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

• URL: http://arxiv.org/abs/2408.12385v1
• Date: Thu, 22 Aug 2024 13:26:41 GMT
• Title: Sharper Bounds for Chebyshev Moment Matching with Applications to Differential Privacy and Beyond
• Authors: Cameron Musco, Christopher Musco, Lucas Rosenblatt, Apoorv Vikram Singh,
• Abstract summary: We prove 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. We illustrate a second application of our new moment-based recovery bound in numerical linear algebra.
• Score: 26.339024618084476
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\tilde{O}(1/n)$ based on a dataset of $n$ points in $[-1,1]$. This bound is optimal up to log factors and matches a recent 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.

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