Projected Generative Diffusion Models

• Projected Generative Diffusion Models are a generative framework that enforces hard constraints on samples via explicit projection operators to maintain physical and statistical integrity.
• The approach reformulates the reverse diffusion process as a constrained optimization problem, interleaving standard updates with tailored projections for convex, nonconvex, and ODE-related constraints.
• PGDM has demonstrated robust performance in applications such as motion synthesis, inverse problems, and remote sensing by balancing computational trade-offs with strong convergence guarantees.

Projected Generative Diffusion Models (PGDM) constitute a class of generative frameworks that explicitly enforce hard constraints on the samples generated by diffusion processes. Unlike standard diffusion models that maximize sample fidelity in an unconstrained setting, PGDM systematically steers the generative trajectory to satisfy specified requirements—ranging from convex feasibility, nonconvex or obstacle-based constraints, differential equation (ODE) compliance, or the maintenance of prescribed coarse-grained statistics. This is achieved by recasting the standard reverse-diffusion process as a constrained optimization at each step, leveraging explicit projection operators to ensure that all intermediate and final outputs remain within the desired set, or respect given physical, statistical, or measurement-induced structure (Christopher et al., 2024, Tsuruyama, 3 Jan 2026, Ravasi, 8 Jan 2025, Zhang et al., 8 Nov 2025, Zheng et al., 27 May 2025).

1. Mathematical Formulation and Core Algorithms

Let $q(x_t \mid x_0)$ denote the forward SDE or discrete Markov kernel in a generative diffusion model. In unconstrained sampling, the reverse process seeks to maximize $\sum_{t=T}^1 \log q(x_{t-1} \mid x_t)$, or equivalently, minimize the negative log-likelihood, by simulating the reverse chain from noise to data. PGDM introduces a feasible set $\mathcal{C} \subset \mathbb{R}^d$, and formulates sampling as

$$\min_{x_T, \dots, x_0} \ F(x_{T:0}) = \sum_{t=T}^1 -\log q(x_{t-1} \mid x_t) \quad \text{subject to} \ x_t \in \mathcal{C} \ \forall t$$

The key step is to interleave each reverse update (Langevin, ODE, or discrete kernel) with a projection:

$$x_t^{i} = \Pi_{\mathcal{C}}(\tilde{x}), \quad \text{where} \ \tilde{x} = x_t^{i-1} + \gamma_t s_\theta(x_t^{i-1}, t) + \sqrt{2\gamma_t} \varepsilon, \; \varepsilon \sim \mathcal{N}(0, I)$$

where $\Pi_{\mathcal{C}}$ denotes the Euclidean or otherwise chosen projection onto $\mathcal{C}$ (Christopher et al., 2024). The number of inner projection steps $M$ and the step size schedule $\{\gamma_t\}$ are hyperparameters.

For constraints on aggregate statistics (e.g., blockwise proportions), the projection is performed in probability space by minimizing a Kullback–Leibler divergence subject to block-mass constraints, based on an information-theoretic Lyapunov function (Tsuruyama, 3 Jan 2026). For measurement-conditioned inverse problems with linear or convolutional operators, the projection takes the form of a Wiener-filter correction or pseudo-inverse update, derived from the marginalization of posterior conditional distributions (Ravasi, 8 Jan 2025, Zheng et al., 27 May 2025).

2. Projection Operators and Constraint Classes

The projection operator is central to PGDM. For constraints of the form $\mathcal{C} = \{x \mid g(x) \leq 0\}$, projection reduces to

$$\Pi_{\mathcal{C}}(x) = \arg\min_{y \in \mathcal{C}} \|y - x\|_2^2$$

Efficient solvers exist for common cases:

• Convex polytopes/balls/affine sets: quadratic programming.
• Nonconvex constraints (e.g., obstacle avoidance): interior-point methods.
• ODE and trajectory constraints: closed-form or custom mappings, such as snapping object centroids to an ODE-predicted trajectory (Christopher et al., 2024).
• Discrete aggregates (block masses, porosity, class proportions): projection via KL minimization and rescaling of mass per block (Tsuruyama, 3 Jan 2026).

PGDM thus generalizes to problems where the constraint set is convex, nonconvex, or defined implicitly (e.g., as the set of solutions to a physical law or measurement equation).

3. Theoretical Guarantees and Convergence

Under convexity of both the negative log-density and the constraint set, projected Langevin/PGDM dynamics admit guarantees analogous to unconstrained Stochastic Gradient Langevin Dynamics (SGLD). Explicitly, projection after each step provably decreases (in expectation) the constraint violation error, and the final sample is constrained almost surely as step size $\gamma_t \to 0$ (Christopher et al., 2024). For block-mass or coarse-grained control, the information-theoretic potential $V_\delta$ acts as an approximate Lyapunov function, decreasing after each projected step and providing stability even under small leakage between blocks (Tsuruyama, 3 Jan 2026).

In measurement-guided scenarios, when the likelihood is approximated via a Gaussian model, the PGDM guidance term leverages the pseudo-inverse of the measurement operator. This correction is shown to outperform adjoint-residual methods (DPS) in terms of SNR in inverse problems, with only a moderate computational overhead (Ravasi, 8 Jan 2025). Theoretical analysis of DMILO-PGD (PGDM with intermediate layer optimization and projected gradient descent) demonstrates robust recovery guarantees under restricted isometry-like conditions on the measurement operator (Zheng et al., 27 May 2025).

4. Application Domains and Empirical Results

PGDM approaches have demonstrated effectiveness across a spectrum of domains:

• Physics-informed video and motion synthesis: Enforces compliance with ODEs governing object motion (e.g., gravity-constrained trajectories), yielding zero constraint violation and competitive FID (Christopher et al., 2024).
• Path/trajectory planning: Generates collision-free paths in obstacle-rich environments with feasibility guarantees and path lengths comparable to conditional baselines, succeeding where prior batch-sampling methods fail to achieve exact feasibility (Christopher et al., 2024).
• Materials design: Achieves exact morphometric constraints (e.g., porosity) differentiating from postprocessing approaches that degrade visual quality (Christopher et al., 2024).
• Explicit macroscopic statistics control: Enforces blockwise or classwise proportions in image generation, with metrics (block-mass error, potential $V_\delta$) constrained to prescribed tolerances and minimal loss in pixel-level fidelity (Tsuruyama, 3 Jan 2026).
• Geophysical inverse problems: PGDM (pseudo-inverse guided) surpasses vanilla and DPS-based diffusion solvers in seismic interpolation and impedance inversion benchmarks, resulting in higher SNR, improved uncertainty quantification, and better alignment with physical measurements (Ravasi, 8 Jan 2025).
• Remote sensing: Land surface temperature downscaling: Physically guided PGDM (SEB-based) achieves state-of-the-art RMSE and SSIM over several benchmarks (e.g., Landsat_CN20), with inherent stochasticity enabling robust uncertainty quantification and strong correlation between estimated and true error (Zhang et al., 8 Nov 2025).
• General inverse problems: DMILO-PGD recovers signals in image inpainting, super-resolution, and nonlinear deblurring across CelebA, FFHQ, LSUN, and ImageNet, consistently outperforming baselines on LPIPS, PSNR, SSIM, and FID (Zheng et al., 27 May 2025).

5. Implementation Strategies and Computational Trade-Offs

PGDM methods wrap pretrained diffusion samplers (score-based networks or standard DDPMs). The computational overhead is dominated by projection steps:

• Cheap projections (e.g., mask shifts, porosity adjustment, aggregate rescaling): lead to minor overhead (20–50% increase per sample).
• Expensive projections (e.g., CPU-based quadratic programming, conjugate gradient solvers for pseudo-inverse updates): can dominate runtime, though these steps are sometimes required only from certain ($t_0$) timesteps onwards, with early iterations using standard updates (Christopher et al., 2024, Ravasi, 8 Jan 2025).
• Intermediate Layer Optimization: DMILO-PGD avoids backpropagation through the full reverse chain by treating inner activations as optimization variables, substantially alleviating memory usage and enabling layerwise sparse corrections (Zheng et al., 27 May 2025).

Projection solvers are selected according to the structure of $\mathcal{C}$: QP for polyhedral constraints, closed-form for ODE alignment, sorting-based or block-wise rescaling for mass constraints, and iterative linear algebra for measurement-driven pseudo-inverse guidance.

6. Extensions, Open Problems, and Evaluation Metrics

Natural extensions of PGDM include:

• Generalizing from explicit spatial block partitions to learned or semantic coarse-grained representations via autoencoders or neural feature maps (Tsuruyama, 3 Jan 2026).
• Integration with continuous-time score-based SDEs and corresponding analysis of Fokker–Planck stability under coarse-grained projections (Tsuruyama, 3 Jan 2026).
• Embedding PGDM into Schrödinger-bridge and entropy-regularized optimal transport formulations (Tsuruyama, 3 Jan 2026).
• Accelerating pseudo-inverse computations by learned solvers and adopting adaptive projection schedules (Ravasi, 8 Jan 2025).
• Extending to nonlinear measurement operators or non-Euclidean constraints via local linearization or custom score-matching approaches (Ravasi, 8 Jan 2025, Zheng et al., 27 May 2025).
• Using Lyapunov-style information-theoretic potentials $V, V_\delta$ as macroscopic evaluation metrics, in addition to FID or IS, to assess global statistical fidelity (Tsuruyama, 3 Jan 2026).

7. Comparative Summary of PGDM Variants

Variant	Constraint Type	Key Algorithmic Feature	Representative Application
Hard Euclidean/ODE	Convex, nonconvex, ODE	Explicit projection in sample space	Physics-informed motion, materials design
KLD/Block-mass	Coarse-grained stats	KL-projection in probability/block-mass space	Class proportion control, checkerboard images
Pseudo-inverse (PI)	Measurement-guided	Wiener-filter/CG solve per step, Jacobian-based	Seismic/interpolation inverse problems
DMILO-PGD	Diffusion-range + sparse	ILO for efficient projection, PGD on data term	Super-resolution, blind deblurring
Physically-guided	Physical priors/SEB	Dual-branch diffusion network with geophysical encoder	LST downscaling, remote sensing

Each PGDM instantiation adapts the projection mechanism to the structure of $\mathcal{C}$ or the necessary fidelity constraints, either in the data, feature, or measurement space, while preserving—or minimally compromising—sample quality relative to unconstrained baselines (Christopher et al., 2024, Tsuruyama, 3 Jan 2026, Ravasi, 8 Jan 2025, Zhang et al., 8 Nov 2025, Zheng et al., 27 May 2025).