Conditional Sampling via Wasserstein Autoencoders and Triangular Transport

Mohammad Al-Jarrah; Michele Martino; Marcus Yim; Bamdad Hosseini; Amirhossein Taghvaei

cs.LGApr 3, 2026

Conditional Sampling via Wasserstein Autoencoders and Triangular Transport

Authors: Mohammad Al-Jarrah, Michele Martino, Marcus Yim, Bamdad Hosseini, Amirhossein Taghvaei

Organizations: Department of Aeronautics & Astronautics, University of Washington, Seattle · Department of Applied Mathematics, University of Washington, Seattle · Department of Mathematics, University of Washington, Seattle

Abstract

We present Conditional Wasserstein Autoencoders (CWAEs), a framework for conditional simulation that exploits low-dimensional structure in both the conditioned and the conditioning variables. The key idea is to modify a Wasserstein autoencoder to use a (block-) triangular decoder and impose an appropriate independence assumption on the latent variables. We show that the resulting model gives an autoencoder that can exploit low-dimensional structure while simultaneously the decoder can be used for conditional simulation. We explore various theoretical properties of CWAEs, including their connections to conditional optimal transport (OT) problems. We also present alternative formulations that lead to three architectural variants forming the foundation of our algorithms. We present a series of numerical experiments that demonstrate that our different CWAE variants achieve substantial reductions in approximation error relative to the low-rank ensemble Kalman filter (LREnKF), particularly in problems where the support of the conditional measures is truly low-dimensional.

Figures & tables

Fig. 1 : Numerical results for the synthetic example in Sec. IV-C . The figure shows the sample distributions for the last three states left to right for

y=\mathbf{1}_{d_{\mathcal{Y}}}

$d_{\mathcal{X}}$	LREnKF	CWAE1	CWAE2	CWAE3
10	3.3855	1.1911	1.1234	1.9790
20	3.7908	1.7968	1.5552	2.4946
30	4.6513	4.0965	3.5174	4.0098
40	5.9057	4.7601	4.4719	5.1238
50	7.4993	5.6551	5.6583	6.5567

TABLE I :

W_{2}

error across methods and problem dimensions for the nonlinear embedded example in Sec. IV-C .

Fig. 2 : Numerical results for the spherical posterior example in Sec. IV-D . Both panels show the sample distributions for each method alongside the true distribution for two distinct values of the observation

y

. The left panel corresponds to

y=1

and the right panel corresponds to

y=2

Fig. 3 : Simulation results for the flow field example in Sec. IV-E . (upper) $u_{x}$ component over time showing a sinusoidal stream and deceleration. (lower) $u_{y}$ component over time showing alternating sign and symmetric vortex formation. Taken with $\text{Re}=281$ and a cylindrical obstruction.

Panel of Figure 5 — Fig. 4 : Numerical results for the flow field example in Sec. IV-E . (a) and (b) are the original field with the $d_{\mathcal{Y}}=9\times 9$ observed pixels boxed in green. The bottom panels are reconstructed from the noisy observation and averaged per-component over 1000 samples.

Panel of Figure 6 — Fig. 5 : Numerical results for the flow field example in Sec. IV-E showing 16 out of 1000 samples. The left panel corresponds to the $u_{x}$ component and the right panel corresponds to the $u_{y}$ component.

$\phi(X)$	LREnKF	CWAE1	CWAE2	CWAE3
$X$	0.0437	0.0110	0.0281	0.0054
$X^{2}$	0.0716	0.0225	0.0584	0.0124

TABLE II : Relative MSE across methods for the flow field example in Sec. IV-E .

Conditional Generative Modeling

Conditional generative modeling learns to produce data distributions steered by labels, inputs, text, or physical and structural constraints.

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Drift Variation Autoencoder: Unifying Generation and Representation Learning through Conditional Posterior Flow Matching

Jiarui Cao

The Chinese University of Hong Kong

Conditional Generative Modeling Representation Learning Conditional Flow Matching Multimodal Generation

Open paper AskPDF arXiv

Stochastic masking, cropping, or modality removal makes deterministic reconstruction an incomplete target: one observation can admit many clean completions. This work takes the corresponding posterior

P(X\mid C)

as the common statistical object for conditional generation and generatively sufficient representation learning. Drift Variation autoencoder trains a masked encoder

Z=E(C)

and a conditional flow decoder with one clean-prediction Flow Matching loss. The analysis first decomposes the ideal conditional KL into generator approximation and the representation deficiency

I(X;C\mid Z)

. It then derives orthogonal risk decompositions for conditional Flow Matching. For an affine Gaussian path, the clean-prediction representation gap is zero if and only if

P(X\mid Z)=P(X\mid C)

. Thus the encoder-dependent excess clean-prediction risk induced by Flow Matching and the profiled ideal conditional KL have the same posterior-sufficient zero set, without being numerically equal objectives. An exact conditional field with a zero-noise endpoint then generates

P(X\mid Z)

and hence

P(X\mid C)

at a joint ideal optimum. The result extends to continuous multimodal product spaces when the complete modality tuple remains the Flow target for every observation mask. On CrossGeom-4, an 18-run controlled benchmark, observable factors have linear-probe

R^2

0.9990

0.9992

, shuffling the joint model's encoder condition increases conditional error by

13.5\times

15.7\times

, and joint target attention reduces disagreement on an unobserved factor shared by two outputs by

90.1

92.8\%

relative to independent target decoders. Visible modalities are also generated and reconstructed, directly validating the full-tuple objective. Unconditional mode balance remains imperfect, delimiting the empirical claim to a controlled multimodal proof of concept.

Abstract

Conditional Generative Modeling

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