We study the problem of differentially private (DP) computation of coreset for the $k$-means objective. For a given input set of points, a coreset is another set of points such that the $k$-means objective for any candidate solution is preserved up to a multiplicative $(1 \pm α)$ factor (and some additive factor). We prove the first computational lower bounds for this problem. Specifically, assuming the existence of one-way functions, we show that no polynomial-time $(ε, 1/n^{ω(1)})$-DP algorithm can compute a coreset for $k$-means in the $\ell_\infty$-metric for some constant $α> 0$ (and some constant additive factor), even for $k=3$. For $k$-means in the Euclidean metric, we show a similar result but only for $α= Θ\left(1/d^2\right)$, where $d$ is the dimension.

翻译：我们研究了针对 $k$-means 目标函数的差分隐私（DP）核心集计算问题。对于给定的输入点集，核心集是另一个点集，使得对于任意候选解，其 $k$-means 目标函数值在乘法因子 $(1 \pm α)$（以及某个加法因子）范围内得以保持。我们首次证明了该问题的计算下界。具体而言，在单向函数存在的前提下，我们证明：即使对于 $k=3$，也不存在多项式时间的 $(ε, 1/n^{ω(1)})$-DP 算法能够为 $\ell_\infty$ 度量下的 $k$-means 问题计算具有某个常数 $α> 0$（及某个常数加法因子）的核心集。对于欧几里得度量下的 $k$-means 问题，我们展示了类似的结果，但仅适用于 $α= Θ\left(1/d^2\right)$ 的情形，其中 $d$ 表示维度。