Random Sampling Inequalities for Polynomials

• Random sampling inequalities for polynomials are precise bounds that measure how well evaluations at random points approximate a polynomial's expected global behavior.
• They utilize hypergraph representations, moment and concentration inequalities, and smoothness parameters to derive explicit exponential tail bounds.
• These techniques have practical applications in combinatorial counting, optimization, random matrix theory, and real algebraic geometry, guiding theoretical and algorithmic advances.

Random sampling inequalities for polynomials quantify, with explicit probabilistic or deterministic bounds, how well the evaluation of a polynomial (or related functional) at finitely many randomly chosen points approximates its expected value, norm, or other global behavior. They form a central class of results in probabilistic combinatorics, high-dimensional analysis, computational mathematics, and theoretical computer science, capturing both the concentration properties of random polynomial systems and the inefficiencies or limitations of random data in polynomial recovery, optimization, and approximation.

1. Moment and Concentration Inequalities for Polynomials

A cornerstone is the framework developed by Schudy and Sviridenko (Schudy et al., 2011), who established sharp moment and concentration inequalities for polynomials $f(Y)$ of degree $q$ in independent (moment-bounded) random variables. The method is based on a hypergraph (or tensor) representation:

$$f(Y) = \sum_{h \in \mathcal{H}} w_h \prod_{v \in V(h)} Y_v^{\tau_{h,v}}$$

where each monomial defines a "hyperedge" $h$ with weights $w_h$ and "power profile" $\tau_{h,v}$.

The approach involves:

• Centering $f$ and expanding the $k$th moment, for even $k$.
• Combinatorial identification of surviving terms by exploiting independence and the structure of overlapping monomials.
• Bounding the resulting moment in terms of "smoothness parameters," denoted $\mu_r$ (related to maxima of partial derivatives), and the moment-boundedness constant $L$ of the variables.

A general exponential tail inequality follows for arbitrary degree $q$:

$$\Pr[\,|f(Y) - \mathbb{E}f(Y)| \geq \lambda\,] \leq \exp\left(-\Omega \left(\min_{1 \leq r \leq q} \frac{\lambda^{1/r}}{R L T \mu_r^{1/r}}\right)\right)$$

where $R$ is an absolute constant, $T$ is the maximal variable power, and the dependence on $n$ (the number of variables) is absent—yielding "dimension-free" bounds.

This framework applies to a wide variety of input distributions: any random variable with exponentially decaying moments, including bounded, Gaussian, exponential, Poisson, laplacian, binomial, and related cases. The analysis handles high degree and small mean regimes previously considered intractable.

Notable applications include combinatorial counting (e.g., triangle counts in random graphs, subgraph statistics), permanents of random matrices (resolving previously open problems for symmetric Bernoulli cases), randomized rounding, and noise sensitivity in threshold functions.

Limitations include the tightness (term-by-term) of the bounds in Boolean settings and a potential blow-up for extremely high degree polynomials or when the smoothness parameters become vanishingly small.

2. Bernstein-like and Central Limit Inequality Regimes

For multilinear polynomials, a Bernstein-type concentration result is established in (Schudy et al., 2011): for $f$ of degree $q$ and variance $\operatorname{Var}f$,

$$\Pr[\,|f(Y) - \mathbb{E}f(Y)| \geq \lambda\,] \leq \exp\left(-\frac{\lambda^2}{R^q\,\operatorname{Var}f(Y)}\right)$$

for $\lambda$ below a regime determined by higher order norms. This matches, up to constant factors, the Gaussian tail predicted by the central limit theorem and holds for centered moment bounded random variables (Gaussian, Rademacher, exponential, Poisson, etc.) (Schudy et al., 2011).

The significance is twofold:

• The variance of $f$ enters directly, unlike prior inequalities that depended on less interpretable smoothness or influence parameters.
• For random sampling and empirical risk minimization, explicit bounds on deviation in terms of the variance enable dimensionally tight probabilistic guarantees for polynomial-based quantities.

3. Multilevel Concentration and Alpha-Subexponential Tails

In the presence of heavier-tailed, $\alpha$-subexponential variables (with tails $\mathbb{P}(|X_i|\geq t)\leq c\exp(-t^\alpha/C)$), (Götze et al., 2019) derives multi-level concentration inequalities for general polynomial functions:

$$\mathbb{P}(|f(X) - \mathbb{E}f(X)| \geq t) \leq 2\exp\left(-\frac{1}{C_{D,\alpha}} \min_{1 \leq d \leq D} \left(\frac{t}{M^d\, \|\mathbb{E} f^{(d)}(X)\|_{HS}}\right)^{\alpha/d}\right)$$

where $D$ is the degree and $M$ bounds the Orlicz $\psi_\alpha$-norms.

For quadratic forms, this reduces to a multilevel Hanson-Wright inequality, involving Hilbert-Schmidt, spectral, row-max, and entrywise norm scales. Such results extend the concentration machinery to settings where sampling from non-Gaussian distributions is needed.

4. Sampling Without Replacement and Dependence Structures

When sampling without replacement (multislice model), dependencies between random variables are strong, but analogous concentration inequalities persist. (Sambale et al., 2020) shows (using log-Sobolev and Beckner inequalities for multislices):

• For Lipschitz $f$: a bounded difference inequality with a finite-sampling correction factor $1-n/N$,
• For polynomials: a multilevel tail bound analogous to the independent case, with explicit dependence on higher order derivatives and symmetry structure.

A sample mean satisfies a Serfling-type inequality:

$$\mathbb{P}\left(\frac{1}{n}\sum_{i=1}^n \omega_i - \mathbb{E} \omega_1 \geq t\right) \leq \exp\left(- \frac{n t^2}{4(1-n/N)|\mathcal{X}|^2}\right)$$

and higher-degree polynomial functionals (e.g., triangle counts in $G(n,M)$) satisfy tail bounds of the same order as in $G(n,p)$ up to logarithmic factors, despite increased dependency.

SCP/SRP negative dependence conditions, as studied in (Adamczak et al., 2021), further allow sub-Gaussian or Bernstein-type concentration for polynomials, using martingale arguments and modified log-Sobolev inequalities, both for scalar and matrix-valued polynomial functionals.

5. Structure-Driven Sampling: Marcinkiewicz–Zygmund and Inequalities on Norms

A parallel stream, essential for discretizing polynomial $L^p$ norms from random (or scattered) sampling, employs Marcinkiewicz–Zygmund inequalities. These establish that for certain families of sample points $\{\xi_j\}$ and appropriate weights $w_j$,

$$A \|p\|_p^p \leq \sum_{j} w_j |p(\xi_j)|^p \leq B \|p\|_p^p,$$

uniform in polynomial degree (up to logarithmic factors), with $A$, $B$ independent of the degree $n$ as long as the number of samples scales like $n^q$ (for polynomials on the $q$-sphere (Filbir et al., 2023)) or as $n \asymp m^2$ for degree-$m$ univariate polynomials (Xu et al., 14 Jul 2024). Random sampling according to the equilibrium measure is often optimal here, and the role of convolution kernels (e.g., de la Vallée–Poussin means) facilitates the operator analysis needed to bridge continuous and discrete norms.

For spaces of analytic functions (Fock space, Bergman, Hardy), discrete quadratic norms (sampling sums) over carefully chosen random or structured points, possibly restricted to a "bulk" region (where the reproducing kernel localizes), provide stability and error control for polynomial recovery (Gröchenig et al., 2020, Gröchenig et al., 2021).

6. Random Polynomials, Topology, and Singularity Theory

Random sampling not only controls scalar norms but also governs higher-level properties, such as the distribution of zeros and the topology of real zero sets or singular loci in random homogeneous polynomials. Under the Kostlan distribution (Breiding et al., 2019):

• Complicated topological features (e.g., large Betti numbers, intricate singular loci) are exponentially rare.
• With high probability, the singular configuration of a random polynomial is captured by its projection to a much lower degree space ($O(\sqrt{d \log d})$ for degree-$d$).
• This defines a "stabilization scale" for random sampling of topology, with applications in enumerative real algebraic geometry.

7. Decoupling and Banach Space Geometry

For vector-valued or high-dimensional polynomial mappings, decoupling inequalities (Carando et al., 2020) provide a geometric toolkit for expressing the $L^p$ norm of a dependent random polynomial $P(\xi)$ in terms of a multilinear operator applied to independent copies, with exponential-in-degree constants if the target Banach space is of finite cotype. Under additional geometric properties (Gaussian Average Property, K-convexity), further optimization is possible, facilitating analysis of random sampling in harmonic analysis, functional analysis, and high-dimensional data settings.

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Summary Table: Key Inequality Types

Setting / Model	Inequality Type	Structural Constant / Scaling	Reference
Polynomials in independent RVs, general degree	Moment-based "hypergraph" bounds	min over derivative orders	(Schudy et al., 2011)
Multilinear polynomials, bounded/gaussian/etc.	Bernstein-type / sub-Gaussian tail	Variance + degree factor	(Schudy et al., 2011)
$\alpha$-subexponential tails	Multilevel, derivative-based	Tensor norm hierarchy	(Götze et al., 2019)
Sampling w/o replacement ("multislice")	Bounded difference + correction	$1-n/N$ factor	(Sambale et al., 2020)
SCP/SRP negative dependence	Log-Sobolev, Talagrand, matrix case	Partition norm minima, etc.	(Adamczak et al., 2021)
$L_p$ norm discretization via sampling	Marcinkiewicz–Zygmund type	Sample size versus degree	(Filbir et al., 2023)
Vector-valued / Banach space polynomials	Decoupling inequalities	Geometry (cotype, GAP, etc.)	(Carando et al., 2020)

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Random sampling inequalities for polynomials thus connect deep algebraic structure (degree, smoothness, dependency) to precise probabilistic behavior (expectation, deviation, concentration, rare event topology), with optimal or near-optimal constants, guiding both theoretical and algorithmic developments in modern analysis and data science.