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 $\mathcal{F}$ generated by a partition of $\mathbb{R}^{m}$ into $k$ sets such that [|\operatorname{Cov}(X) - \operatorname{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.