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arxiv: 2606.24090 · v1 · pith:J3ZZ4ZYOnew · submitted 2026-06-23 · 🧮 math.PR

Sparsity-adaptive concentration inequalities for random polynomials

Guozheng Dai , Ke Wang This is my paper

Pith reviewed 2026-06-25 23:47 UTC · model grok-4.3

classification 🧮 math.PR
keywords concentration inequalitiesrandom polynomialssparse sub-exponential variablesmoment boundsHanson-Wright inequalityrandom tensorssingular value bounds
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The pith

Sparsity-weighted partition norms of expected derivative tensors control the moments of centered random polynomials.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves L_r moment bounds for the deviation of any degree-D polynomial evaluated at a vector whose coordinates are independent sparse α-sub-exponential random variables. The bounds are expressed via norms over all partitions of the derivative multi-indices, each term multiplied by the product of the sparsity probabilities belonging to the distinct coordinates in that block. This weighting separates the scaling that arises when many coordinates contribute modestly from the scaling that arises when a few coordinates contribute strongly. A reader would care because high-dimensional models routinely involve sparse inputs, and non-adaptive bounds become loose precisely when most variables are zero with high probability.

Core claim

For any polynomial f of degree at most D and any 0<α≤1, the L_r norm of f(X)−Ef(X) is bounded in terms of partition norms of sparsity-weighted expected derivative tensors, where the weights count distinct coordinates rather than multiplicities and therefore distinguish diagonal, partially diagonal, and off-diagonal contributions. This captures the sparse scaling in both collective fluctuation regimes and extreme-coordinate regimes. When all sparsity parameters equal one the bound recovers the Götze-Sambale-Sinulis polynomial inequality; in degree two it recovers sparse Hanson-Wright bounds.

What carries the argument

partition norms of sparsity-weighted expected derivative tensors, where each weight is the product of the sparsity parameters of the distinct coordinates appearing in a given block of the partition

If this is right

  • When every sparsity parameter equals one the bound reduces exactly to the Götze-Sambale-Sinulis polynomial concentration inequality.
  • For quadratic polynomials the result specializes to sparse versions of the Hanson-Wright inequality.
  • The bounds imply deviation inequalities for the Euclidean distance between a sparse simple random tensor and any fixed subspace.
  • They also yield lower bounds on the smallest singular value of a matrix whose columns are independent sparse simple random tensors.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same partition-weighting construction may produce adaptive concentration statements for non-polynomial functions of sparse vectors via polynomial approximation.
  • The distinction between diagonal and off-diagonal blocks suggests that analogous weighting could tighten other moment inequalities for sparse random matrices or tensors.

Load-bearing premise

Weighting each partition block by the product of the sparsity parameters of its distinct coordinates correctly separates the collective and extreme fluctuation regimes of the sparse variables.

What would settle it

A direct calculation of the L_r norm for a quadratic form in two or three sparse variables that exceeds the numerical value of the proposed partition-norm bound for some choice of p_i and α.

read the original abstract

We prove concentration inequalities for polynomials of independent, sparse $\alpha$-sub-exponential random variables. Specifically, we consider $X_i=\delta_i\xi_i$, where the Bernoulli selectors $\delta_i$ are independent with parameters $p_i$, and the variables $\xi_i$ are independent \(\alpha\)-sub-exponential random variables (not necessarily centered). For any polynomial $f:\mathbb R^n\to\mathbb R $ of degree at most $D$ and any $0<\alpha \le 1 $, we establish an $L_r$-moment bound for \(f(X)-\mathbb E f(X)\) in terms of partition norms of sparsity-weighted expected derivative tensors. The weights count distinct coordinates rather than multiplicities and therefore distinguish diagonal, partially diagonal, and off-diagonal contributions. This captures the sparse scaling in both collective fluctuation regimes and extreme-coordinate regimes. When all sparsity parameters are equal to one, our result recovers the polynomial concentration inequality of G\"otze, Sambale, and Sinulis. In degree two, it recovers sparse Hanson-Wright bounds. As applications, we derive deviation inequalities for the distance between a sparse simple random tensor and a fixed subspace, and obtain lower bounds for the smallest singular value of matrices whose columns are independent sparse simple random tensors.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript establishes L_r-moment bounds for f(X)−Ef(X), where f:R^n→R is a polynomial of degree at most D and X_i=δ_i ξ_i with independent Bernoulli selectors δ_i of parameters p_i and independent α-sub-exponential ξ_i (0<α≤1, not necessarily centered). The bounds are expressed in terms of partition norms of sparsity-weighted expected derivative tensors; the weights count distinct coordinates (rather than multiplicities) to separate diagonal, partially diagonal, and off-diagonal contributions. The result recovers the Götze–Sambale–Sinulis inequality when all p_i=1 and recovers sparse Hanson–Wright bounds in degree 2. Applications include deviation inequalities for the distance from a sparse simple random tensor to a fixed subspace and lower bounds on the smallest singular value of matrices whose columns are independent sparse simple random tensors.

Significance. If the central derivation holds, the work supplies a unified, sparsity-adaptive extension of polynomial concentration that distinguishes collective and extreme-coordinate regimes through the distinct-coordinate weighting. The explicit recovery of the dense Götze–Sambale–Sinulis result and the sparse Hanson–Wright inequality provides a useful consistency check. The applications to random tensors and singular-value bounds indicate immediate relevance to high-dimensional probability and random matrix theory.

minor comments (2)
  1. The abstract introduces 'partition norms of sparsity-weighted expected derivative tensors' without a one-sentence gloss or forward reference; adding a brief parenthetical definition or citation to the relevant section would improve accessibility for readers outside the immediate subfield.
  2. The range of r for which the L_r bound is stated is not indicated in the abstract or the opening paragraphs; clarifying whether the result holds for all r≥1 or only for r in a specific interval would remove ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No major comments were listed in the report, so we have no specific points to address point-by-point. The manuscript stands as submitted.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation establishes L_r-moment bounds via sparsity-weighted partition norms on expected derivative tensors, with weights defined to count distinct coordinates and thereby separate diagonal/off-diagonal regimes. This construction is shown to recover the dense Götze-Sambale-Sinulis inequality when all p_i=1 and the sparse Hanson-Wright bound in degree 2; both recoveries are external benchmarks rather than tautological. No step reduces a claimed prediction to a fitted input by construction, no load-bearing premise rests on self-citation, and the weighting mechanism is introduced directly from the problem statement rather than smuggled via prior ansatz. The result is therefore self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of sub-exponential random variables and independence assumptions; no new entities or fitted parameters are introduced in the abstract.

axioms (2)
  • domain assumption Properties of α-sub-exponential random variables
    The paper relies on the definition and properties of α-sub-exponential variables for the concentration.
  • domain assumption Independence of the δ_i and ξ_i
    Assumed for the X_i = δ_i ξ_i construction.

pith-pipeline@v0.9.1-grok · 5749 in / 1414 out tokens · 39320 ms · 2026-06-25T23:47:32.191229+00:00 · methodology

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Reference graph

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