Coefficients-Preserving Sampling (CPS)

• Coefficients-Preserving Sampling (CPS) is a framework that retains key empirical and model parameters despite challenges like finite sampling, noise, or high-dimensionality.
• It optimizes sampling strategies in applications such as Polynomial Chaos Expansion and PDE simulations, reducing sample complexity and maintaining coefficient integrity.
• CPS is also applied in neural network interpolation and generative modeling, enabling robust recovery and accurate performance under perturbations and constraints.

Coefficients-Preserving Sampling (CPS) encompasses a set of methodologies designed to ensure that the essential coefficients—be they empirical parameters in time series, expansion coefficients in polynomial spaces, model parameters in neural subspaces, flow solver correction terms, or stochastic process descriptors—are accurately preserved or recoverable, despite complexities introduced by finite sampling intervals, high-dimensionality, constraints, noise injection, and evolving data distributions. CPS appears across disciplines such as stochastic process inference, uncertainty quantification, PDE simulation, signal processing, machine learning, and image/video synthesis, serving as a principled framework for extracting, maintaining, or reconstructing latent coefficients without distortion due to sampling artifacts, model mismatch, or algorithmic operations.

1. Foundations in Stochastic Process Inference

CPS first emerges in the context of empirical estimation of Kramers–Moyal (KM) coefficients for stochastic processes sampled at discrete intervals (Anteneodo et al., 2010). For a process with continuous dynamics $P(x, t | x_0, 0)$, the intrinsic $k$-th order KM coefficient is $$D_k(x_0) = \lim_{\tau \to 0} \mathcal{D}_k(x_0, \tau)$$, where the finite-time estimate is

$$\mathcal{D}_k(x_0, \tau) = \frac{1}{k! \, \tau} \langle [x(\tau) - x_0]^k \rangle = \frac{1}{k! \, \tau} \int dx \, P(x, \tau | x_0, 0)\,[x - x_0]^k$$

A CPS approach must compare the sampling interval $\tau$ against the process’s correlation/relaxation timescale. If $\tau$ is too large, finite-time estimates degenerate to independence-limit polynomials dominated by stationarity rather than dynamics, thereby obfuscating the recovery of drift and diffusion coefficients. Conversely, when noise is negligible or sampling resolution is low, deterministic limits may overshadow genuine stochastic features. Exact finite-$\tau$ expressions for linear and quadratic dynamics, such as $$\mathcal{D}_1(x_0, \tau) = -(x_0 - b/a) (1 - e^{-a \tau})/\tau$$, offer practical inverting mechanisms for CPS: tuning $\tau$ or post-processing empirical coefficients ensures preservation of the dynamical parameters.

2. Sparse Recovery and Polynomial Chaos Expansion

In compressive sensing and uncertainty quantification, CPS is tightly linked to coherence-controlled sampling strategies in Polynomial Chaos (PC) expansions (Hampton et al., 2014). For a sparse basis $\{\psi_k(\xi)\}$, the coherence parameter $\mu(Y) = \sup_{k,\xi} |w(\xi)\psi_k(\xi)|^2$ controls the number of Monte Carlo samples needed to recover coefficient vectors $c$ by $\ell_1$-minimization:

$N \geq C\, \mu(Y)\, s\, \log P$

CPS mandates adopting sampling measures—standard, asymptotic (e.g., Chebyshev or Hermite ball), or coherence-optimal (MCMC-sampled densities proportional to $f(\xi)B(\xi)^2$)—which minimize $\mu(Y)$, thus reducing sample complexity and preserving recoverability of the true coefficients. Numerical studies and PDE/ODE applications demonstrate that coherence-optimal sampling sharply transitions recovery probability and dramatically reduces error, particularly in high-order/high-dimensional regimes. CPS here entails constructing or selecting sampling strategies with theoretical guarantees for coefficient identifiability.

3. CPS in Multidimensional PDE Simulation

In the simulation of PDEs with random coefficients, CPS refers to unbiased sampling strategies that maintain the influence of all stochastic coefficients in high-dimensional settings (Blanchet et al., 2018). For a parabolic PDE with random drift $\mu(x)$ expanded as $\mu(x) = \sum \lambda_i n^{-q} V_i \psi_i(x)$, unbiased estimators are constructed via multilevel Monte Carlo schemes with randomization and antithetic pairing:

• Antithetic MLMC increments $\Delta_n$ cancel bias
• A geometric random variable randomizes resolution levels
• A second outer randomization debiases nonlinear functionals of the solution
• Rough path theory provides uniform-in-dimension control of variance and bias, overcoming the challenge of exponential growth in error terms

CPS in this context guarantees that all spatially heterogeneous coefficients are preserved in the sampling distribution, enabling robust uncertainty quantification in thermodynamics, porous media flow, finance, and engineering.

4. Signal Recovery in Shift-Invariant and Evolving Spaces

The dynamical sampling framework extends CPS to shift-preserving operators in finitely generated shift-invariant function spaces (Aguilera et al., 2019). Let $$V = S(\mathcal{G}) = \operatorname{span}\{T_k g : g \in \mathcal{G}, k \in \mathbb{Z}^d\}$$ and $L$ a shift-preserving operator ($L T_k = T_k L$). Necessary and sufficient conditions for CPS—ensuring that integer translates of iterates $\{L^j f_i\}$ form a frame for $V$—are established via fiberization and range operator analysis. Spectral gap properties and frame generator sets are required for coefficients-preserving recovery via time evolution. This framework supports robust reconstruction under evolving dynamics, relevant to sensor networks, time-varying signal processing, and inverse problems.

5. CPS in Neural Parameter Subspace Interpolation

Compressed Parameter Subspaces (Datta et al., 2022) represent a neural network instantiation of CPS, where endpoint parameters $\{\theta_i\}$ trained on shifted input distributions are compressed so that convex linear combinations $\theta(\alpha) = \sum_i \alpha_i \theta_i$ (with $\sum \alpha_i = 1$) remain low-loss and robust under distributional shift. Pairwise cosine regularization during joint training enforces proximity, ensuring that sampled coefficients (ensemble or interpolated) map back to high-performing models across adversarial, backdoor, permutation, and stylization perturbations. CPS in this setting describes architectures and procedures that preserve the interpolation integrity of weights, supporting applications in robust ensembling, task shift adaptation, and continual learning without catastrophic forgetting.

6. CPS for Constrained Posterior Sampling in Diffusion Models

In diffusion-based time series generation (Narasimhan et al., 16 Oct 2024), Constrained Posterior Sampling modifies DDIM sampling by projecting the posterior mean estimate onto a multi-constraint set after each denoising update:

$$\bar{z} = \arg\min_{z} \Big\{ \tfrac{1}{2}\|z-\bar{z}\|^2_2 + \gamma(t)\,\Pi(z) \Big\}$$

with $\Pi(z) = \sum_{i=1}^n \max(0, f_{c_i}(z))$ enforcing hard constraints and $\gamma(t)$ tuning the strength across diffusion steps. CPS here is hyperparameter-free, scalable (up to $\sim100$ constraints), and empirically yields substantial improvements in Fréchet distance, TSTR, DTW, and SSIM metrics for realism and constraint adherence. Applications include synthetic time-series generation for power grid stress-testing and privacy-preserving data synthesis.

7. CPS in Flow Solvers and Reinforcement Learning for Generative Models

In high-fidelity Euler equation solvers, CPS appears as a robustification of the HLL-family Riemann solvers via pressure- and Mach-dependent anti-diffusion coefficients (Gogoi et al., 14 Nov 2024). By adaptively adjusting correction coefficients $\delta_2$, $\delta_3$, $\delta_n$, schemes maintain sharp resolution of contact and shear waves at low Mach numbers while suppressing instabilities (e.g., carbuncle) at high Mach numbers:

$$\delta_{2,\text{new}} = \delta_{3,\text{new}} = \left[\frac{\bar{a}}{\bar{a}+|u_n|}\right] f_{p1}, \qquad \delta_{n,\text{new}} = [1 - f(M)] f_{p1}$$

Pressure scaling $f_{p1}$ and Mach scaling $f(M)$ ensure coefficients preservation in appropriate flow regimes, verified by spectral and matrix stability analysis.

In RL-based flow matching generative models (Wang et al., 7 Sep 2025), CPS is formulated to ensure that at every timestep, the sum of deterministic and noise coefficients adheres strictly to a scheduler, thereby preventing excess stochasticity artifacts. The sampling update enforces:

$$x_{t-\Delta t} = (1-(t-\Delta t))\hat{x}_0 + (t-\Delta t)\cos(\eta\pi/2)\hat{x}_1 + (t-\Delta t)\sin(\eta\pi/2)\epsilon$$

with all coefficients quadratic-summed to match the scheduled noise level. CPS yields cleaner outputs, more reliable reward modeling, and faster, more stable RL convergence compared to SDE-based sampling, particularly in image and video synthesis.

8. Implications, Applications, and Future Directions

CPS principles pervade an expanding array of scientific and engineering contexts:

• Ensuring sampling rate selection faithfully retains latent process coefficients in time series inference
• Optimizing sample distributions for compressed sensing and uncertainty quantification
• Preserving spatially heterogeneous coefficients in simulation of stochastic PDEs
• Enforcing reconstruction and coverage in evolving or translation-invariant function spaces
• Enabling robust interpolation and adaptation in neural parameter subspaces and hypernetworks
• Enforcing hard domain constraints in synthetic time series generation
• Tuning numerical solver fluxes for physical accuracy across Mach regimes
• Prescribing exact stochasticity schedules in generative modeling pipelines for reinforcement learning

Ongoing research targets refinement of CPS for minimal-sample recovery, extension to non-orthogonal polynomial bases, adaptive projection mechanisms for complex constraints, hybrid stochastic-deterministic samplers with exact coefficient preservation, and domain-specific adaptations in physical modeling, secure data synthesis, and continual learning scenarios.

A plausible implication is that as models and data streams become more complex and multi-modal, the strategic preservation of key coefficients—via the tailored sampling, interpolation, and constraint satisfaction strategies articulated in CPS methodologies—will remain central to both mathematical guarantees and practical utility in inference, simulation, and generative modeling.