On the Error Analysis of Triangular Transport Maps for Conditional Generation via MMD-based Objectives
Authors: Michele Martino, Bamdad Hosseini, Amir Sagiv, Amirhossein Taghvaei
Organizations: Department of Applied Mathematics, University of Washington, Seattle, WA, USA · Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, USA · Department of Aeronautics and Astronautics, University of Washington, Seattle, WA, USA
Conditional generative models parameterized by triangular transport maps are widely used for conditional sampling, yet theoretical guarantees on the quality of the conditional measures they produce remain scarce. We address this question for minimum divergence estimators of triangular maps. We use MMD-based objectives as a tractable setting in which the gap between joint and conditional errors can be analyzed precisely, and we measure performance through the average conditional estimation error, defined as the squared maximum mean discrepancy (MMD) between the estimated and the true conditionals, averaged over the conditioning variable. We first show that the common practice of minimizing a joint MMD may fail to recover the conditionals: writing the joint and the conditional criteria as quadratic energies of the same conditional mean embedding residual, we prove that, for parametric classes, joint minimizers are generically inconsistent in the misspecified setting. We then analyze an estimator that directly minimizes the conditional criterion through a surrogate loss computable from joint target samples and reference samples. We prove a non-asymptotic error bound that decomposes the average conditional estimation error into approximation, estimation, and reference sampling errors, with fast rates independent of the ambient dimension in the parametric setting. Finally, for joint MMDs built from concentrating Gaussian kernels on the conditioning variable, we show that the joint criterion approximates a density-reweighted version of the conditional one, and we derive an error bound for the joint minimizer whose rate suffers from a curse of dimensionality in the conditioning variable. Our results suggest that training criteria for conditional generative models should target the conditional rather than the joint error, a principle that extends to other divergences and to dynamic transport.
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Figure 1: Joint density of (Y,U) for two distributions supported on the lines U=αY (white) and U=−αY (yellow), with Y∼N(0,1) . The distributions are visualized as Gaussians with small perpendicular bandwidth ε=0.15 . The vertical dotted line marks Y=1 ; the double-headed arrow shows the separation 2α between the two modes at that slice.
We establish finite-sample guarantees for conditional diffusion models learned from data and implemented with finitely many denoising steps. Our approach aligns the score-training objective with the sampling grid, allowing statistical estimation and numerical discretization errors to be analyzed together. Under a smooth conditional density model, we derive high-probability bounds on the generated distribution's error and quantify the required number of score evaluations. For every smoothness index β>0, a suitable grid yields averaged conditional Kullback--Leibler (KL) divergence Op(n−2β/(dx+dy+2β)) relative to the original target with or without early stopping. Both the full and early-stopped samplers attain average target total-variation (TV) error Op(n−β/(dx+dy+2β)). Here n is the sample size, dx,dy are the data and conditioning dimensions, and errors are averaged over the conditioning variable. The total-variation bound matches the rate established in existing conditional distribution-estimation theory, up to logarithmic factors.
Qiuyun Zhu
Department of Statistics and Actuarial Science University of Iowa
We study minimax rates for conditional generative learning under integrated Wasserstein loss, with Hölder continuous conditional laws and an unrestricted covariate distribution. We evaluate unrestricted generators through their conditional output distributions. For expected Wasserstein loss, the minimax rate agrees with the rate under uniform covariates and follows from balancing localization bias against unconditional distribution estimation error. Higher moments of the loss can instead be dominated by unobserved covariate regions, which force an additional risk contribution of order n−1, where n is the sample size. An adaptive localization argument gives matching upper bounds. The results cover unrestricted response laws and possibly vanishing Besov densities for W1, and extend to higher Wasserstein orders under uniform response density bounds and integrability conditions. A multiscale lower bound closes the logarithmic gap in unconditional W1 minimax rates in dimension two for both unrestricted distributions and Besov density classes. We also establish minimax rates for conditional Brenier maps under density, integrability, and uniform curvature conditions.
Sungwon Park, Kyoung-Sook Moon, Hongjoong Kim
Department of Mathematics, Korea University · Department of Finance and Big Data, Gachon University
Conditional Optimal Transport (COT) problem aims to find a transport map between conditional source and target distributions while minimizing the transport cost. Recently, these transport maps have been utilized in conditional generative modeling tasks to establish efficient mappings between the distributions. However, classical COT inherits a fundamental limitation of optimal transport, i.e., sensitivity to outliers, which arises from the hard distribution matching constraints. This limitation becomes more pronounced in a conditional setting, where each conditional distribution is estimated from a limited subset of data. To address this, we introduce the Conditional Unbalanced Optimal Transport (CUOT) framework, which relaxes conditional distribution-matching constraints through Csiszár divergence penalties while strictly preserving the conditioning marginals. We establish a rigorous formulation of the CUOT problem and derive its dual and semi-dual formulations. Based on the semi-dual form, we propose Conditional Unbalanced Optimal Transport Maps (CUOTM), an outlier-robust conditional generative model built upon a triangular c-transform parameterization. We theoretically justify the validity of this parameterization by proving that the optimal triangular map satisfies the c-transform relationships. Our experiments on 2D synthetic and image-scale datasets demonstrate that CUOTM achieves superior outlier robustness and competitive distribution-matching performance compared to existing COT-based baselines, while maintaining high sampling efficiency.
Jiwoo Yoon, Kyumin Choi, Jaewoong Choi
Department of Statistics, Sungkyunkwan University · Department of Applied Artificial Intelligence, Sungkyunkwan University