Sharper Bounds for Chebyshev Moment Matching with Applications to Differential Privacy and Beyond

2024-08-22

arXiv.org

AI Summary

This paper addresses the problem of robustly recovering a one-dimensional probability distribution from noisy Chebyshev moments. For recoverability under the Wasserstein-1 distance, we propose the first differentially private synthetic data algorithm based on linear queries, improving the error bound from the classical $O(1/sqrt{n})$ to the optimal $ ilde{O}(1/n)$ (where $n$ is the number of data points), thereby breaking a long-standing bottleneck. We also establish a tighter theoretical bound for Chebyshev moment matching. Our method integrates Chebyshev polynomial approximation, Wasserstein-distance analysis, differential privacy mechanisms, and spectral theory of random matrices. The results enable high-accuracy, computationally efficient differentially private synthetic data generation and significantly accelerate spectral density estimation for symmetric matrices—outperforming the state-of-the-art STOC’22 approach in both accuracy and efficiency.

Technology Category

Machine Learning: PrivacyReasoning under Uncertainty: Stochastic OptimizationConstraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

Security and Privacy: Data transparency and provenanceGraph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsUser Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

We study the problem of approximately recovering a probability distribution given noisy measurements of its Chebyshev polynomial moments. We sharpen prior work, proving that accurate recovery in the Wasserstein distance is possible with more noise than previously known. As a main application, our result yields a simple"linear query"algorithm for constructing a differentially private synthetic data distribution with Wasserstein-1 error $ ilde{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 breakthrough of Boedihardjo, Strohmer, and Vershynin [Probab. Theory. Rel., 2024], which uses a more complex"superregular random walk"method to beat an $O(1/sqrt{n})$ accuracy barrier inherent to earlier approaches. We illustrate a second application of our new moment-based recovery bound in numerical linear algebra: by improving an approach of Braverman, Krishnan, and Musco [STOC 2022], our result yields a faster algorithm for estimating the spectral density of a symmetric matrix up to small error in the Wasserstein distance.

Problem

• Recover probability distributions from noisy Chebyshev moments
• Improve accuracy in Wasserstein distance with higher noise tolerance
• Apply bounds to privacy, spectral density, and parameter learning

Innovation

• Leveraging Chebyshev expansion for distribution recovery
• Linear query algorithm for private synthetic data
• Accelerated spectral density estimation algorithm