhibits male-bias, consistently
39

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Published as a conference paper at ICLR 2026
Task DropRegion, Robots
METHOD
Metric
SU(%)
RS1
CS(%))
DAT
Max Vel. = 0.928
vanilla DIFFUSER
60
0.61
(11432:9.544)
(0.8940.32.
0.88+0.32
MMD-CBS
100
1.0
(22 2-1)
(1.0 1.0_
DIFFUSER + projection
63
0.6710.47
(10040; 100
(0.92 0.92...27
CD-LB (w/o proj.)
100=0
(9.6430: 0. 9429)
(0.59
0.60+0.49)
CD-SHD (w/o proj.)
100_0
1.040.07
(7.2+20. 5.7
(0.9610.19, 0.9640.
PCD-LB
100_
0.91
1000:0,100+00
(0.86+a.35. 0..74
PCD-SHD
100=0
1.040.044
(100+0,100g) 100
(0.97 0.96+u.18)
Max Vel. = 1.13
vanilla DIFFUSER
60.-19
0.61
(37+18.35 35+48)
(0.8940.32. 0.88+0.32
MMD-CBS
100
1.0
(51 58 19)
(1.010.0:
1.0)
DIFFUSER + projection
61 1:4
0.65
(10010: 100
10.92
0.91 10.28)
CD-LB (w/o proj.) CD-LB
100=0
0.97
(18439: 19
(0.5940.49:
0.60+0.49
CD-SHD (w/o proj.)
100+0.0
1.040.07
(24+43, 22
(0.96+0.19. 0.96+0.19
PCD-LB
10040.0
0.90
(100 100
(0.85 0.86.1.35)
PCD-SHD
100
1.040.042
1:0020:100:::
(0.9640.19. 0.964u.18)
Max Vel. = 1.34
vanilla DIFFUSER
60
0.61
(78+ 73
(0.8940.32. 0.88
MMD-CBS
100
1.0
(71 2139)
(1.020.0. 1.0_
DIFFUSER + projection
59
0..3
(100 100
(0.91 0.90
CD-LB (w/o proj.) CD-LB
100=o
0.97
311+16:33+41
00.59 0.60
CD-SHD (w/o proj.)
100+0.0
1.0+0.07
(50+50, 48
(0.96 19, 0.96
PCD-LB PCD-SHD
100
0.90
(100 100.o)
(0.85 0.85
100
1.0.0.033
(100 100
(0.9610.19. 0.961..19)
Table 9: Task DropRegion on, 2 robots, 100 random tests, sample size 128 except MMD-CBS.
producing more male samples, thus yielding 100% male gender satisfaction (M) and generally lower female gender satisfaction (F) across both runs.
C.6.3 ABLATION STUDY
We ablate both the coupling strength y and the noise-scaling factor k. For PCD, we vary y (50, 150, 250, 350, 450, 500} with M E {2. 6} exemplar images per model. Separately. for SD+P we vary l E (1, 2, 5, 10, 20) (again with M E {2, 6)) to examine how amplifying the noise standard deviation impacts diversity and other metrics. Figures 27 and summarize the trends for XOR%, M/F%, SE-CLIP, SE-LPIPS, SE-FID and IS-LPIPS as y and & increase, respectively.
As y increases, observe that XOR% increases monotonically, M/F% (gender constraint satis- faction) is maintained perfect by design, SE-CLIP maintains in place, SE-LPIPS and SE-FID drops almost monotonically. while IS-LPIPS increases. This suggests that (i) larger " strengthens the coupling signal, yielding clearer young/old contrast (higher XOR%), (ii) generated sample pairs resemble more of the exemplars provided (denoted by gradually decreasing SE-LPIPS and SE-FID) due to gradients from the coupling loss, (iii) stronger coupling also injects modest additional diver- sity (higher IS-LPIPS). and (iv) projection ensures the gender constraint is satisfied regardless of
In the independent study of k we observe that increases, the trend across all metrics are very much similar to that of 7, with one notable difference: IS-LPIPS rises sharply and monotonically as k grows. This suggests that increasing the noise scaling factor boosts sample variation, and yet does not violate the attribute constraints provided by the exemplars (perfect M/F%, high SE-CLIP, low SE-LPIPS and FID).
40

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Published as a conference paper at ICLR 2026
Task Empty, 4 Robots
METHOD
Metric
SU(S)
RS1
CS(%)
DAT
Max Vel. 0.703
vanilla DIFFUSER
65
.624.49
(69246.
654448, 64248, 62248)
(.992.073, 1.2000055, 2.240014, 1. 0016)
MMD-CBS
100+0
1.20
(16136,
17 11 17238)
(.992.044, 1. 0035. 1. 10.)
DIFFUSER proj.
65148
.614.49
(100zo, 100gz. 100 100go)
(.99 1= 00055, 1= 0014,, 0016))
CD-LB (wfo proj.)
100+.
-994.us0
(.03141.8, 149.9,
3417, 149.,,
3.1
(.81 .88 28.4+.2...6............ .86
.85 ( (.99 + ouss, u0a))
CD-SHD (w/ proj.)
100+0
1+0
(38 188444,,84440 38 35 37 474
PCD-LB PCD-SHD
96420
.92:.28
(1004a,
100 100 100
2. 51. 51 22 49 37)
100+0
.994.us. .004
(100 100 100 100
(.96 ,.6, .97 009, 207.
Max Vel. 0.692
vanilla DIFFUSER
65
.624.40
(42+49, 42 43 41
(.99+.073, 1.2.00055, 0.04 1. o016)
MMD-CBS
100
1.
(15+36,
17 11,31, 16g37)
(.992.044, L.4. 1.19. 1,
DIFFUSER+ + proj.
64148
.614.49
(100
100go, 100 100ga)
(.99 0.050, 1.1.0014, 1. 0916)
CD-LB (w/o proj.)
100+0
.994.055
(040, L49,9, 2.9,9,2.541. 3.1)
(.81 18.888.... .86 .85 24)
CD-SHD (w/o proj.)
100
1.40.
(28+-55. 49, 274 27
(.99+.076, 1.4.001, 1.4 1.4.00s5, .003)
PCD-LB PCD-SHD
24
.924.28
(100+a,
100 100 100+a)
(.6+.39, 564..7, .8 as, .55+.a)
100+0
.994.um .995
(100 100 100 100
(.96+.1, .8 4.7 .94..77)
Max Vel. = 0.675
vanilla DIFFUSER
65+48
.624.49
(2 21 26
(.99+.073, 1.0.00055, 0.140 1. .016)
MMD-CBS
100+0
1.
(15436,
17 11 14
(.994.044, 1. 019, ).)
DIFFUSER proj.
64448
.63..48
(100go, 100ga, 100ga, 100ga)
(.99 0.000, L.4.0014, 1. .016)
CD-LB (w/o proj.)
100.o.
1994.065
(0..o.,1..9..
1.7 3.217)
(.812.33, .88 .881.88,
.85 28.
CD-SHD (w/o proj.)
100
1.100.
121
19 21 21
(.992.076, 1.4.001, 1.2.0085, 1. 003)
PCD-LB PCD-SHD
92+27
.83..3
(100aa,
100 10 100
(.73 .79. 33. .7, #.34, 77. ..34)
100
.994.099
(1004a, 100 100 100
(.96+.1, .984.04, .97 009, .96 46777
Table 10: Task Empty, 4 robots, 100 random tests, sample size 128 except MMD-CBS.
Success Rate ISU)
Inter Robot Saftty Score (RS)
Velecity Condtraltt Salifaction (CS)
Dota Adherence (DA)
Mae selacity 0.075: Projection ADAM
Max Velceity 072; Prouection ADNH
Veiclty 872; Projection AINH
Mac Watecty 0.87: Projecter AZNM
3.0
3.2
Successee
Seeelre
0.6
SDD
L2
0.6
Ni only
0/6
Sapcline
Sucire
Musci nn Pre. only
Tmil zrly
0.4
only
0.5
(a) SU (N 2)
2)
(b) RS 7 (N = 2)
CS t (N 2)
(d) DA (N 2)
Success Rate 150)
Inter-Rabee Robot Safety Score (SS)
Wecty Conatra nt Salifaction (CS)
Data Adherence IDA)
SHD
2.0
Bacediné only
0.0
crly
0.5
6.7
0..
3.2
5.5
06.
5110
3.5
05
Ranel
5.0
Pro, only
20
10
16
20t2
10
10
20th
100
10
20
107
(e) SU (N = 4)
(1) RS 1 (N 4)
(g) CS 1 (N 4)
(h) DA 1 (N 44
Figure 18: Coupling Strength Ablation of PCD on task Highways with velocity limit Umax 0.878. (a,b,cdd) N = 2 robots; (e.f.g.h) N 4 robots.
41

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Published as a conference paper at ICLR 2026
Task Highways, Robots
METHOD
Metric
SU(S)T
RST
CS(%)
DA
Max Vel.= 0.878
vanilla DIFFUSER
53450
2.14.1
473+44, 47 48 67747
(1.+.04, +. 1.4.04, 122)
MMD-CBS
100+0
1.10.
(68247, 64 63 65
(,82.14, 20 .9
DIFFUSER proj.
54450
2:11
(10020. 100 100g0. 100
(1.1.052.
28, 1. 1.277.1....
CD-LB (wlo proj.)
10040
1.2.032
(040, 2. 0.
1992.12. 1.2.0777 1. 066
CD-SHD (wo proj).
100
1.2.012
(35+48, 355444,35444 36448. 35
.1. 1. 0.4)
PCD-LB PCD-SHD
100+.
495
(100
100 100 100 (100 10 100 100
16
.9, .99 098
100
1.6.063
(.98 19. 4., 077
Max 0.781
vanilla DIFFUSER
53
211 #.41
(62+-49; 57 58 49; so
(1.4.04, 98 4; t..004 1.2.028)
MMD-CBS
100
1.20.
(582-19
550 58 59
14, 20. .9, .99
DIFFUSER + proj.
53
.222.42
(100go. 100
100 100
1.
CD-LB (w/o proj.)
100+0
1.2.069
07.7 .0078..88)
(.992.11, 984. 1.2.067 1.
CD-SHD (w/o proj.)
100+0
1.4.051
(49
42, 45 430)
(.99-.77.. .8 1.4.07,, 1.4.05
PCD-LB PCD-SHD
100
2.1
(100
100 100 100 (100
.984.15, 954.21
.9 99.
494..99
100+0
14.064
100 100 100
.98 964.19, .98 .9 082)
Max Vel. = 0.647
vanilla DIFFUSER
53
2.11
(46=50, 43. 39.2 45.9
(1.1.04, .81. 04, 1.1.028)
MMD-CBB
100+0
1.40.
41 46 51))0)
DIFFUSER proj.
49450
.284.45
(10040.
100+o. 100 100
(1._.059, 18
CD-LB (w/o proj.)
100
.97
(,072.7,.2144
25. .055g23)
L.
CD-SHD (w/ proj.)
100+0
1.4.012
(18+39, 16437, 22. 21)
(1.+..7..98+.111 .9 L..07, 1.4.044)
PCD-LB PCD-SHD
87434
.92+.27
(100
100#a, 100 100
(.98 .95 .8 .9++..9 (.964.2, .9. 97
100
.994.1
(100 100 100 100
Table 11: Task Hi ghways, 4 robots, 100 random tests, sample size 128 except MMD-CBS.
Obstacle Safe Rate
Obstacle Safe Rate
Max Velocity 0.878: Projection ADMM
Max Velocity 0.878; Projection ADMM
1.2
1.0
1.0
Rate
0.8
0.8
0.6
0.6
Obstacle
Obstacle
0.4
0.4
SHD LB
SHD
0.2
0.2
LB
Baseline
Baseline
0.0
Proj. only
0.0
Proj. only
10 3
10 2
10
10c
10
10 2
10a
(a) Obstacle Safe Rate t (N 2)
(b) Obstacle Safe Rate + (N 4)
Figure 19: Obstacle safe rates for the coupling strength ablation of PCD on task Highways with velocity limit Umax 0.878. The obstacle safe rate is defined as the average an indicator of whether all trajectories not colliding into static obstacles.
42

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Published as a conference paper at ICLR 2026
Task Conveyor, 4 Robots
METHOD
Metric
SU(%))
RST
CS(%)f
DAT
Max Vel. = 1.21
vanilla DIFFUSER
1.19.9
.074
(.5347.3, w,7.6, .3 .3....)
(.961.21,
95 25
2.
MMD-CBS
79_41
19
(0.10, 0. 0. 0.10.) )
199
.94.
194
DIFFUSER projection
0.
.11
(100 100 100 100
[73+.44, .224.65 754.44
CD-LB (w/o proj.)
100+0
.99
a0. 002 1.s)
.90 .9.
CD-SHD (w/o proj.)
100
1.:0.
(.078g2.s, .4. 03942., 023
(.984.14. .84 18.
PCD-LB
95_22
.84..6
(100
100 100 100
(880
88
80
.9. .1
PCD-SHD
100+0
.3 28
(100 100 100 100+0)
(.91 .90 300000000000
30. ..1.31)
Max Vel. = 1.46
vanilla DIFFUSER
1.29.9
.074..26
(3.2-18: 4.1 1.9 1.6
(.96 21, .9.. 2, .95 .97
MMD-CBS
79_41
10
(2. 2. 4. 4.220)
(.992.10: .6...066.... .944. .9.....9..99111 19)
DIFFUSER projection
1.49.9
.092+.29
(100 100 100 100
(.90+.au.
.94..2 91
CD-LB (w/o proj.)
100+a
19
(1443.7.001.411 0078 13
999 .0.9..0.................. .9..32))
CD-SHD (w/o proj.)
100+a
1.40.
(.684x.2,
33 30.
(.98 .9 .84. 984.11)
PCD-LB
100+0
894.32
(100
100 100
100+a)
(.904.29. .90
.9 90.
PCD-SHD
10010
.97..18
(100 100 100 100+o)
(.971.17, .9. .9.. 19.
Max Vel. 1.76
vanilla DIFFUSER
1.99.0
.0744.26
113447,322:00 30.46. 30+48)
(.96 196+.21..66..2. .6 .5 197+.11
MMD-CBS
79_41
10
(48450. 53 40g49 43450)
(.994.10:
.96 .944. .94
DIFFUSER projection
1.49.9
.08
(100+0. 100 100 100+0)
(.95+.22, .94 .94 97
CD-LB (w/o proj.)
100+e
.994.1
(.35.s.9, .34 4-.4..4. 42...o)
(.894.31, 99, 91,..1+......... .90 .9, ..).
CD-SHD (w/o proj.)
100+e
1.40.
(13+33, 14 12+32, 10+30)
(.984.14, .84. .8 .8. .14.1
PCD-LB PCD-SHD
100+.
.92.27
(100 100 100 100
(.95 .64.. .64. .5.
10040
.994.097
(100 100 100 100
(.981.13, .8.. 98. 18.16.111....1... 98.114
Table 12: Task Conveyor, 4 robots, 100 random tests, sample size 128 except MMD-CBS. "MMD-CBS yielded no trajectories when failed to find an inter-rohor-coolisson-ffee solution within time limit; therefore, the CS metric for MMD-CBS is calculated only with inter-robot- collision-free trajectories.
43

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Published as a conference paper at ICLR 2026
Task DropRegion, 4 Robots
METHOD
Metric
SU(%)
RST
CS(%)*
DAT
Max 0.928
vanilla DIFFUSER
3.0+17
.053
(11 9.7 13 11
1904..3,,-94..0
89.. .9
MMD-CBS
100
1.20.
(18_38,
16:37, 17138, 19439)
14,82.14,.982,44,, 1.20, .99_
DIFFUSER + projection
2.0
.0844.28
(10040:
100 100 100
(.931.26. .92,,77,
CD-LB (w/o proj.)
100
38
3.0+17)
CD-SHD (w/o proj.)
100+e
.3
(5.3
6.1 4.6
PCD-LB PCD-SHD
98-14
.72
(100+a.
100 100 100+a) (100 100 100 100
42,
2114.1 264 21,.,,,,,
42
10040
19
(.22.27. 9924.9. 22, 27.
Max Vel. = 1.13
vanilla DIFFUSER
3.0
.053
(40=49, 37 48, 37 40
(.90 -89.
32, +..1.11
MMD-CBS
100
1.20.
(43_30,
39, 550, 50
1898 194,,,
DIFFUSER projection
3.0
.0741.26
(100
100 100 100
CD-LB (w/o proj.)
100
.872.34
(9.6
9.9 8.7 9.0
(.24 994..76..24.47..... 2.. .3.
24,
CD-SHD (w/ proj.)
100+.
.93
(18=as; 15430, 18439, 18438)
PCD-LB PCD-SHD
99
.72+.45
(100
100 100+o, (100
100+a)
(.22+ 42, 22.4, 25 43, .2_241)
100+.
.98
100 100 100
(.92 .914.29 .2. 28, .2).
Max Vel. 1.34
vanilla DIFFUSER
3.0
.053
(78 75 76, 77442)
(.90..30- .9. 08.999..3....... 22, 89..311
MMD-CBS
100
1.40.
(65_48,
69+46, 75443, 75243)
14, 149
1.20. .99_.1)
DIFFUSER + projection
2.0
.068
(100 100 100 100
(.92 .9.227..111. 211.2.111........
.9.
CD-LB (w/o proj.)
100+
.874.34
(15_36.
16437, 16437, 15236)
(.241.43. .24..22.4. 114442,,264...0 2.4
CD-SHD (w/o proj.)
10040
..3+.2
(39-49. 344 39, 48)
(.931.26 .24.272 .9. .934.22
PCD-LB PCD-SHD
99
.74+.4a
(100+0,
100 100 100+a)
(.21 (+.
2+.4 4..4.224 21
100+e
.99 .99
(100 100 100 100+a)
(.91 914.29 -1 ..18)))
Table 13: Task DropRegion, 4 robots, 100 random tests, sample size 128 except MMD-CBS.
44

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Published as a conference paper at ICLR 2026
DP
(b) DP + projection
(c) C.D (w/o proj.)
(d) PCD (Ours)
(e) DP
(I) DP projection
(g) CD (w/o proj.)
(h) PCD (Ours)
(i) DP
(j) DP projection
(k) CD (w/o proj.)
(I) PCD (Ours)
DP
(n) DP + projection
(o) CD (w/o proj.)
(p) PCD (Ours)
BAN
(q) DP
(r) DP projection
(s) CD (w/o proj.)
(t) PCD (Ours)
Figure 20: Additional qualitative results of the PushT experiment.
Rows correspond to initial
observations; columns correspond methods. Robot trajectories colormap (warmer colors indicate later time steps); red crosses mark velocity violations. Only the first few dozen steps are shown for clarity.
45

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Published as a conference paper at ICLR 2026
PushT Experiment Results
METHOD Metric
DTW1
DFD1
CS(%))
TC1
Max Vel. 6.2
DP
3.16
0.46544.119
17
(0.927=0.19) 0.931 00.114 1888)8)..))))..
DP + proj.
2.95
0.422
(100 100
(0.86210.22)
0.860 1.860....55)
CD-DPP
1.74
0.544
(44.4116.6: 44.5
(0.923 20.206; 1.922210.204
CD-DPP-PS
4.48_0.911
0.64840.114
(39.6115.6.39 39.7411..
(0.912 208, 0.917 199
CD-LB
1.13
0.59640.151
(41.6415.4, 41.344
(0.910 208, 0.912
CD-LB-PS
4.50
0.64640.12
411441164,,44441110
(0.921 199, 0.925 189)
PCD- DPP
4.6341107
0.643+0.15
(100 100
(0.776 3010 0.777
PCD-DPP-PS
1.34
0.61010.118
(100 100
(0.856 1..5680.258. 0.8554..25 0.253)
PCD-LB
5.07
0.696 132
(10040, 100
(0.747 10.3, 0.750.n.3)
PCD-L.B-PS
4.32+1.09
0.60540. 138
(100 100
(0.865 0.861 259)
Max =
DP
3.1641.13
0.46540.118 149
164.8_13.8, 65
(0.927 195 195, 0.931 188
DP + proj.
2.96
0.418
(100 100=a)
(0.896=0.227 222 0.888 237
CD-DPP
1.04
0.544
(62.5113.9.
62.4413.6)
(0.923 20.206 0.922
CD-DPP-PS
4.48
00.14
(57.4414.0. 57.3413.5
(0.912 00.12 208, 0.917
CD-LB
4.13
0.59640.151
388.9413.8.88.6411.
(0.910 00.910=0.200 208, 0.912 201
CD-LB-PS
4.50+1.01
0.64640.129
(58.8 58.7414.3)
(0.921 190, 0.925 189)
PCD-DPP
4.55+1.00
0.63840.18
(100 100
(0.829 0.834 0.2.7 .71
PCD-DPP-PS
4.39+1.05
0.62240.133
(100 100=a)
(0.885 236, 0.885+0 233
PCD-LB
5.12
0.708 135
(100 100
(0.778 1 288
0.79111.22
PCD-LBB-PS
1.02
1o.18
(100 100
(0.89010.234,
1.88210.240
Max Vel. = 10.7
DP
3.16
0.445
(77.619.97, 77.619.81)
(0.92710.195,
0.931 1.931_10.18 10.188
DP + proj-
3.00
0.43540.155
(100 100_a)
(0.905 222, 0.90640.210 2.2
CD-DPP
3.74+1.04
0.54440.
(76.2+10.3,
16.2
(0.923 200. 0.922
CD-DPP-PS
4.48
0.64840.114
(71.8410.s, 11.7+10.8)
20. (0.912 208, 0.917 7 199
CD-LB
4.13+1.11
0.596440.15
(0.910 0.912 201
CD-LB-PS
4.50+1.01
0.64644.129
(72.6+1.1.72.6411))
(0.921 199, 0.925 0.19))
PCD-DPP
4.52-10996
0.63711..12
(100 100
(0.852...266 0.855L10.251
PCD-DPP-PS
4.40
0.626
10001,,00000
(0.892 228, 0.900 220
PCD-LB 5.16+1.09
0.71644.188
(100 100
(0.804 0.80340.200 0.273)
PCD-LB-PS 4.39+1.01
0.6222+.11 129
1000+01100-1.
(0.896 227. 0.900 222)
Table 14: Results of PushT task by all compared methods with three velocity limits. DP refers DIFFUSION POLICY; CD denotes DP+coupling only; PS denotes posterior sampling variants of cost functions.
METHOD
XOR%*
M/F
SE-CLIP
SE-LPIPS
SE-FIDJ
IS-LPIPST
SD
20
71/14
A400..06// 441
.77 77....43/
.42.43
.75
/.75
CNet
8
100/56
.68 -7540000 .8 1.00000//000000000
.29/.44
A5..0.043/
SD+P
44
100/100
181 20.00
.1540.022/
.11/.17
.0610.043/ 10.40.004
Syne'
48
95/17
.2.4 052/ .6140.066
.67=n.01/ /.68
.39/81
.70 /.69
SD+C
48
51/47
057/ .45
.4640.067
o76 0..33
.47-54
/.75 4..50.008 0..051 .69 1..40.
.75 0.
SD+C
64
47/37
.55 /0.../.0///// 0..0
.70
33/.42
PCD
96
100/100
1000
288 /9240.004
.1
.05/.09
.11
Table 15: Paired face-generation results with projection and coupling applied. - exemplars used are as per Figure 4a Boldface indicates the best score(s) for each metric.
46

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Published as a conference paper at ICLR 2026
Dynamic Time Whrping (DTW)
Discrete Rrechet Distance (DFD)
Ve lockt inn Safisfattion ICSI
Trss Completion (TC)
Plax Wiccky 0.2; Profectio ADMM
Hox Wercty Projccion AD9M
Not Veccty 92 Proection A2191
Mac Prleet en ADVM
100
1.00
7.00
only
only
0.00
5.00
5.0
0.80
4.00
2.00
2.00
5.70
2.00
5.10
0.50
0.70
0.90
200-15
287
18-P5
0.05
0:42-
0.25
[mo) orly
DTP
2.50
5.30
(a) DTW1 1
(b) DFD f
(c) CS 1
(d) TC f
Dynamic Time Worping (DTW)
Discrete Peechet Distance (DFO)
Velocity Constro in Sttsfaction (CSI
Tast Completion (TC)
3.50
Taneline pm
Trg. only
5.00
100
Pay only
Bareline
0.05
3.00
6.00
0.50
0.30
Seacire
0.50
5.00
5.70
0.05
0.50
Pmuj
3.50
0:50
2,40
0.70
0.75
D7P-25
D7F
LS-5
10
10
10
10"
10
(e) DTW 1
(1 DFD f
(g) CS t
(h) TC f
Dynamic Time Whiping (DTW)
Diccnte trechet Distance (Dm)
Velocity Condrmant Satistaction
Tast Completion (TC)
12.7: PMo ROM11
1300
Meudlly 10.7 Pooedie
Not Mordey 15:3: Poodirr 42NM
Velelly 18th hojedler
5.00
Baseiin
7.00
Tro, only
2P--P5
5.00
280
Pm: Pnl.
DPPIPS
DAP
0:95
Susciie
0.92
1.00
3.00
0.00
0.50
Buscline
5.50
4.50
2.50
0.55
0:40
5,00
0.50
0.20
2,00
0.30
LS75
109
10n0
10
10
10
10c4
10
1041
10-1
10
10-1
(i) DTW
(i) DFD
(k) CS
TC
Figure 21: Coupling Strength Ablation of PCD: Evaluation metrics results at different Smax as coupling stength y and coupling function vary. (a,b,c,d) Umax 6.2; (e.f.g.h) Umax 8.4; (i.j.k,l) 10.7.
Umax
METHOD X
M/P%
SE-CLIP
SE-LPIPS.
SF--FID
IS-LPIPST
SD
71/14
40 41.00.
77 TT=20043/.7620..m 7620.026
.39/.40
.7540.060/
/.75
CNet
100 100/29
/.7320.055 .64
.7 / 20.050
.24/.30
.4440.043/
/.1220004
SD+P
44
100 100
8440.014/
22.2..00/ .2720.023
.11/.20
.1440.051
111.0.039
Sync*
48
98/17
.64 00../.9//00..... /0.59
67=20.00 oa0.oas//666
.41/.74
774...002
.6940.060
SD+C
48
51/46
.4
/.4//444.
7622..012/ / 77 0.09
.43/.49
.75.0.0s8
7.5 10.005
SD+Ct
64
47/38
.54 154.0.064/ 10.10.
.70.0.0335/ 10.044
.35/.37
.69...000 60..0.00///. 10.006
PCD
100/100
.85
.19 0.004 2224222...000
00.041
.09/.14
.18
16400.06
Table 16: Paired face-generation results with projection and coupling applied. -- exemplars used are as per Figure 25
Boldface indicates the best score(s) for each metric.
47

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Published as a conference paper at ICLR 2026
Werpii
Hox beaedty 42, Proection WYNE
Now Wabily 22 Pruector MONE
Maw Weydit 6.2 Prgecior HOME
Nox Whetly 6.2. Priection NONE
150
0.50
DPP
D00
1.00
30.00
D7P-P5
D05.5
0.0
1.20
0.50
D70
8.00
18:55
1,00
L3-25
0.40
6.00
5.00
Readin
0.40
4.00
0.0
0.23
0.0
0.00
10
13
10
10
30
30
10
10
DTW 1
(b) DFD f
(c) CS
(d) TC f
Dynamic Warping (DTW)
Discreto Frechet Distance (DFD)
Velsctty Constrant Sausractio (CS)
Completion (TC)
M4c U4 edty 3:l: Procetion yoM
Nao Veldiiy 3.4; Projectton NOME
Nac Velecty E-4; Projection NONE
Map Veltly 8-4; Projectonn NONC
1,60
0.70
1.05
30.00
1.20
0.50
0.02
DTR
8.00
1.00
LS-75
0.02
6.00
Dacel
Dasii Ire
DPF
2.00-
6:25-
0.00
0.00-
(c) DTW 1
(f) DFD f
(g) CS f
(h) TCTCT f
Dynamic Time Warping (DTW)
Discrate frechet Distance (DFU)
Welocit Constrant ann Satisfatiion (CSI 12.7 Indectioo NOME
Task Completion (TC)
Nox 20.7. Trolect HOME
Max Wecdlt 12.7 Projection WoMe
Mar Ve 10.7 Prstccionn NOW
12.00
Boseline
3.40
0.00
10.00
DPP-P5
3.20
0.82
37M
R.00
0.50
2.80
Boodinc
0.40
DTP
0.50
D7P-55
D7P-7
0.40
2.00
6:00
(i) DTW1
(i) DFD f
(k) CS t
()) TC 1
Figure 22: Coupling Strength Ablation of CD (coupling-onl) only, without projection): Evaluation met- ries results at different Umax S as coupling stength y and coupling function vary. (a,b.c.d) Umax = 6.2; (e.f.g.h) Vmmax 8.4; (i.j...) Umax
10.7.
Dynamic Time Worping (DTN)
Discrete Rrcchet Distance (DFD)
Velocity Condtraint Saftsfaction ICS)
Task Complction (TC)
OPF p0ppe1t
1.20
0.80
1.00
5.50
0.00
0 0.80
10
6.50
0.40
5.40
3.50
0.40
3PAP2--62
0.20
520
1305 E2
2.50
Juadin
db
Serpping Stope
Sampling Steps
Samp ing Stepe
Samplng Stape
(a) DTW T
(b) DFD
(c) CS 1
(d) TCTTC 1
Figure 23: Sampling steps ablation of PCD: Evaluation metrics results with different number of sampling steps T given the same coupling strength 7 for each type of cost.
48

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Published as a conference paper at ICLR 2026
Figure 24: Sample pairs generated with both projection and coupling applied. The two exemplars used for each model (each row) are as per Figure 4a
Figure 25: Six exemplar images for each model (one row per model), generated via ChatGPT Ope-
nAl 2023) The exemplars differ only by minor visual details, ensuring close structural alignment.
Figure 26: Sample pairs generated with both projection and coupling applied. The six exemplars
used for each model (each row) are as per Figure 25
49

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Published as a conference paper at ICLR 2026
SE-CLIP
100
2 #zmmpiers exemplors
100
0.90
80
80
XOR (%0
60
MVF 195)
60
SEPCUP
0.85
eccmmllara (90
sowmplers IM)
20
20
exemplers (7)
exxomaaars (M)
commptans
eccmmclrra (f)
commplars [F]
0.75
10
10a
10?
10a
10c
10
Y
(a) XOR% 1
(b) M/F%
(c) SE- CLIP 1
SE-LPIPS
SE-FID
LPIPS
0.30
0.30
0.30
0.25
0.25
exarm plara
plars
0.25
cxcmolors
woumslars (?)
0.20
0.20
0.20
0.15
0.10
exomplan IF)
0.10
0.10
0.05
eeemmppe
0.05
0.05
10a
10a
10a
10a
(d) SE-LPIPS
SE-FID
(1 IS-LPIPS
Figure 27: Ablation of coupling strength for PCD, using M E (2,6) exemplars per model. We evaluate y {500 150, 250, 350, 450, 5003.
M/F
SE-CLIP
100
ecemplars crmmptors
100
0,900
80
80
0.875
XOR (9))
MA/F (95)
60
0.850
0.825
evemolors (1)
0.800
exomplars
20
20
sommolar (t)
examplass
eemmmleess (90
0.775
exempias (M)
6 exematars (F)
cxampiars IF)
0.750
10a
10a
10a
10
10a
10:
(a) XOR% f
(b) M/F%
(c) SE-CLIP
IS-LPIPS
SE-LPIPS
SE-FID
0.20
0.30
0.30
evemptans
exemplars
0.25
0.25
0.15
+ 6 soumplars IF)
axemplars IM)
0.20
S-LPPS
0.1.0
0.15
0.10
+ eexemplars (M)
oremplars (MI
2xxmplan plars IF)
0.10
examp IF)
0.05
eexompla planmppassii (M)
exemp INI
0.05
woemplers
0.00
10a
10a
10a
10a
10:
&
(d) LE-PS
(e) SE-FID
(1) IS-LPIPS
Figure 28: Ablation of noise scaling factor & for SD+P, using M E (2, 6} exemplars model. We
evaluate & {1, 2, 5, 10, 20).
50


Reviews:
[{"review_id": "NAzLyaks3D", "reviewer": "Reviewer_Emdc", "summary": "The paper studies the problem of combining pretrained diffusion models while enforcing task-specific constraints. The authors present a projected coupled diffusion framework for constrained joint generation. This framework introduces two generative dynamics with coupled  guidance terms, and use projection to impose hard constraints. Several experiments are used to show the effectiveness.", "questions": "See comments in Weaknesses.", "limitations": "", "rating": 2, "confidence": 2, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- This paper studies the problem of generating samples from pretrained diffusion models while satisfying task-specific constraints. This problem is important because pretrained diffusion models are often available, but sample constraints are typically not enforced during training.\n\n- The authors formulate a new problem of generating correlated samples under hard constraints.\n\n- They propose a generation method based on coupled dynamics, combining a coupled cost with projection onto hard constraints. This approach generalizes projected diffusion to coupled dynamics.\n\n- The authors further show that several existing methods can be viewed as special cases of the proposed coupled dynamics framework.\n\n- Several experiments show benefits from both projected diffusion and cost guidance.", "weaknesses": "- The projected coupled dynamics are intuitively designed by combining cost and projection. However, the conditions under which this method converges have not been studied, and to where. \n\n- The effect of coupled costs has not been analyzed, and costs may not always be differentiable.\n\n- The projection step is not discussed in detail. It can be infeasible when the constraints are non-convex. It is also not clear how to do projection for latent diffusion models. \n\n- All special cases correspond to degenerate forms of the projected coupled dynamics. New application scenarios for the projected coupled dynamics have not been explored.\n\n- Experiments demonstrate the effectiveness of combining cost and projection, which is expected since it benefits from both projected diffusion and reward guidance. However, it remains unclear to what extent the experiments reveal the advantages of the coupled dynamics.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Q4hN4vArxp", "reviewer": "Reviewer_fLqz", "summary": "The submission proposes a test-time framework to sample from multiple pre-trained diffusion models while (1) promoting some notion of joint \"correlation\" across variables and (2) enforcing constraints on the individual outputs of each model. PCD augments the usual Langevin / DDPM updates with (1) a user-specified  coupling cost between variables and (2) projections at every diffusion step to guarantee individual constraint satisfaction. Their framework  several existing methods (e.g., classifier guidance, projected diffusion, and some forms of compositional diffusion) as special cases. Empirical demonstrations cover three domains: multi-robot navigation (collision avoidance coupling cost, velocity constraints), robot manipulation on PushT (diverse, non-intersecting trajectories  as a coupling cost with velocity constraints), and ``paired'' face generation (age-contrast coupling with gender/attribute constraints).", "questions": "Can you scale the approach beyond two variables/agents? \n\nCan you numerically verify that the two coupling limits match known methods (classifier guidance, projection), at least in a toy experiment?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The paper is well structure and clearly written. The problem of composing several trained diffusion models at test time under constraints is indeed relevant, and prior work has aimed to tackle similar problems when sampling a single variable (e.g. a single image). The main novelty, in my understanding, is the ability to compose diffusion models potentially defined over different variables. The approach is widely applicable and requires minimal modifications of standard sampling algorithms. The use case in muti-robot systems is well motivated, since constraints arise naturally in physical systems and training joint distributions can be computationally costly as the number of agents grows. The toy-example in images -- although artificial -- showcases the applicability of the framework in a totally different setting.", "weaknesses": "I think that the notion of \"correlated\" variables, which is emphasized throughout the paper, could be better defined/explained/motivated. Since the framework is flexible to accommodate arbitrary coupling costs, correlations are in my view an understatement with respect to the practical utility and applicability of the proposed method and underlying problem it is tackling.\n\nIn PushT trajectory dissimilarity appears to me as a contrived objective/task, there might be manipulation examples (e.g. bimanual over multiple objects) that lend themselves more naturally/directly to the framework.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "3VWLF78VAu", "reviewer": "Reviewer_XYUt", "summary": "This paper introduces Projected Coupled Diffusion (PCD), a novel test-time framework for generating jointly correlated samples from multiple pre-trained diffusion models while simultaneously enforcing hard, task-specific constraints. The core problem is that generating from a joint distribution $p(x, y)$ is difficult, especially when $x$ and $y$ must be correlated in a specific way and satisfy some given hard constraints. PCD addresses this by modifying the reverse diffusion sampling process. At each step, the update for each variable is guided by three components:\n\n1. The score from its own pre-trained model (e.g., $s_X^\\theta(x_t, t)$).\n\n2. A gradient from a coupling cost function $c(x, y)$ that encourages the desired correlation between variables.\n\n3. A projection operator $\\Pi_{\\mathcal{K}}$ that forces the updated sample back into the feasible set of hard constraints.\n\nThis approach requires no retraining and unifies compositional generation with hard-constraint enforcement at test time. The authors demonstrate PCD's effectiveness across three distinct domains: multi-robot motion planning, diverse robot manipulation, and constrained image-pair generation.", "questions": "See the Weakness above.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. This paper unifies two important aspects in diffusion sampling: coupled generation and constrained generation through a clear and effective way. The proposed PCD provides a general, test-time-only framework to address this. PCD operates over multiple pre-trained models and costs (analytic or learned), requiring no retraining of the base diffusions, which enables the method applicable to a wide variety of settings, for example, where paired data and costs are scarce or proprietary.\n\n2. This paper is well written, easy-to-follow, and conducts extensive studies across various domains, including multi-robot planning, Push-T trajectory pairs, and paired face generation.", "weaknesses": "1. PCD relies on projection and gradient-based updates to enforce constraints. How does it handle test-time constraints that are non-differentiable, for instance, a logic-based rule where a sample is accepted only if it passes some non-differentiable verification? A concrete example would help.\n\n2. The performance of the proposed method might require the estimated Tweedie to be of high-quality. Otherwise, the further guidance term might likely be inaccurate or even compromise the overall sampling process.\n\n3. This paper introduces gradient-based guidance and per-step projections, which can increase wall-clock latency compared with non-gradient based baselines. In appendix C.1, the authors also mention \"PCD is approximately 4 ∼ 7× slower than vanilla diffusion mainly due to the per-step projection operation.\" Are there any potential ways to enable faster sampling with PCD?\n\n4. Could the authors provide a curve of performance v.s. the number of sampling steps to illustrate how will the quality of the estimated Tweedie term affect performance? An intuition is that with more sampling steps, the Tweedie estimate will be more accurate, which could facilitate better guidance.\n\n\n5. I would encourage the authors to include discussion on the limitations of PCD along with failure analysis with qualitative examples.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "hmcWI5zFuQ", "reviewer": "Reviewer_DJcn", "summary": "The paper proposes a method to jointly sample two independent variables coupled using a joint cost function. Practically, given the independent denoising score functions of each variable, the goal is to couple their denoising process directly at test time to sample them jointly. The paper introduces some variants: (1) standard LMC/DDPM based: perform one step-denoising, use classifier guidance using the gradient of the cost function and then perform projection to map the noisy latents to the safe-set. (2) use tweedie clean estimates to calculate the gradient of the cost function. Overall the paper claims that the proposed method generalizes the formulation of classifier-guidance, projected diffusion, compositional diffusion and joint diffusion.\n\nI am familiar with a concurrent work in this line of research: https://arxiv.org/abs/2509.08775\nWhile my concerns are based on the concurrent work, my judgment of this paper will be independent of it.", "questions": "1. How feasible is designing a projection operator for every task? How sensitive is the overall quality of samples to projection hyperparameters?\n\n2. How is the cost function in general defined for noisy latents? It seems from the algorithms that the method always uses the clean estimates for projection and using the same for cost function also empirically results in the best performance.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. The paper's unified formulation of classifier-guidance, projected diffusion, compositional diffusion and joint diffusion is very impactful and timely. The authors have provided exhaustive results to analyze the performance of their algorithm on toy-domain, a planar robotics task and an image generation task as well.\n\n2. PCD can impose hard constraints in addition to joint diffusion while being computationally efficient.", "weaknesses": "1. Data fidelity vs projection: Since the cost function and projection operation hold for the clean distribution, data fidelity is of primary importance here. The more realistic the clean estimates are, the better evaluation and guidance can be done. However, as the authors acknowledge as well, the projection operation hurts data fidelity, as also observed by the concurrent work, when applied to low-quality clean estimates (particularly at higher noise levels). \n\n2. Differentiability of cost function: This is a considerably strong assumption (which is also used by many prior works like MPD, DPCC). For example, the signed distance based indicator function in SHD might not be differentiable everywhere. This is a case, especially for collision checking objectives for real robot executions. I agree that effective engineering design can mitigate this, but this limits the scalability of the approach.\n\n3. convexity of constraints: Since two experiments in the paper deal with navigation and manipulation, it is worth noting that non-convex safe sets are pretty common in these two settings, most commonly arising from obstacle avoidance constraints. For example, in the highway task if the trajectories are trained without the rectangle in between and forced to avoid it at test time, the resulting safe-set becomes non-convex. This again limits the scalability.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)