---
title: Random Average Sampling
url: https://www.emergentmind.com/topics/random-average-sampling
type: topic
---

# Random Average 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 [2401.03683].

For shift-invariant and quasi shift-invariant spaces, the function space $V_K(D)$ is the span of translates of finitely many compactly supported generators $d_i$, localized by restriction to $K$. The resulting space has finite dimension $d$ and supports a basis in which any $f\in V_K(D)$ can be expressed as $f = \sum_{i,s,t} c_i(s,t)\, d_i(\cdot - x_s, \cdot - y_t)$ 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 $\mathbb{R}^d$,
\[
X_t = \int_{\mathbb{R}^d} f(t-s)\, dL(s),
\]
where $L$ is a Lévy basis [1902.01255].

## 2. Sampling Regimes and Operators

Random average sampling can be performed over:

- **Deterministic (non-random) sampling sets**: For sequences $\Gamma_n\subset\mathbb{Z}^d$, classic conditions such as the Følner property ensure ergodicity and convergence of sample means and covariances. For example, $\Gamma_n = (-n, n]^d\cap\mathbb{Z}^d$ forms a canonical Følner sequence [1902.01255].

- **Random sampling sets**: An i.i.d. or ergodic field $(Y_t)_{t\in\mathbb{Z}^d}$ with $P(Y_0=1)>0$ specifies random inclusion in the sampling domain, yielding $\Gamma_n = \{ t \in [-n,n)^d \cap \mathbb{Z}^d : Y_t = 1\}$. As $n\to\infty$, sampling density and ergodicity are preserved almost surely.

- **Randomized sampling positions in continuous domains**: Sampling locations $(u_j, v_k)$ in $K$ are drawn i.i.d. from a suitably regular probability density [2401.03683, 2111.14321].

The sampling operator $S:V_K(D)\to\mathbb{C}^{n\times m}$ is
\[
S(f) = \{ S_{j,k}(f) \}_{j=1,\ldots,n;\; k=1,\ldots,m}.
\]
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., [2401.03683, 2111.14321]) prove that with high probability (exponentially close to one in the number of samples), the samples encode any $f$ in a “well-conditioned” subset of $V_K(D)$ in a stable manner, i.e., there exist constants $A, B>0$ such that
\[
A\|f\|_{L^{p,q}(K)} \leq \|\{S_{j,k}(f)\}\|_{\ell^{p,q}} \leq B \|f\|_{L^{p,q}(K)}.
\]
These are uniform over all signals in the subspace (unit ball or constrained as in $V_{p,q}^*(D, \delta)$). The keys to such inequalities are:

- Covering number bounds for the compact unit sphere in $V_K(D)$ or its subsets.
- Uniform application of Bernstein’s or Bennett's inequalities to arrays of centered, Lipschitz, bounded variance random variables
\[
Y_{j,k}(f) := |(f*w)(u_j, v_k)| - \int_K p(u,v)|(f*w)(u,v)| du dv
\]
with $E[Y_{j,k}(f)] = 0$ for each $f$ [2401.03683].

Sample complexity bounds follow: the number of samples needed is $O(d)$, where $d$ 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 [2401.03683]).

## 4. Reconstruction and Inversion

When the sampling operator $S$ is injective and well-conditioned, reconstruction is given by explicit matrix inversion methods:
- Form the “moment matrix” $M_{(j,k),(i;s,t)} = (d_i*w)(u_j-x_s, v_k-y_t)$.
- Under injectivity and suitable sample counts, $M$ is invertible with high probability and the coefficients $c$ can be stably recovered by solving $M c = S(f)$ via e.g., $c = (M^*M)^{-1} M^* S(f)$ [2401.03683].

The signal $f$ is then reconstructed as
\[
f(u,v) = \sum_{j = 1}^n \sum_{k=1}^m (f*w)(u_j, v_k)\, h_{j,k}(u,v)
\]
for explicitly computable $h_{j,k}\in V_K(D)$. 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 [2111.14321]. For modest sample sizes, reconstruction errors may be driven to numerical precision (see Table in [2111.14321] and simulations in [2401.03683]).

| Sample size $(n,m)$ | $L^\infty$-error    | $L^1$-error       | $L^2$-error      |
|---------------------|--------------------|-------------------|------------------|
| (5,5)               | $2.3\times10^{-15}$| $7.7\times10^{-15}$| $8.0\times10^{-30}$|
| (7,7)               | $2.2\times10^{-15}$| $3.9\times10^{-15}$| $2.5\times10^{-30}$|
| (10,10)             | $1.3\times10^{-15}$| $3.2\times10^{-15}$| $1.5\times10^{-30}$|

*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 $X_t = \int_{\mathbb{R}^d} f(t-s)\, dL(s)$, random average sampling is used to estimate mean and autocovariance statistics on randomly or deterministically sampled lattices $\Gamma_n \subset \mathbb{Z}^d$. Under appropriate regularity and mixing conditions:

- The sample mean $\bar{X}_{\Gamma_n}$ satisfies a central limit theorem:
\[
\sqrt{|\Gamma_n|}(\overline{X}_{\Gamma_n} - E[X_0]) \xrightarrow{d} N(0, \sigma^2_\infty)
\]
where $\sigma^2_\infty$ is an explicit sum of weighted covariances [1902.01255].

- The sample autocovariance $\widehat{\gamma}_{\Gamma_n}(h)$ 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., $(\mu - \Delta)X = dL$ in $\mathbb{R}^3$), where random average sampled means yield consistent, asymptotically normal estimators for $\mu > 0$ [1902.01255].

### 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 $\mathbb{R}^d$, LCA groups, and higher-dimensional product spaces. The methodology covers both univariate and multivariate settings [2401.03683, 2111.14321].

## 6. Extensions and Outlook

Random average sampling generalizes naturally to:
- Arbitrary product group settings (e.g., mixed spatial-temporal data indexed by $\mathbb{R}$ and $\mathbb{R}^d$).
- 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 [2102.04688].

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 [2102.04688].

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 [2111.14321, 2401.03683]. 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.

Source: https://www.emergentmind.com/topics/random-average-sampling