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Optimal Minimization of the Covariance Loss | IEEE Journals & Magazine | IEEE Xplore

Optimal Minimization of the Covariance Loss


Abstract:

Let X be a random vector valued in \mathbb {R}^{m} such that \|X\|_{2} \le 1 almost surely. For every k\ge 3 , we show that there exists a sigma algebra $...Show More

Abstract:

Let X be a random vector valued in \mathbb {R}^{m} such that \|X\|_{2} \le 1 almost surely. For every k\ge 3 , we show that there exists a sigma algebra \mathcal {F} generated by a partition of \mathbb {R}^{m} into k sets such that \|\mathrm {Cov}(X) - \mathrm {Cov}(\mathbb {E}[X\mid \mathcal {F}]) \|_{\mathrm {F}} \lesssim \frac {1}{\sqrt {\log {k}}} . This is optimal up to the implicit constant and improves on a previous bound due to Boedihardjo, Strohmer, and Vershynin. Our proof provides an efficient algorithm for constructing \mathcal {F} and leads to improved accuracy guarantees for k -anonymous or differentially private synthetic data. We also establish a connection between the above problem of minimizing the covariance loss and the pinning lemma from statistical physics, providing an alternate (and much simpler) algorithmic proof in the important case when X \in \{\pm 1\}^{m}/\sqrt {m} almost surely.
Published in: IEEE Transactions on Information Theory ( Volume: 69, Issue: 2, February 2023)
Page(s): 813 - 818
Date of Publication: 13 October 2022

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