Inverse Design for Conditional Distribution Matching
Introduces Conditional Distribution Matching (CDM) with MLGD-F algorithm, enabling inference-only distributional control in generative models, validated on synthetic, MNIST, and image editing tasks.
Key Findings
Methodology
This paper formulates the Conditional Distribution Matching (CDM) problem, defining two variants: sampling (CDMS) and optimization (CDMO). The core algorithm, MLGD-F, combines a pretrained score-based diffusion model with a fast conditional sampler, enabling gradient-based inference without additional training. It leverages single-step conditional sampling within the reverse diffusion process to estimate gradients of a distributional loss based on metrics like MMD or SWD. The method iteratively guides the input x towards a target distribution G(Y) by minimizing the divergence between P(Y|X=x) and G, validated across synthetic Gaussian mixtures, MNIST rotations, and large-scale image editing with Stable Diffusion, demonstrating robustness in high-dimensional, complex output spaces.
Key Results
- In synthetic Gaussian mixture experiments, MLGD-F achieved an 11Γ speedup over traditional slow samplers, with a 20% reduction in divergence metrics at high dimensions. On MNIST, the model generated digit classes matching target distributions with SWD reduced by 25-30%. In image editing, the approach maintained high visual quality while aligning outputs with complex distributional targets, outperforming baseline methods in MMD scores by over 20%. Ablation studies confirmed that single-step conditional sampling significantly improves gradient stability and computational efficiency, especially in high-dimensional tasks.
- Across all scenarios, MLGD-F demonstrated superior target distribution matching, with faster inference and lower memory footprint compared to multi-step diffusion models. The method effectively handles discrete mixtures and continuous low-rank supports, showcasing broad applicability and scalability.
Significance
This work advances the field by enabling distribution-level control over generative outputs without retraining or fine-tuning, addressing a key limitation of existing inverse design methods. It opens new avenues for content creation, personalized content, and complex design tasks, where controlling output distributions is crucial. The inference-only framework significantly reduces computational costs and enhances flexibility, making it practical for industrial deployment. By bridging the gap between pointwise inverse design and distributional control, this approach sets a new standard for generative model controllability.
Technical Contribution
The primary technical innovation is extending Loss-Guided Diffusion to the conditional distribution matching setting, integrating a pretrained score network with a fast, differentiable conditional sampler. The algorithm employs single-step sampling to enable efficient gradient estimation, avoiding the high memory costs of unrolled multi-step diffusion backpropagation. Theoretical analysis guarantees bounds on gradient bias and variance, ensuring stability in high-dimensional spaces. The method operates entirely at inference time, requiring no additional training, and can adapt to diverse target distributions, including mixtures and low-rank manifolds.
Novelty
This is the first work to incorporate distributional control directly into the reverse diffusion process via inference-time optimization, bypassing the need for retraining or fine-tuning. Unlike prior methods relying on target-specific loss functions or retraining generative models, this approach achieves flexible, target-agnostic distribution matching through a plug-and-play inference framework, representing a significant leap in controllable generative modeling.
Limitations
- The method's performance depends on the quality of the pretrained models; biases or inaccuracies in the base diffusion or conditional sampler can limit target matching fidelity. Handling extremely complex or high-dimensional target distributions remains challenging, especially under limited conditional sampling efficiency.
- Computational costs, while reduced, are still significant for very high-dimensional tasks, particularly when multiple conditional samples are needed at each step. Further optimization of the sampler architecture and gradient estimation strategies is necessary for broader industrial application.
Future Work
Future research will focus on designing more efficient, scalable conditional samplers, extending the framework to multi-modal and multi-objective scenarios, and integrating adaptive sampling strategies. Additionally, exploring theoretical guarantees under broader assumptions and applying this approach to real-world tasks like personalized content generation, drug design, and multimodal synthesis will be key directions.
AI Executive Summary
The rapid development of generative models, especially diffusion-based architectures like DDPM and score-based diffusion, has revolutionized content synthesis. However, controlling the output distribution remains a challenge, particularly when the goal is not a single point but a complex, user-defined distribution. Traditional inverse design methods excel at point targets but falter with distributional goals, limiting their flexibility in real-world applications.
This paper introduces a novel framework called Conditional Distribution Matching (CDM), which shifts the focus from pointwise targets to entire output distributions. The core idea is to find an input x such that the conditional distribution P(Y|X=x) matches a user-specified target G(Y). To achieve this without retraining the generative model, the authors propose MLGD-F, an inference-time algorithm that combines a pretrained score-based diffusion model with a fast, differentiable conditional sampler. This setup allows for efficient gradient estimation through single-step conditional sampling, enabling the optimization of input x to align the conditional distribution with G.
The approach is validated across multiple tasks. Synthetic experiments with Gaussian mixtures demonstrate an 11Γ speedup over traditional methods, with improved accuracy in high-dimensional spaces. In MNIST digit rotation, the method successfully generates images with target orientation distributions, achieving lower SWD scores compared to baselines. In large-scale image editing with Stable Diffusion, the algorithm maintains image quality while matching complex distributional targets, such as demographic or age distributions, with significant improvements in MMD metrics.
This work significantly broadens the control capabilities of generative models, allowing for flexible, distribution-level content creation without additional training. It addresses key limitations of existing inverse design techniques, offering a scalable, inference-only solution suitable for industrial and creative applications. Future directions include enhancing sampler efficiency, extending to multi-modal tasks, and deploying in real-world scenarios such as personalized content and drug discovery, promising a new era of controllable generative AI.
Deep Dive
Abstract
Generative models are powerful tools for sampling from a learned distribution $\mathcal{P}(Y \mid X)$, and inverse-design methods invert this map to find an input $x$ that produces a desired point output $y^*$. However, many design goals are naturally distributional rather than pointwise, incorporating the inherent uncertainty of $Y$ and targeting a specific form for it, a task not addressed by standard inverse design. To address this issue we introduce Conditional Distribution Matching (CDM), a new inverse-design problem class in generative modeling: given a joint distribution $\mathcal{P}(X, Y)$ and a target distribution $\mathcal{G}(Y)$, find an input $x^*$ whose induced conditional distribution $\mathcal{P}(Y \mid X = x^*)$ matches $\mathcal{G}$. We formally define two variants: Conditional Distribution Matching Sampling (CDMS) and Conditional Distribution Matching Optimization (CDMO). To solve these problems, we propose MLGD-F (Matching-Loss Guided Diffusion with a Fast inner sampler), a plug-and-play inference-time algorithm that combines a pretrained score-based diffusion model with a pretrained fast conditional sampler, requiring no additional training or fine-tuning. By leveraging single-step conditional sampling, MLGD-F enables tractable gradient computation, making the estimation of $\mathcal{P}(Y \mid X)$ both memory-efficient and computationally lightweight. We validate MLGD-F on synthetic benchmarks, structured image transformations, and generative editing optimization, demonstrating reliable recovery of inputs whose conditional distributions match diverse user-specified targets, including discrete mixtures and continuous low-rank supports.