We study the minimax estimation error for distributed covariance matrix estimation in the vertical-split (feature-split) setting, where two agents each observe different coordinates of $m$ i.i.d. sub-Gaussian samples and communicate a limited number of bits to a central server. While Rahmani et al. [2025] established nearly tight bounds for dense (unstructured) cross-covariance matrices, we investigate whether imposing elementwise $s$-sparsity on the cross-covariance $C_{21}$ can reduce the required communication and sample complexity. In contrast to the horizontal-split setting, where Braverman et al. [2016] showed that sparsity does not reduce communication cost for mean estimation, we prove that sparsity does help for cross-covariance estimation in the vertical split. Specifically, we establish minimax lower bounds showing that the communication budget per agent scales as $B_k = Ω(σ^4 d_k\, s' \log(d_1 d_2/s')/\varepsilon^2)$ and the sample complexity for cross-covariance estimation as $m = Ω(σ^4\, s' \log(d_1 d_2/s')/\varepsilon^2)$, where $s' = s \wedge d_{\min}$. For the $1$-sparse case, this yields an exponential improvement from $d_1 d_2$ to $\log(d_1 d_2)$ compared to the dense rate. Our lower bounds are established via Fano's method with an explicit sparse packing using a Varshamov--Gilbert-type argument for signed partial permutation matrices combined with the Conditional Strong Data Processing Inequality of Rahmani et al. [2025]. We show the bounds are tight with a matching achievable scheme, based on covering-net quantization and entry-wise hard thresholding, that attains the $s$-sparse lower bound up to polylogarithmic factors.

翻译：我们研究了垂直分裂（特征分裂）设置下分布式协方差矩阵估计的极小化极大估计误差，在该设置中，两个智能体各自观测 $m$ 个独立同分布的子高斯样本的不同坐标，并向中央服务器传输有限比特数的信息。虽然Rahmani等人[2025]针对稠密（无结构）互协方差矩阵建立了几近紧的界，我们探究了对互协方差 $C_{21}$ 施加逐元素 $s$-稀疏性是否能够降低所需的通信量和样本复杂度。与水平分裂设置（Braverman等人[2016]证明稀疏性不会降低均值估计的通信成本）相反，我们证明在垂直分裂中，稀疏性确实有助于互协方差估计。具体而言，我们建立了极小化极大下界，表明每个智能体的通信预算规模为 $B_k = Ω(σ^4 d_k\, s' \log(d_1 d_2/s')/\varepsilon^2)$，互协方差估计的样本复杂度为 $m = Ω(σ^4\, s' \log(d_1 d_2/s')/\varepsilon^2)$，其中 $s' = s \wedge d_{\min}$。对于 $1$-稀疏情况，这相比稠密速率实现了从 $d_1 d_2$ 到 $\log(d_1 d_2)$ 的指数级改进。我们的下界通过Fano方法建立，其中使用Varshamov–Gilbert型论证对有符号部分置换矩阵构造显式稀疏堆积，并结合Rahmani等人[2025]的条件强数据处理不等式。我们证明了这些界是紧的，并给出了一个匹配的可实现方案，该方案基于覆盖网量化和逐元素硬阈值化，在多项式对数因子范围内达到 $s$-稀疏下界。