Random Average Sampling

• Random average sampling is a probabilistic framework that encodes and reconstructs functions from local convolution-type measurements taken at random sample locations.
• It employs rigorous probabilistic inequalities and explicit inversion formulae to ensure stable, often exact, recovery with sample counts scaling linearly with the local signal dimension.
• Applications of this method span signal processing, spatial random fields, and stochastic PDEs, providing efficient alternatives to traditional deterministic grid sampling.

Random average sampling is a modern probabilistic framework for encoding and reconstructing functions or fields from convolution-type measurements (local averages) taken at random sample locations. It has found prominent applications in signal processing, stochastic fields, shift-invariant function spaces, and the analysis of spatial random fields—particularly in regimes where traditional deterministic grids are unsuitable or computationally expensive. Random average sampling guarantees stable and, in appropriate cases, exact reconstruction with high probability, provided sufficient sampling density relative to the signal complexity, and is underpinned by rigorous probabilistic inequalities and explicit inversion formulae in finite-dimensional settings.

1. Mathematical Framework for Random Average Sampling

Let $(G_1, +)$ and $(G_2, +)$ be locally compact abelian groups (LCA groups), each equipped with Haar measures. The ambient signal space is typically a mixed Lebesgue space $L^{p,q}(G_1\times G_2)$ with norm

$$\|f\|_{L^{p,q}(G_1\times G_2)} = \left(\int_{G_1} \left( \int_{G_2} |f(u,v)|^q dv \right)^{p/q} du \right)^{1/p}.$$

Sampling is restricted to a compact domain $K = K_1 \times K_2 \subset G_1\times G_2$. An averaging (window) function $w\in L^\infty(G_1 \times G_2)$ of bounded support $W\subset K$ defines the average (convolutional) sampling operation: $$S_{j,k}(f) := (f*w)(u_j, v_k) = \int_{G_1 \times G_2} f(u', v')\, w(u_j - u', v_k - v') du' dv'$$ at sample points $(u_j, v_k)\in K$ drawn independently according to a probability density $p(u,v)$ which is bounded away from zero and infinity (Garg et al., 2024).

For shift-invariant and quasi shift-invariant spaces, the function space $(G_2, +)$0 is the span of translates of finitely many compactly supported generators $(G_2, +)$1, localized by restriction to $(G_2, +)$2. The resulting space has finite dimension $(G_2, +)$3 and supports a basis in which any $(G_2, +)$4 can be expressed as $(G_2, +)$5 for separated shift sets.

Random average sampling also plays a critical role in random fields: for example, in Lévy-driven spatial moving average fields on $(G_2, +)$6,

$(G_2, +)$7

where $(G_2, +)$8 is a Lévy basis (Berger, 2019).

2. Sampling Regimes and Operators

Random average sampling can be performed over:

• Deterministic (non-random) sampling sets: For sequences $(G_2, +)$9, classic conditions such as the Følner property ensure ergodicity and convergence of sample means and covariances. For example, $L^{p,q}(G_1\times G_2)$0 forms a canonical Følner sequence (Berger, 2019).
• Random sampling sets: An i.i.d. or ergodic field $L^{p,q}(G_1\times G_2)$1 with $L^{p,q}(G_1\times G_2)$2 specifies random inclusion in the sampling domain, yielding $L^{p,q}(G_1\times G_2)$3. As $L^{p,q}(G_1\times G_2)$4, sampling density and ergodicity are preserved almost surely.
• Randomized sampling positions in continuous domains: Sampling locations $L^{p,q}(G_1\times G_2)$5 in $L^{p,q}(G_1\times G_2)$6 are drawn i.i.d. from a suitably regular probability density (Garg et al., 2024, Garg et al., 2021).

The sampling operator $L^{p,q}(G_1\times G_2)$7 is

$L^{p,q}(G_1\times G_2)$8

Essential properties derived include injectivity, well-conditioned inversion, and the concentration of sampling norms about their expected value with explicit probability estimates.

3. Probabilistic Sampling Inequalities

Recent results (e.g., (Garg et al., 2024, Garg et al., 2021)) prove that with high probability (exponentially close to one in the number of samples), the samples encode any $L^{p,q}(G_1\times G_2)$9 in a “well-conditioned” subset of $$\|f\|_{L^{p,q}(G_1\times G_2)} = \left(\int_{G_1} \left( \int_{G_2} |f(u,v)|^q dv \right)^{p/q} du \right)^{1/p}.$$0 in a stable manner, i.e., there exist constants $$\|f\|_{L^{p,q}(G_1\times G_2)} = \left(\int_{G_1} \left( \int_{G_2} |f(u,v)|^q dv \right)^{p/q} du \right)^{1/p}.$$1 such that

$$\|f\|_{L^{p,q}(G_1\times G_2)} = \left(\int_{G_1} \left( \int_{G_2} |f(u,v)|^q dv \right)^{p/q} du \right)^{1/p}.$$2

These are uniform over all signals in the subspace (unit ball or constrained as in $$\|f\|_{L^{p,q}(G_1\times G_2)} = \left(\int_{G_1} \left( \int_{G_2} |f(u,v)|^q dv \right)^{p/q} du \right)^{1/p}.$$3). The keys to such inequalities are:

• Covering number bounds for the compact unit sphere in $$\|f\|_{L^{p,q}(G_1\times G_2)} = \left(\int_{G_1} \left( \int_{G_2} |f(u,v)|^q dv \right)^{p/q} du \right)^{1/p}.$$4 or its subsets.
• Uniform application of Bernstein’s or Bennett's inequalities to arrays of centered, Lipschitz, bounded variance random variables

$$\|f\|_{L^{p,q}(G_1\times G_2)} = \left(\int_{G_1} \left( \int_{G_2} |f(u,v)|^q dv \right)^{p/q} du \right)^{1/p}.$$5

with $$\|f\|_{L^{p,q}(G_1\times G_2)} = \left(\int_{G_1} \left( \int_{G_2} |f(u,v)|^q dv \right)^{p/q} du \right)^{1/p}.$$6 for each $$\|f\|_{L^{p,q}(G_1\times G_2)} = \left(\int_{G_1} \left( \int_{G_2} |f(u,v)|^q dv \right)^{p/q} du \right)^{1/p}.$$7 (Garg et al., 2024).

Sample complexity bounds follow: the number of samples needed is $$\|f\|_{L^{p,q}(G_1\times G_2)} = \left(\int_{G_1} \left( \int_{G_2} |f(u,v)|^q dv \right)^{p/q} du \right)^{1/p}.$$8, where $$\|f\|_{L^{p,q}(G_1\times G_2)} = \left(\int_{G_1} \left( \int_{G_2} |f(u,v)|^q dv \right)^{p/q} du \right)^{1/p}.$$9 is the (local) dimension of the subspace, multiplied by explicit stability constants and inversely proportional to prescribed accuracy and confidence (see Theorems 3.2–3.3 in (Garg et al., 2024)).

4. Reconstruction and Inversion

When the sampling operator $K = K_1 \times K_2 \subset G_1\times G_2$0 is injective and well-conditioned, reconstruction is given by explicit matrix inversion methods:

• Form the “moment matrix” $K = K_1 \times K_2 \subset G_1\times G_2$1.
• Under injectivity and suitable sample counts, $K = K_1 \times K_2 \subset G_1\times G_2$2 is invertible with high probability and the coefficients $K = K_1 \times K_2 \subset G_1\times G_2$3 can be stably recovered by solving $K = K_1 \times K_2 \subset G_1\times G_2$4 via e.g., $K = K_1 \times K_2 \subset G_1\times G_2$5 (Garg et al., 2024).

The signal $K = K_1 \times K_2 \subset G_1\times G_2$6 is then reconstructed as

$K = K_1 \times K_2 \subset G_1\times G_2$7

for explicitly computable $K = K_1 \times K_2 \subset G_1\times G_2$8. This result extends to more general “coercive” settings.

In infinite-dimensional mixed Lebesgue spaces, consistent approximation is achieved by finite-dimensional projections and leveraging stability and convergence rates that vanish as sample count increases (Garg et al., 2021). For modest sample sizes, reconstruction errors may be driven to numerical precision (see Table in (Garg et al., 2021) and simulations in (Garg et al., 2024)).

Sample size $K = K_1 \times K_2 \subset G_1\times G_2$9	$w\in L^\infty(G_1 \times G_2)$0-error	$w\in L^\infty(G_1 \times G_2)$1-error	$w\in L^\infty(G_1 \times G_2)$2-error
(5,5)	$w\in L^\infty(G_1 \times G_2)$3	$w\in L^\infty(G_1 \times G_2)$4	$w\in L^\infty(G_1 \times G_2)$5
(7,7)	$w\in L^\infty(G_1 \times G_2)$6	$w\in L^\infty(G_1 \times G_2)$7	$w\in L^\infty(G_1 \times G_2)$8
(10,10)	$w\in L^\infty(G_1 \times G_2)$9	$W\subset K$0	$W\subset K$1

This suggests that practical implementations can yield essentially perfect recovery under the model assumptions.

5. Applications and Central Limit Theorems

Lévy-Driven Moving Average Random Fields

For strictly stationary fields $W\subset K$2, random average sampling is used to estimate mean and autocovariance statistics on randomly or deterministically sampled lattices $W\subset K$3. Under appropriate regularity and mixing conditions:

• The sample mean $W\subset K$4 satisfies a central limit theorem: $W\subset K$5 where $W\subset K$6 is an explicit sum of weighted covariances (Berger, 2019).
• The sample autocovariance $W\subset K$7 also satisfies a joint multivariate CLT under higher-order moment assumptions.

A canonical application is to parameter estimation in stochastic partial differential equations (e.g., $W\subset K$8 in $W\subset K$9), where random average sampled means yield consistent, asymptotically normal estimators for $$S_{j,k}(f) := (f*w)(u_j, v_k) = \int_{G_1 \times G_2} f(u', v')\, w(u_j - u', v_k - v') du' dv'$$0 (Berger, 2019).

Shift-Invariant and Spline Spaces

With appropriate averaging kernels and quasi shift-invariant generators, random average sampling enables stable and exact encoding of functions in spline, Sobolev, and general shift-invariant subspaces over $$S_{j,k}(f) := (f*w)(u_j, v_k) = \int_{G_1 \times G_2} f(u', v')\, w(u_j - u', v_k - v') du' dv'$$1, LCA groups, and higher-dimensional product spaces. The methodology covers both univariate and multivariate settings (Garg et al., 2024, Garg et al., 2021).

6. Extensions and Outlook

Random average sampling generalizes naturally to:

• Arbitrary product group settings (e.g., mixed spatial-temporal data indexed by $$S_{j,k}(f) := (f*w)(u_j, v_k) = \int_{G_1 \times G_2} f(u', v')\, w(u_j - u', v_k - v') du' dv'$$2 and $$S_{j,k}(f) := (f*w)(u_j, v_k) = \int_{G_1 \times G_2} f(u', v')\, w(u_j - u', v_k - v') du' dv'$$3).
• Fields on lattices, spatial or temporal stochastic processes, and operator-theoretic settings.
• Random batch methods in molecular dynamics simulations for efficient thermal average computation by randomizing interaction terms (Ye et al., 2021).

In the context of random fields, central limit theorems established under random and deterministic sampling guarantee statistical inference validity for quantities beyond means: higher-order autocovariances and moment-based estimators are covered. Variations include random batch partitionings of computational sums or forces, which accelerate high-dimensional inference and simulation (Ye et al., 2021).

A plausible implication is that random average sampling schemes, due to their minimal dependency on regular sample grids and robust probabilistic guarantees, provide powerful tools for large-scale, data-adaptive signal processing, statistical estimation, and numerical simulation.

7. Numerical Performance and Empirical Observations

Numerical studies using B-spline tensor products and compactly supported averaging kernels demonstrate that for moderate sample counts, exact reconstruction can be achieved essentially to machine precision. Empirical reconstruction errors decay rapidly with sample count, matching the theoretical predictions for exponential concentration and stable injectivity of the sampling operator (Garg et al., 2021, Garg et al., 2024). The required sample size scales linearly with the local dimension of the signal space and only logarithmically with failure probability.

Such alignment of empirical and theoretical performance highlights the practicality and reliability of random average sampling schemes in computational and applied settings where randomization affords computational and statistical advantages.