We study differentially private (DP) estimation of a rank-$r$ matrix $M \in \RR^{d_1\times d_2}$ under the trace regression model with Gaussian measurement matrices. Theoretically, the sensitivity of non-private spectral initialization is precisely characterized, and the differential-privacy-constrained minimax lower bound for estimating $M$ under the Schatten-$q$ norm is established. Methodologically, the paper introduces a computationally efficient algorithm for DP-initialization with a sample size of $n \geq \wt O (r^2 (d_1\vee d_2))$. Under certain regularity conditions, the DP-initialization falls within a local ball surrounding $M$. We also propose a differentially private algorithm for estimating $M$ based on Riemannian optimization (DP-RGrad), which achieves a near-optimal convergence rate with the DP-initialization and sample size of $n \geq \wt O(r (d_1 + d_2))$. Finally, the paper discusses the non-trivial gap between the minimax lower bound and the upper bound of low-rank matrix estimation under the trace regression model. It is shown that the estimator given by DP-RGrad attains the optimal convergence rate in a weaker notion of differential privacy. Our powerful technique for analyzing the sensitivity of initialization requires no eigengap condition between $r$ non-zero singular values.

翻译：我们研究在高斯测量矩阵的迹回归模型下，对秩为$r$的矩阵$M \in \RR^{d_1\times d_2}$进行差分隐私（DP）估计。理论上，我们精确刻画了非私有谱初始化的灵敏度，并建立了在Schatten-$q$范数下估计$M$的差分隐私约束极小化下界。方法上，本文引入了一种计算高效的算法用于DP初始化，所需样本量为$n \geq \wt O (r^2 (d_1\vee d_2))$。在特定正则条件下，该DP初始化落入$M$附近的局部球内。我们还提出了一种基于黎曼优化的差分隐私算法（DP-RGrad），该算法在DP初始化及样本量$n \geq \wt O(r (d_1 + d_2))$条件下实现了近乎最优的收敛速率。最后，本文讨论了迹回归模型下低秩矩阵估计的极小化下界与上界之间的非平凡差距。结果表明，在一种更弱的差分隐私概念下，DP-RGrad给出的估计量达到了最优收敛速率。我们分析初始化灵敏度的强大技术无需依赖于$r$个非零奇异值之间的特征间隙条件。