Diffusion-based Null-space Projection

• Diffusion-based null-space projection is a method that projects iterative diffusion trajectories onto feasible manifolds derived from physical models, data manifolds, or acquisition processes.
• It employs techniques such as local tangent subspace approximation and analytical null-space enforcement to ensure measurement consistency and artifact suppression.
• Empirical results highlight significant improvements in planning success and imaging metrics, demonstrating enhanced robustness in high-dimensional generative tasks.

Diffusion-based null-space projection comprises algorithmic techniques for constraining the trajectory of iterative generative or inverse solvers, based on diffusion models, so that the trajectory remains feasible with respect to constraints arising from physical models, data manifolds, or data acquisition processes. These projection strategies are integrated into the reverse process of diffusion models to eliminate components that would otherwise drive the iterates off the target manifold or violate explicit equality constraints. The null-space or off-manifold directions eliminated at each step are determined by either analytical properties of the measurement operator (in inverse problems) or locally approximated tangent spaces derived from offline datasets (in planning). This family of projection methods includes data-driven local tangent projections, analytical null-space enforcement via pseudo-inverses, and manifold-aware subspace filtering, unified by their aim to preserve feasibility, robustness, and realism in high-dimensional generative procedures.

1. The Guidance Gap and Manifold Deviation in Guided Diffusion

Standard guided diffusion methods, e.g., classifier-guided policies for planning or measurement consistency guidance in inverse problems, apply a gradient-driven correction to the reverse diffusion dynamics. In the context of generative planning, an unconditional diffusion model learns a score network $\epsilon_\theta$ and a return predictor $\mathcal{J}_\phi$; at generation time, classifier guidance is imposed via an added gradient term: $$\tilde{p}_\theta(\tau^{i-1}|\tau^i) = \mathcal{N}\big(\mu_\theta(\tau^i) + \omega\,\Sigma^i\,\nabla_{\tau}\mathcal{J}_\phi(\mu_\theta(\tau^i)), \Sigma^i\big)$$ However, there exists a fundamental discrepancy between the "ideal" energy-guided term $\nabla_{\tau^i}\mathcal{J}_t(\tau^i)$ and the surrogate MSE-learned term $$\nabla_{\tau^i}\mathcal{J}_\phi^{\mathrm{MSE}}(\tau^i)$$, given by: $$\Delta_{\rm guidance}(\tau^i) = \|\nabla_{\tau^i}\mathcal{J}_t(\tau^i) - \nabla_{\tau^i}\mathcal{J}_\phi^{\mathrm{MSE}}(\tau^i)\|_2$$ A lower bound holds for some $\tau^i$: $$\Delta_{\rm guidance}(\tau^i) \ge \frac{c}{\sqrt{1-\alpha_i}\sqrt{d}}$$ with $c>0$ independent of dimension $d$. This bound implies that, particularly in high-dimensional spaces, guidance steps inevitably push the generated sample off the intrinsic data manifold $\mathcal{M}_i$, generating so-called "manifold deviation" and resulting in unphysical or infeasible outputs, such as state-sequence violations in trajectories or perceptually implausible reconstructions in imaging (Lee et al., 1 Jun 2025).

2. Local Subspace Approximation and Projection Techniques

To counteract off-manifold deviation, several manifold-based null-space projection methodologies employ a local approximation of the data manifold at each reverse step, typically using low-rank subspace techniques derived from the available dataset:

• Trajectory Planning (LoMAP): For a noisy sample $\tau^{i-1}$, a local low-rank tangent subspace is estimated:
  • Compute a "denoised" surrogate via Tweedie's formula using the current score network.
  • Retrieve $k$ nearest neighbors from an offline corpus under cosine distance.
  • Apply the forward diffusion process to these neighbors to match noise level $i-1$.
  • Assemble a covariance matrix from these $k$ samples, perform eigendecomposition, and retain the principal subspace spanning a prescribed fraction $\lambda$ of local variance.
  • The resulting $U \in \mathbb{R}^{d \times r}$ columns form an approximate tangent basis.
• Projection Operator: Any sample $z \in \mathbb{R}^d$ is projected onto the tangent subspace by $\mathcal{P}_{\mathcal{T}}(z) = UU^\top z$, thereby purging all normal components that would otherwise move $z$ off the manifold (Lee et al., 1 Jun 2025).
• Manifold-Aware Gradient Filtering (DiffStateGrad): In inverse imaging problems, the guidance gradient is projected onto the dominant singular subspace (SVD/ PCA, retaining a specified energy fraction $\tau$), ensuring measurement consistency and artifact suppression. This is done by decomposing $x_t$ into $Z_t$, computing $U_{t,k}$, and filtering the gradient as $g_t^{\mathrm{proj}} = U_{t,k}U_{t,k}^\top g_t$ at each diffusion step (Zirvi et al., 2024).

3. Analytical Projection for Explicit Constraints: Classical Null-Space Methods

When explicit linear (or nonlinear) equality constraints are present, as in inverse problems or dynamics-constrained planning, analytical null-space projections are defined by the (pseudo-)inverse of the operator or the Jacobian of the constraint map:

• Linear Inverse Problems (DDNM, Plug-and-Play): For measurement model $y = A x + n$, the affine projector onto $A x = y$ is

$$P_{\mathrm{null}}(u) = u + A^\top (A A^\top)^{-1} (y - A u)$$

or equivalently, $A^\dagger y + (I - A^\dagger A)u$, which separates range and null components. The entire reverse diffusion chain alternates between denoising, null-space projection, and stochastic sampling, with strict consistency $A x_{t} = y$ enforced at every step (Wang et al., 2022, Wang et al., 11 Sep 2025).

• Nonlinear Dynamics Constraints (PAD-TRO): For sequential state-control trajectories, the nonlinear equality constraints encoding system dynamics are enforced via a first-order projection: $x^+ = x - J_c(x)^\top (J_c(x)J_c(x)^\top)^{-1}c(x)$ where $J_c(x)$ is the Jacobian of the constraints. This update is inserted at each denoising step in the reverse diffusion chain, yielding zero dynamic feasibility error and provable constraint satisfaction under full row rank assumptions (Chen et al., 6 Oct 2025).
• Hybrid Data-Consistency Modules: For ill-posed inverse problems, hybrid projections combine several plug-and-play null-space or proximal projections (e.g., generalized alternating projection (GAP) and half-quadratic splitting (HQS)) in a convex combination, scheduled across the diffusion chain to balance hard and soft constraint enforcement (Wang et al., 11 Sep 2025).

4. Model-Free and Data-Driven Null-Space Projection

For problems where the underlying manifold is unknown and only observed via samples (without an explicit measurement operator), model-free null-space projection is attainable by leveraging score-based diffusions:

• Metric Projection via Score-Based Diffusion: The estimator of the metric projection $\pi_\mathcal{M}(y)$ is constructed by:
  1. Learning a score function $b(y) = \nabla_y \log h(y)$ by minimizing a denoising-score objective over noise-perturbed dataset samples.
  2. Simulating a reverse diffusion SDE with drift $b(y)$, initialized at $y$, until the trajectory intersects the data manifold $\mathcal{M}$.
  3. The output approximates $\pi_\mathcal{M}(y)$; in high dimensions, the induced posterior concentrates exponentially near the true projection, yielding consistent recovery without access to a measurement operator (Christensen et al., 27 Oct 2025).

This statistical null-space projection exploits high-dimensional geometry, with the learned score field enforcing a geometry-driven contraction toward the data manifold.

5. Algorithmic Patterns: Pseudocode Templates

Null-space projection modules are realized as modular subroutines within standard diffusion sampling loops. A representative structure for LoMAP-based manifold projection, followed by more general pseudocode for inverse problems, is as follows:

for i = M, M−1,…,1: mu = mu_theta(tau^i) g = grad_tau J_phi(mu) tau^{i−1} ~ N(mu + omega * Sigma^i * g, Sigma^i) tau^{i−1} = LoMAP(tau^{i−1}; dataset, k) # enforce initial state observation

for t = T, ... ,1: x_{0|t} = (x_t - sqrt(1−α_t) * eps_theta(x_t, t)) / sqrt(α_t) x_{0|t}' = A_pinv * y + (I - A_pinv * A) * x_{0|t} mu = (1/ sqrt(α_t)) * (x_t - (β_t/ sqrt(1−α_t)) * eps_theta(x_t, t; x0=x_{0|t}')) x_{t-1} ~ N(mu, σ_t^2 I)

6. Empirical Performance and Applications

Diffusion-based null-space projection achieves substantial improvements in critical metrics across planning and inverse problems:

• Planning: LoMAP reduces artifact ratio in Maze2D from ≈50% to ≈10% and raises success rates in high-dimensional AntMaze from single-digit percentages to >30% (non-hierarchical) and >85% (hierarchical) (Lee et al., 1 Jun 2025).
• Inverse Imaging: Null-space-projected diffusion models (DDNM, RN-SDE) attain strict measurement consistency, higher PSNR/SSIM, and lower FID/Loss compared to unconstrained or naïvely guided schemes. For example, DDNM matches or exceeds state-of-the-art zero-shot scores in super-resolution, deblurring, and inpainting tasks (Wang et al., 2022, Guo et al., 2024). DiffStateGrad and its relatives provide robustness to step-size/noise perturbations and control worst-case failures by artifact filtering (Zirvi et al., 2024).
• Trajectory Optimization: PAD-TRO eliminates dynamic feasibility error (infinity norm $\approx 10^{-12}$ vs. $10^{-2}$–$10^{-1}$ for unconstrained baselines) and improves trajectory success rates by a factor of 4 in quadrotor planning (Chen et al., 6 Oct 2025).

A summary of representative modules and contexts is provided below:

Application Area	Null-Space Projection Method	Key Manifestation
Offline RL Planning	LoMAP: local tangent projection (PCA/knn)	Manifold-constrained trajectory plans
Inverse Problems	Pseudoinverse range-null projection (GAP, DDNM)	Strict measurement consistency
Imaging Robustness	DiffStateGrad: SVD subspace filtering	Artifact and outlier suppression
Trajectory Opt.	Jacobian-based nonlinear constraint projection	Zero dynamic feasibility error
Manifold Filtering	Score-based geometric projection (no $A$)	Distributional proximity to manifold

7. Theoretical Guarantees and Limitations

In problems with linear measurement operators, null-space projection preserves strict data consistency at every denoising step, as the projection onto the affine set $A x = y$ is enforced exactly via the Moore–Penrose pseudo-inverse. For nonlinear constraints, the projection update yields $c(x^+)=0$ (with $c$ the constraint map) when $J_c(x)$ is full row rank, ensuring exact feasibility (Chen et al., 6 Oct 2025).

In model-free manifold projection, high-dimensional geometric concentration ensures that with overwhelming probability, the sampled output lies exponentially close to the true metric projection onto the data manifold (Christensen et al., 27 Oct 2025).

Limitations include the computational overhead of local SVD or Jacobian inversion at each step, particularly in very high dimensions or long planning horizons, and the need for representative offline datasets for effective local subspace approximation. Several approaches propose adaptive rank or hybrid projections to mitigate efficiency issues (Zirvi et al., 2024, Lee et al., 1 Jun 2025, Chen et al., 6 Oct 2025). The plug-and-play property of these modules allows seamless integration into both flat and hierarchical (multi-stage) diffusion planners or solvers.

In summary, diffusion-based null-space projection enables reliable, data-consistent, and constraint-satisfying solutions in generative and inverse settings by systematically removing off-manifold and infeasibility-inducing components within the probabilistic reverse diffusion process.