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arxiv: 2408.12385 · v3 · pith:MSOCYB5Rnew · submitted 2024-08-22 · 💻 cs.DS · cs.LG

Sharper Bounds for Chebyshev Moment Matching, with Applications

Cameron Musco , Christopher Musco , Lucas Rosenblatt , Apoorv Vikram Singh This is my paper

Pith reviewed 2026-05-23 22:17 UTC · model grok-4.3

classification 💻 cs.DS cs.LG
keywords Chebyshev momentsdistribution recoveryWasserstein distancemoment matchingLipschitz functionsdifferential privacyspectral densitynoisy moments
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The pith

A global decay bound on Chebyshev coefficients of Lipschitz functions enables distribution recovery from noisier moments in Wasserstein distance.

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

The paper shows that any Lipschitz function has Chebyshev coefficients that decay globally at a rate sufficient to tolerate higher noise levels when matching moments to recover an unknown probability distribution. This sharpening of prior recovery guarantees matters because it directly improves error bounds in three concrete settings without changing the algorithms themselves. The approach applies the same bound to produce an optimal linear-query method for private synthetic data, a faster matrix spectral density estimator, and a wider regime for a maximum-likelihood estimator on populations of parameters. An extension of the bound to multivariate distributions is also given.

Core claim

By leveraging a global decay bound on the coefficients in the Chebyshev expansion of any Lipschitz function, accurate recovery in the Wasserstein distance is possible with more noise than previously known.

What carries the argument

global decay bound on the coefficients in the Chebyshev expansion of any Lipschitz function, which controls the contribution of high-order moments under additive noise

If this is right

  • A linear-query algorithm produces a differentially private synthetic distribution with Wasserstein-1 error ilde{O}(1/n) that is optimal up to logs.
  • An ilde{O}(n^2/\epsilon) algorithm estimates the spectral density of an n by n symmetric matrix to ilde{O}(\epsilon) Wasserstein error.
  • The maximum-likelihood estimator for learning populations of parameters achieves sample-optimal results in an extended parameter regime.
  • The same decay bound extends to distribution recovery in d greater than 1 dimensions.

Where Pith is reading between the lines

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

  • The bound may apply to moment-matching problems that use other orthogonal polynomial bases beyond Chebyshev.
  • Similar coefficient decay arguments could tighten analyses for distribution recovery under different noise models such as multiplicative perturbations.
  • The private synthetic data result suggests that linear queries suffice for near-optimal Wasserstein utility in other privacy settings.

Load-bearing premise

The global decay bound on Chebyshev coefficients for Lipschitz functions is tight enough and applies directly to the recovery problem without requiring extra conditions on the target distribution or the noise model.

What would settle it

A concrete Lipschitz function on [-1,1] whose Chebyshev coefficients decay slower than the claimed global bound, or a distribution and noise level where Wasserstein recovery error exceeds the sharpened guarantee despite using the bound in the analysis.

Figures

Figures reproduced from arXiv: 2408.12385 by Apoorv Vikram Singh, Cameron Musco, Christopher Musco, Lucas Rosenblatt.

Figure 1
Figure 1. Figure 1: Experimental validation of Algorithm 2 for private synthetic data. For each dataset, we collect subsamples of size n for different n. We plot the W1 distance between the uniform distribution, p, over the subsample and a differentially private approximation, q, constructed by Algorithm 2 with privacy parameters ϵ = 0.5 and δ = 1/n2 . As predicted by Theorem 4, the Wasserstein-1 error scales as O˜(1/n). The … view at source ↗
read the original abstract

We study the problem of approximately recovering a probability distribution given noisy measurements of its Chebyshev polynomial moments. This problem arises broadly across algorithms, statistics, and machine learning. By leveraging a global decay bound on the coefficients in the Chebyshev expansion of any Lipschitz function, we sharpen prior work, proving that accurate recovery in the Wasserstein distance is possible with more noise than previously known. Our result immediately yields a number of applications: 1) We give a simple "linear query" algorithm for constructing a differentially private synthetic data distribution with Wasserstein-$1$ error $\tilde{O}(1/n)$ based on a dataset of $n$ points in $[-1,1]$. This bound is optimal up to log factors, and matches a recent result of Boedihardjo, Strohmer, and Vershynin [Probab. Theory. Rel., 2024], which uses a more complex "superregular random walk" method. 2) We give an $\tilde{O}(n^2/\epsilon)$ time algorithm for the linear algebraic problem of estimating the spectral density of an $n\times n$ symmetric matrix up to $\epsilon$ error in the Wasserstein distance. Our result accelerates prior methods from Chen et al. [ICML 2021] and Braverman et al. [STOC 2022]. 3) We tighten an analysis of Vinayak, Kong, Valiant, and Kakade [ICML 2019] on the maximum likelihood estimator for the statistical problem of "Learning Populations of Parameters'', extending the parameter regime in which sample optimal results can be obtained. Beyond these main results, we provide an extension of our bound to estimating distributions in $d > 1$ dimensions. We hope that these bounds will find applications more broadly to problems involving distribution recovery from noisy moment information.

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 / 3 minor

Summary. The paper claims that a global O(1/k) decay bound on Chebyshev coefficients of any 1-Lipschitz function on [-1,1] yields sharper noise tolerance for recovering distributions from noisy moments in Wasserstein-1 distance. This improves prior moment-matching results and directly yields three applications: an optimal (up to logs) linear-query DP synthetic data algorithm matching Boedihardjo et al. (2024), an Õ(n²/ε)-time algorithm for spectral density estimation improving Chen et al. (ICML 2021) and Braverman et al. (STOC 2022), and a tightened analysis of the MLE from Vinayak et al. (ICML 2019) that enlarges the sample-optimal regime. A d>1 extension is sketched.

Significance. If the decay bound and its application to the dual W1 formulation hold, the work supplies a clean, parameter-free improvement to noisy moment recovery that simplifies and accelerates several downstream tasks while achieving near-optimal rates. The explicit credit for a global Lipschitz-derived bound (rather than fitted parameters) and the matching of a recent complex construction with a simpler linear-query method are strengths. The result is likely to be cited in DP, numerical linear algebra, and statistical estimation literature.

minor comments (3)
  1. [§3] §3 (or wherever the decay bound is proved): the statement that |a_k| = O(1/k) holds uniformly for all 1-Lipschitz f should include the explicit constant and the precise normalization of the Chebyshev weight; this constant propagates into the final noise tolerance and should be tracked.
  2. [Application 2] Application 2: the claimed Õ(n²/ε) runtime should be compared quantitatively with the prior Chen et al. and Braverman et al. runtimes in a single table so the speedup is immediately visible.
  3. [d>1 extension] The d>1 extension paragraph is only a sketch; if it is intended as a contribution, it needs a precise statement of the dimension dependence and the modified decay bound.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive review, accurate summary of our contributions, and recommendation for minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper's central contribution is a new global decay bound on Chebyshev coefficients for any 1-Lipschitz function, derived directly from the Lipschitz property on [-1,1]. This bound is applied to control truncation and noise errors in the dual formulation of Wasserstein-1 distance. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the three applications substitute the improved noise tolerance into external frameworks (Boedihardjo et al., Chen et al., Vinayak et al.) without circular dependence. The argument is internally consistent and externally falsifiable via the stated Lipschitz assumption.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The contribution rests on proving and applying a mathematical bound on Chebyshev coefficients rather than introducing fitted parameters or new entities.

axioms (1)
  • domain assumption There exists a global decay bound on the coefficients in the Chebyshev expansion of any Lipschitz function that can be leveraged for distribution recovery.
    This bound is the central technical device invoked to obtain the improved noise tolerance.

pith-pipeline@v0.9.0 · 5882 in / 1099 out tokens · 31572 ms · 2026-05-23T22:17:32.712615+00:00 · methodology

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Reference graph

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