Diffusion Differentiable Resampling

• Diffusion differentiable resampling is a technique leveraging reverse-time diffusion processes to create smooth, gradient-friendly mappings from Gaussian reference distributions to complex targets.
• It employs ensemble score estimation with weighted particles to achieve unbiased and consistent resampling within sequential Monte Carlo frameworks.
• Empirical results demonstrate superior statistical accuracy and computational scalability compared to traditional resampling methods in filtering tasks.

Diffusion differentiable resampling refers to a class of resampling algorithms, most prominently within sequential Monte Carlo (SMC) and particle filtering, that utilize diffusion-based stochastic processes to construct smooth, pathwise-differentiable mappings from easy-to-sample reference distributions to complex target distributions. By leveraging the theory of diffusion processes and denoising diffusion probabilistic models (DDPMs), these methods achieve unbiased or consistent resampling, enable effective gradient-based parameter optimization, and outperform traditional differentiable resampling approaches in both statistical accuracy and efficiency (Andersson et al., 11 Dec 2025, Wan et al., 21 Jul 2025).

1. Motivating Problems in Differentiable Resampling

In SMC frameworks, the resampling step converts a weighted sample ensemble $\{(w_i, X_i)\}_{i=1}^N$ into an unweighted set that still approximates the posterior $\pi$. Classical schemes such as multinomial, stratified, or systematic resampling are not pathwise differentiable, since the mapping from continuous weights to resampled indices is discrete—making $\partial X_i^*/\partial\theta$ undefined when weights depend on model parameters $\theta$. This breaks end-to-end gradient flow required for variational-EM or deep learning on latent state-space models.

Several prior remedies exist:

• Expectation-based sensitivity estimators: Differentiation under reparameterizable noise, suffering high variance.
• Soft relaxations (e.g., Gumbel–Softmax, Soft resampling, OT-based relaxations): Introduce bias, instability, or high computational cost.

Diffusion differentiable resampling bypasses these limitations by constructing a smooth transport map from a tractable reference $\mu$ (typically Gaussian) to the target mixture $\pi$ via a reverse-time diffusion SDE, yielding pathwise differentiability as all randomness is reparameterizable (Andersson et al., 11 Dec 2025).

2. Core Methodology: Diffusion-Based Transport Maps

The foundation is a reverse-time diffusion process that transports $\mu$ to $\pi$. Consider a forward SDE (e.g., Ornstein-Uhlenbeck):

$$dX(t) = b^2\nabla\log\mu(X(t))\,dt + \sqrt{2}\,b\,dW(t), \quad X(0)\sim \pi$$

whose marginal converges to $\mu$. The time-reversed SDE transporting $\mu$ back to $\pi$ is:

$$dU(t) = b^2 \left[ -\nabla\log\mu(U(t)) + 2\nabla\log p_{T-t}(U(t)) \right] dt + \sqrt{2}\,b\,dW(t), \quad U(0)\sim\mu$$

Here, $p_t$ denotes the marginal of the forward process at time $t$. Importantly, $\nabla\log p_{T-t}$ is unknown in general.

To circumvent this, an ensemble score estimator is constructed using weighted particles:

$$s_N(x, t) = \sum_{i=1}^N \alpha_i(x, t) \nabla\log p_{t \mid 0}(x | X_i)$$

$$\alpha_i(x, t) = \frac{w_i\,p_{t \mid 0}(x | X_i)}{\sum_{j=1}^N w_j\,p_{t \mid 0}(x | X_j)}$$

This provides a self-normalized, importance-weighted estimate of the score at $x$, using all available weighted samples. Substituting this approximation produces the practical reverse SDE:

$$d\widetilde U(t) = b^2\left[ -\nabla\log\mu(\widetilde U(t)) + 2s_N(\widetilde U(t), T-t) \right] dt + \sqrt{2}\,b\,dW(t), \quad \widetilde U(0)\sim\mu$$

Diffusion resampling thus renders the entire mapping from $(\{X_i, w_i\}, \text{Gaussian noise})$ to $\{X_i^*\}$ smooth and differentiable with respect to $\{X_i, w_i\}$ and, by chain rule, any parameter $\theta$ on which weights depend (Andersson et al., 11 Dec 2025).

3. Algorithmic Implementation and Differentiability

Simulation of the reverse SDE is typically performed via an Euler–Maruyama discretization. For each output particle, a trajectory is integrated, at each time step updating by the reference drift, estimated ensemble score, and injected Gaussian noise:

1. Initialize $U_{i,0} \sim \mu$
2. For $k=1,\dots,K$:
   • Compute step size $\Delta_k$
   • Calculate ensemble scores $s_N$
   • Draw $\xi \sim N(0, 2b^2 \Delta_k I)$
   • Update:

   $$U_{i, t_k} = U_{i, t_{k-1}} + b^2\left[ -\nabla\log\mu(U_{i, t_{k-1}}) + 2 s_N(U_{i, t_{k-1}}, T-t_{k-1}) \right]\Delta_k + \xi$$

3. Output $X_i^* = U_{i,T}$

All operations are composed of differentiable primitives (log-density, score, linear algebra, Gaussian sampling with fixed noise seeds), so $\partial X_i^*/\partial\theta$ can be computed via automatic differentiation.

This subroutine replaces classical discrete resampling in SMC or particle filtering pipelines, preserving all statistical guarantees and enabling gradient flow for parameter learning (Andersson et al., 11 Dec 2025).

4. Statistical Properties: Consistency and Bias

The key theoretical result is that, under standard Lipschitz and dissipativity assumptions, the error between the ideal reverse SDE law and the ensemble-score–based approximation $\mathsf W_1(\widetilde q_t, q_t)$ can be made arbitrarily small as the number of input particles $N \to \infty$ and integration time $T \to \infty$:

$$\mathsf W_1(\widetilde q_t, q_t) \leq e^{b^2(C_{\rm ref}-2\,C_p)\,t}\, \mathsf W_1(p_T, \mu) + 2 b^2 N^{-r} \int_0^t C_e(T-\tau) e^{b^2(C_{\rm ref}-2C_p)(t-\tau)} d\tau$$

With suitable $N$ and $T$, the output law of the diffusion resampling converges to the target $\pi$, implying unbiasedness and consistency in the limit (Andersson et al., 11 Dec 2025).

In practice (finite $N$, $K$, $T$), empirical results show variance and bias competitive with or superior to entropic OT and Gumbel–Softmax relaxations, especially in high-dimensional, multimodal, or highly nonlinear filtering tasks.

5. Practical Performance and Experimental Results

Diffusion differentiable resampling has been evaluated on both classical and neural SSM scenarios:

• Gaussian mixture importance resampling: Achieves sliced Wasserstein distance (SWD) of $0.80\pm0.21$, outperforming entropic OT at $\varepsilon=0.3$ ($0.84\pm0.22$).
• Linear Gaussian SSM: Log-likelihood error $2.61\pm2.08$, closely matching the best OT results; competitive in average KL divergence and estimation error.
• Neural Lotka–Volterra: RMSE $1.05\pm0.39$, substantially better than all alternatives (OT, Soft, Gumbel).
• Vision-based pendulum tracking: SSIM $0.866\pm0.039$, PSNR $21.4\pm1.23$, exceeding or matching classical soft or OT-based resamplers.

Computationally, the method is parallel in $N$ (particle size), with cost $O(TN)$; it is more parallelizable and scalable than entropic OT ($O(N^2 \log N/\varepsilon)$), while matching the $O(N)$ scaling of soft relaxations for moderate $N$ (Andersson et al., 11 Dec 2025).

6. Extensions and Related Approaches

Several connections exist to other diffusion-based differentiable sampling frameworks:

• DiffPF adopts parameterized conditional diffusion processes for flexible posterior sampling in particle filtering, training DDPMs on predicted particle trajectories and observations. All reverse diffusion steps are smoothly differentiable by reparameterization, enabling equal-weight (no importance-weight) posterior approximations. DiffPF demonstrates strong gains in multimodal visual odometry and localization benchmarks, providing a direct application of diffusion-based differentiable resampling in practical SSMs (Wan et al., 21 Jul 2025).
• Example-based diffusion samplers for point sets (Doignies et al., 2023) leverage similar differentiable reverse diffusion constructions to learn complex spatial point set distributions given only sample data, with full differentiability enabling gradient-based refinement under custom error metrics.
• Diffusing differentiable representations (diffreps) (Savani et al., 9 Dec 2024) adapt probability flow ODE formulations of diffusion processes to the parameter space of neural representations, enforcing differentiable sampling of structured objects such as images and NeRFs, subject to parameter space manifold constraints.

While the above methods instantiate differentiable resampling for distinct types of model spaces or tasks (probabilistic state sequences, geometric point sets, differentiable neural parameterizations), all share a reliance on diffusion-based transport and gradient-preserving stochastic mapping.

7. Implementation, Complexity, and Practical Guidance

Key implementation points include:

• Reference $\mu$ selection: A mean-reverting Gaussian matched to current posteriors (in SMC) is preferable for stability; its log-score is explicit.
• SDE solver: Euler–Maruyama is sufficient, but higher-order exponential integrators or probability-flow ODE schemes may reduce bias.
• Gradient stability: Adjoint or reversible–adjoint solvers are recommended for robust automatic differentiation along diffusion trajectories.
• Computational scaling: $O(TN)$ steps per batch, with parallel evaluation of ensemble scores; cost dominated by kernel score evaluations but still lower than OT-based relaxations for large $N$.
• Extensions: Alternative ensemble score approximations (kernelized, learned), offline variational inference, smoothing, or integration with forward–backward Gibbs bridges for improved convergence.

The pathwise differentiability of diffusion-based transport not only enables efficient parameter learning but also yields empirically robust and statistically accurate particle resampling in diverse filtering contexts (Andersson et al., 11 Dec 2025, Wan et al., 21 Jul 2025).