Generative Prior-Guided Optimization

• The paper introduces a novel framework that integrates deep generative models as priors into optimization, reducing search space and enhancing solution realism.
• It employs architectures such as feedforward ReLUs, GANs, diffusion, and neural operators to enforce regularization in inverse problems and design tasks while ensuring convergence.
• The approach demonstrates improved theoretical guarantees, reduced sample complexity, and versatile applications across imaging, physical design, and creative generation.

Generative Prior-Guided Optimization refers to a class of methodologies in which the optimization process for an inverse or design problem is explicitly constrained or regularized by a generative model trained to capture the distributional or structural properties of the target data. Rather than relying on hand-crafted or analytical priors, generative prior-guided optimization leverages the expressive power of deep neural networks—such as GANs, diffusion models, or neural operators—trained as surrogates for the true data distribution. The approach is increasingly central for addressing high-dimensional inverse problems, structured signal recovery, engineering design, image manipulation, property-guided molecular generation, and more, where conventional priors (e.g., sparsity, smoothness) are too restrictive or inadequate.

1. Foundations and Model Structures

Generative prior-guided optimization relies on deep generative models to encode the manifold of natural signals or feasible designs. Various architectures have been employed:

• Feedforward Neural Networks with ReLU Activations: In phase retrieval and compressive sensing settings, the prior is implemented as the range of a multi-layer fully-connected network with ReLU activations and no bias: $$G(x) = \operatorname{relu}(W_d \cdots \operatorname{relu}(W_1 x))$$, mapping a low-dimensional latent $x \in \mathbb{R}^k$ to the natural signal space $y \in \mathbb{R}^n$ (Hand et al., 2018, Huang et al., 2018).
• Pretrained Generative Adversarial Networks and Diffusion Models: For high-level semantics and image restoration, pretrained GANs (e.g., DCGAN, StyleGAN2, BigGAN) or DDPM-style diffusion models are used as explicit priors in both forward and inverse tasks, enabling complex structural constraints (Pan et al., 2020, Fei et al., 2023, Yang et al., 2 Dec 2024).
• Generative Neural Operators: In physical inverse design, neural operators learn distributions over physical system parameters, acting as generative priors for PDE-constrained optimization (Yang et al., 28 Apr 2025).
• Score-Based and Latent-Diffusion Models: Score models enable posterior sampling via Langevin dynamics in the latent or intermediate feature space, regularizing solutions to lie on the learned manifold (Daras et al., 2022, Blasingame et al., 11 Feb 2025, Yao et al., 5 Jul 2024).

The key advantage is that the generative model reduces the search to a subspace of realistic or feasible instances, facilitating both strong regularization and high representational capacity.

2. Optimization Formulations with a Generative Prior

Classical inverse problems are commonly formulated as:

$$\underset{x}{\rm min}~ \mathcal{L}_{\text{data}}(A x, y) + \lambda \cdot \mathcal{R}(x),$$

where $\mathcal{L}_{\text{data}}$ is a data-fidelity term and $\mathcal{R}$ is a regularizer encoding prior knowledge (e.g. $\ell_1$ norm for sparsity).

In generative prior-guided methods, the solution is constrained to the range of a generative model $G$: $$\underset{z}{\rm min}~ \mathcal{L}_{\text{data}}(A G(z), y),$$ where $z$ is the low-dimensional latent code. In more expressive recent frameworks: $$\underset{z, \theta}{\rm min}~ \mathcal{L}_{\text{data}}(A G(z; \theta), y) + \mathcal{R}(z, \theta),$$ where $\theta$ denotes fine-tuned generator parameters (Pan et al., 2020).

Variations exist depending on data and task:

• Measurement Constraints: Quadratic, phaseless, or linear (compressive) measurements, e.g., $f(z) = \frac{1}{2} \| |A G(z)| - b \|^2$ for phase retrieval (Hand et al., 2018).
• Property Guidance: For multi-objective design, guidance via property gradients is added in the generative sampling process, e.g., $$g(x_t) = \epsilon_\theta^*(x_t, t) + \sum_{i=1}^{m} \lambda_{i,t} \nabla f_i(x_t)$$ in PROUD, with per-objective gradient terms encoding Pareto optimality (Yao et al., 5 Jul 2024).
• Hybrid Architectures: The generator can be surgically truncated at test time to expand latent space and improve representation error (Smedemark-Margulies et al., 2021), or regularized via a neural score model on intermediate features (Daras et al., 2022).

Optimization is typically performed by gradient descent in latent space, possibly with sign-flip or subgradient strategies to address nonconvexity and symmetries (Huang et al., 2018).

3. Theoretical Guarantees and Geometric Properties

A significant body of work demonstrates that generative prior-guided optimization can have fundamentally better properties than classical approaches:

• Benign Nonconvexity: Under Weight Distribution Condition (WDC) and Range Restricted Concentration Property (RRCP), the landscape for empirical risk over the latent space is globally well-behaved: any local minima are within a small neighborhood of the ground truth or a negative scalar multiple, and spurious local minima are statistically absent (Hand et al., 2018).
• Sample Complexity: For deep generative priors, theoretical sample complexity can match the intrinsic data dimension (latent space), i.e., $m = \Omega(k d^2 \log n)$, as opposed to the $O(k^2 \log n)$ required by sparse methods (Hand et al., 2018, Huang et al., 2018).
• Provable Convergence: For compressive sensing, subgradient descent in the latent space is established to converge to ground truth under realistic noise and initialization (Huang et al., 2018).
• Posterior Sampling and Fast Mixing: Langevin dynamics in score-guided frameworks are proven to mix rapidly under random weight assumptions, providing sample-based uncertainty quantification along with optimization (Daras et al., 2022).

For more advanced use cases, constraints are enforced in disentangled style spaces of semantic generative models (e.g., StyleGAN2's $\mathcal{W}^+$) to enable semantically controlled reconstructions with theoretical guarantees on stability (Kelkar et al., 2021).

4. Practical Applications and Algorithmic Variants

Generative prior-guided optimization has been applied in a wide range of domains:

• Inverse Imaging and Compressed Sensing: Improved image recovery, robust to noise and incomplete measurements, with empirical superiority over classical sparse recovery (DPR, SGILO, AIPO) (Hand et al., 2018, Huang et al., 2018, Daras et al., 2022, Liu et al., 2022).
• Image Restoration and Manipulation: Fine-grained manipulation (jittering, morphing, inpainting), colorization, and super-resolution via optimization over both latent codes and generator parameters, with discriminator-feature regularization maintaining fidelity to natural images (Pan et al., 2020, Fei et al., 2023).
• Interactive Generative Design: Latent-space navigation and subspace blending, guided by human inputs and Bayesian preference learning, enabling user-driven controlled synthesis (Hin et al., 2019).
• Dataset Distillation and Generalization: Improved dataset condensation by optimizing latent inputs to a pre-trained generator, yielding synthetic data with enhanced generalizability across unseen architectures and high resolutions (Cazenavette et al., 2023).
• Creative Generation and Concept Mixing: Diffusion Prior constraints integrated with VLM feedback to systematically optimize for novel or hybrid concepts in text-to-image synthesis, bypassing degenerate convergence into existing classes (Richardson et al., 2023).
• Topology and Engineering Design: Topology optimization using conditional diffusion models with low-cost physical field approximations, complemented with iterative classic optimization (SIMP) for manufacturability and performance (Giannone et al., 2023).
• Reinforcement Learning and Planning: Diffusion planners with learnable, behavior-regularized priors in latent space to efficiently generate high-reward trajectories under offline RL constraints (2505.10881).

5. Comparative Analysis and Methodological Trade-offs

Generative prior-guided optimization yields strong empirical and theoretical advantages over traditional sparsity- or analytic-prior-based approaches:

Method	Sample Complexity	Convergence Guarantees	Expressivity
Classical $\ell_1$	$O(k^2 \log n)$	No general global guarantees	Low–Moderate
Generative Prior	$O(k d^2 \log n)$	Benign geometry, provable convergence	High

• Representation Error: Fixed low-dimensional latent priors can induce high representation error for out-of-distribution images. Generator surgery increases latent dimensionality at test time, reducing this error (Smedemark-Margulies et al., 2021).
• Optimization Heuristics: Amortized strategies that gradually ramp up prior strength (e.g., via $\lambda$ scheduling in AIPO) improve convergence robustness to initializations over direct nonconvex gradient descent (Liu et al., 2022).
• Sample-Based vs. Deterministic Optimization: Langevin or posterior sampling (score-based) approaches provide uncertainty quantification and help avoid spurious minima; deterministic gradient-based methods are efficient but are point estimates (Daras et al., 2022).
• Guidance in Generative Planning: In RL and molecular generation, directly optimizing prior distributions in latent space is computationally more efficient and less prone to OOD error than repeated inference-time search or multi-candidate selection (2505.10881, Blasingame et al., 11 Feb 2025, Yao et al., 5 Jul 2024).

6. Extensions, Limitations, and Future Directions

Recent trends highlight several research directions and open challenges:

• Multiobjective and Property-Guided Generation: Explicitly encoding Pareto optimality in sampling by dynamically adjusting diffusion gradients enables principled trade-off discovery and diversity in objective space (Yao et al., 5 Jul 2024).
• Adaptive Prior Design and Non-Parametric Priors: Non-parametric and task-adapted priors, carefully optimized for distributional invariance under interpolation or function composition, address mismatches seen in standard parametric choices (Singh et al., 2019).
• Constraint Satisfaction and Semantic Control: Fine-grained control via partial freezing of semantically mapped latent space variables allows robust transfer of known features from prior images in ill-posed reconstructions (Kelkar et al., 2021).
• Generative Priors for First-Order Physical Optimization: The Deep Physics Prior unifies neural-operator surrogate modeling and generative modeling, supporting first-order optimization in physical design with unknown priors (Yang et al., 28 Apr 2025).
• Limitations: Challenges include model misspecification (if the generative prior does not cover the true data manifold), optimization nonconvexity when moving beyond idealized conditions, and sensitivity to the accuracy of the pretrained generator or surrogate operator.

A plausible implication is that as generative models grow more expressive and better capture complex distributions, the scope and fidelity of generative prior-guided optimization will continue to expand, particularly in disciplines where incorporating high-level domain knowledge and semantic constraints is essential.

7. Summary Table of Key Research Contributions

Reference	Application Domain	Generative Prior Integration	Key Result
(Hand et al., 2018)	Phase Retrieval	Deep network range, latent code optimization	Global geometry; $O(k d^2 \log n)$ samples
(Huang et al., 2018)	Compressive Sensing	Gradient/subgradient in latent, benign landscape	Provable convergence; $O(k)$ samples
(Pan et al., 2020)	Image Restoration	GAN prior with generator fine-tuning	High PSNR/SSIM; semantic manipulations
(Smedemark-Margulies et al., 2021)	Compressive Sensing	Generator surgery (increased latent dimension at test time)	Lower representation error, improved OOD
(Kelkar et al., 2021)	Semantic Imaging	Style-based, partially constrained latent optimization	Structure-/expression-preserving recovery
(Giannone et al., 2023)	Topology Optimization	Conditional diffusion + kernel relaxation, plus SIMP	Fast, high-quality, manufacturable designs
(Daras et al., 2022)	Inverse Problems	Latent-space score model + Langevin mixing	Fast convergence, strong artifacts removal
(Liu et al., 2022)	Inverse Imaging	Amortized MAP (gradual prior annealing)	Robust, initialization-insensitive MAP
(Cazenavette et al., 2023)	Dataset Distillation	Pretrained generator latent optimization	Improved generalization, high-res scaling
(Richardson et al., 2023)	Creative Synthesis	Diffusion Prior + VLM adaptive constraint growth	Unique and hybrid visual concept creation
(Yao et al., 5 Jul 2024)	Multi-objective Generation	Diffusion with per-step Pareto guidance in denoising	High FID + Pareto set coverage/image, prot.
(Yang et al., 28 Apr 2025)	Physical Inverse Optimization	Neural operator prior constrained optimization	Physically realistic, fast inverse solvers
(2505.10881)	RL Diffusion Planning	Latent prior guidance; latent regularized objective	Efficient high-value, OOD-robust plans
(Blasingame et al., 11 Feb 2025)	Guided Generation Theory	Unified greedy/posterior and end-to-end guidance	Explicit compute/accuracy trade-off
(Yang et al., 2 Dec 2024)	Driving Scene Synthesis	Video diffusion prior in iterative 3D model fitting	Artifact-free, extrapolated scene synthesis

These methodologies collectively define the state of generative prior-guided optimization as a rapidly developing paradigm for structured, high-dimensional, and controllable inverse and design problems across science and engineering.