Discretization-Invariant Sampling

• Discretization-Invariant Sampling is a framework that guarantees accuracy and statistical consistency irrespective of the sample grid or mesh.
• It employs techniques like Marcinkiewicz–Zygmund bounds, one-sided discretization inequalities, and universal discretization for reliable error control.
• Applications span signal processing, numerical PDEs, operator learning, control systems, and statistical inference, while addressing challenges in sample complexity and adaptive sampling.

Discretization-invariant sampling refers to a diverse set of mathematical frameworks, algorithms, and design principles in which the quality, error, or statistical properties of numerical sampling or discretization are provably independent of—or robust to—the specific sampling mesh, grid, or point set. Across applications in signal processing, numerical analysis, operator learning, and statistical inference, discretization-invariance quantifies sampling schemes whose performance guarantees, statistical expectation, or approximation error are unaffected (or change in a precisely controlled manner) when the discretization pattern or mesh is changed. This paradigm stands in contrast to classical methods that are tuned to specific grids and can suffer dramatic failures under mesh or sampling perturbation.

1. Core Concepts and Definitions

At its core, discretization-invariant sampling addresses the challenge of approximating or reconstructing continuous objects (functions, operators, geometric manifolds, distributions) from finite samples, in a manner where the sample set layout does not fundamentally alter the outcome, up to a provably controlled error. Formal definitions vary by context:

• In numerical approximation and signal recovery, a discretization-invariant sampling set for a function space $X$ on a domain $\Omega$ is one for which the recovery, interpolation, or norm estimation error for all $f \in X$ depends only on intrinsic properties of $X$, not details of the points $\{x_j\}$ (Dai et al., 2021, Limonova et al., 2024).
• In operator learning, a neural operator or network $H$ is discretization-invariant if, for all points sets $X$, the error between the true operator $H[f]$ and its sampled approximation $\hat H^X[f]$ is upper-bounded by a factor that vanishes as the discrepancy of $X$ decreases; this is central to the design of DI-Nets and discretization-invariant deep operator networks (Wang et al., 2022, Zhang et al., 2023).
• In control theory, discretization-invariant design requires that the set of admissible initial states for a sampled-data system decreases monotonically (nesting property) with coarser sampling, independent of the particular sampling interval or method (Schutz et al., 17 Mar 2025).
• In statistics of discrete frequency distributions, invariant sampling refers to statistics (such as certain moments) whose expectation does not depend on the sample size, thereby eliminating dependence on the discretization of the sampling process (Rossi, 2013).

A unifying thread is the abstraction from the particular realization of a sampling grid to intrinsic, mesh-independent properties supported by rigorous constants or invariant moment identities.

2. Mathematical Frameworks for Discretization-Invariant Sampling

Several mathematical frameworks have formalized and extended discretization-invariance:

Norm Discretization and Sampling Recovery

• Marcinkiewicz–Zygmund Discretization: For a $\Omega$0-dimensional subspace $\Omega$1, a set of $\Omega$2 points $\Omega$3 yields a discretization-invariant Marcinkiewicz bound if

$\Omega$4

where $\Omega$5 depend only on $\Omega$6, and structural constants, not on the specific sample locations (Dai et al., 2021).

• One-Sided Discretization Inequalities: Left discretization inequalities (LDI) guarantee that

$\Omega$7

where $\Omega$8 is independent of the sampling pattern. Such inequalities are central to sampling recovery and sparse approximation, allowing discretization-invariant error bounds (Limonova et al., 2024).

• Universal Discretization: A sampling set is universal if it provides MZ or LDI bounds simultaneously for a large family of subspaces, enabling recovery and approximation guarantees for multiple models with a single mesh (Limonova et al., 2024).
• Sharp Sample Complexity: Under entropy number or Nikol’skii-type bounds,

$\Omega$9

suffices for uniform discretization-invariant norm equivalence, demonstrating minimal sampling rates independent of node placement (Dai et al., 2021).

Operator Learning and Neural Architectures

• Discretization-Invariant Networks (DI-Nets): For $f \in X$0 in a Banach space of functions on $f \in X$1, a map $f \in X$2 is discretization-invariant if for any finite sample $f \in X$3, the norm error satisfies

$f \in X$4

where $f \in X$5 is the total variation and $f \in X$6 is the discrepancy of the sampling set. The Koksma–Hlawka inequality underpins these nonasymptotic error bounds, and quasi–Monte Carlo sampling provides low-discrepancy point sets (Wang et al., 2022).

• Discretization-Invariant Operator Networks (BelNet): In BelNet, the architecture includes a projection network $f \in X$7 mapping arbitrary sensor locations and values to coefficients in a learned basis. Operators are universally approximated independently of input or output discretization, allowing for arbitrary, non-aligned sensor grids during both training and inference (Zhang et al., 2023).

3. Algorithms and Constructions Satisfying Discretization-Invariance

Several algorithms and schemes exemplify discretization-invariant principles across multiple domains.

Signal Manifold Discretization

• Registration-Efficient Manifold Discretization (REMD): For a parametrizable signal manifold $f \in X$8, REMD iteratively relocates sample points to minimize the average squared error between the true manifold distance and discretized distance, partitioning the input domain into Voronoi cells and relocating samples to their centroidal projections. The resulting sample set achieves near-optimal registration error, independent of sample arrangement, and can be extended to a joint sampling scheme for multiple class manifolds (Vural et al., 2011).

Numerical PDE Schemes

• Invariant Meshless and Finite Difference Schemes: Using equivariant moving frames, one constructs discretizations of differential equations that exactly preserve Lie group symmetries. The invariantization process acts at the level of both meshless and mesh-based stencils, ensuring that the accuracy and stability properties of the numerical schemes do not depend on the grid arrangement, only on group-theoretic invariants (Bihlo, 2012, Bihlo et al., 2012).
• Evolution-Projection Techniques: For moving mesh schemes, the solution and grid are coevolved under invariant discretizations, then projected (by invariant interpolation) back to a reference grid. This process preserves symmetry and convergence rates irrespective of the instantaneous mesh (Bihlo et al., 2012).

Digital Control and Sampling in Feedback Systems

• $f \in X$9-Step Hold Control Invariance: The $X$0-step hold control invariant set $X$1 for digital control systems shrinks monotonically as $X$2 increases, providing sets of initial conditions from which all state constraints are satisfied for any admissible input, independent of discretization. This property underpins adaptive-sampling feedback policies that guarantee safety for all permissible sampling intervals (Schutz et al., 17 Mar 2025).

Statistical Invariant Moments

• Sampling of Discrete Frequency Distributions: Linear combinations of observed frequency counts, specifically the invariant moments

$X$3

have expectation values depending only on the underlying population probabilities, not the sample size. These discretization-invariant statistics are critical for unbiased inference in population genetics and scale-free systems (Rossi, 2013).

4. Theoretical Guarantees and Error Bounds

Discretization-invariance is underpinned by explicit and sharp quantitative guarantees:

• Norm Equivalence and Recovery Bounds: If a finite-dimensional subspace admits an LDI or MZ discretization, then optimal interpolation or recovery is achieved for any function in the space, with error determined by the best approximation error and constants independent of node placement (Dai et al., 2021, Limonova et al., 2024).
• Discrepancy-Controlled Error: In DI-Nets, the deviation in model outputs is provably bounded by the product of the input function’s smoothness (total variation) and the discrepancy of the sampling set, with quasi-Monte Carlo sequences achieving near-optimal rates $X$4 (Wang et al., 2022).
• Universal Approximation: Discretization-invariant operator networks can approximate any continuous operator on compact function classes, with convergence rate set by sensor count, input variation, and the universal approximation capability of the internal neural basis (Zhang et al., 2023).
• Monotonicity in Sampling Time: In sampled-data control, $X$5-step hold invariants provide a nested family, restoring the desired monotonic containment of admissible sets as sampling is coarsened, and thus invariance under the coarseness of the sampling grid (Schutz et al., 17 Mar 2025).

5. Applications Across Domains

The impact of discretization-invariant sampling spans several fields:

• Signal Analysis and Registration: Efficient manifold discretization and joint sampling enable robust template matching and classification with strong invariance to transformations or sampling non-uniformities (Vural et al., 2011).
• Neural-Field Models and Machine Learning: Discretization-invariant neural networks achieve classification, segmentation, and operator learning accuracy that is unaffected by the type or distribution of spatial samples, thereby generalizing across sampling strategies unseen at training time (Wang et al., 2022, Zhang et al., 2023).
• Numerical Solution of PDEs: Invariant meshless and finite-difference schemes preserve qualitative behavior, accuracy, and symmetry even on irregular, adaptive, or moving meshes. Empirical results confirm reductions in root-mean-square error by an order of magnitude or more relative to noninvariant discretizations (Bihlo, 2012, Bihlo et al., 2012).
• Statistical Inference in Discrete Systems: Invariant moments permit unbiased inference about population structure in genetics and scale-free systems, free from artifacts imposed by changing sample sizes (Rossi, 2013).
• Safety-Critical Control Systems: Discretization-invariant feedback design enables adaptive sampling while guaranteeing inter-sample constraint satisfaction under modeling and discretization error (Schutz et al., 17 Mar 2025).

6. Limitations and Open Problems

While discretization-invariant sampling provides a powerful framework, several limitations and open research problems remain:

• Dependence on Input Regularity and Discrepancy: The strongest theoretical error bounds rely on bounded variation of the input functions and on low-discrepancy sampling; highly clustered or random samples may violate these conditions, rendering the guarantees ineffective (Wang et al., 2022).
• Optimal Sample Complexity: For universal discretization of high-dimensional spaces, sharp sample complexity bounds (e.g., $X$6 for $X$7) are not fully resolved, and reductions under additional structure or coherence are the subject of current research (Limonova et al., 2024, Dai et al., 2021).
• Practical Construction of Low-Discrepancy Sets: While QMC and related algorithms exist, generating optimally low-discrepancy and universal/invariant sampling sets in high dimensions or for large subspace families can be computationally intensive.
• Dynamic and Adaptive Sampling: Universality proofs and invariance properties are typically established for static or a priori known sequences; extending these to fully adaptive data-driven schemes demands further theoretical development (Wang et al., 2022).
• Operator Network Conditioning: For discretization-invariant deep operators, understanding and controlling the sensitivity of the learned projection/interpolation layers to ill-conditioning in sensor arrangements is an active area of theoretical and empirical study (Zhang et al., 2023).

7. Illustrative Examples and Comparative Results

Empirical studies confirm the theoretical advantages of discretization-invariant sampling strategies:

Context	Invariant Method	Comparative Error Reduction
Meshless PDEs	Moving frame-based	RMS error reduction by factor 6–20×, often uniform
Classification	REMD + cmd/dmd	Higher registration/classification rates than random
Neural-Field Nets	DI-Net, BelNet	Robust out-of-sample accuracy under grid shifts
Control Invariant Set	M-step hold sets	Monotonic, safe sets for all sampling intervals
Statistics	Invariant moments	Sample-size-independence for expectation values

Across these domains, discretization-invariant sampling not only delivers theoretical robustness, but also outperforms traditional schemes in terms of error, robustness to mesh perturbations, and generalization, thus establishing it as a central concept in modern computational and statistical analysis (Vural et al., 2011, Bihlo, 2012, Wang et al., 2022, Zhang et al., 2023, Limonova et al., 2024, Schutz et al., 17 Mar 2025).