We study $\left(ε,δ\right)$-differentially private algorithms for the problem of approximately computing the top singular vector of a matrix $A\in\mathbb{R}^{n\times d}$ where each row of $A$ is a data point in $\mathbb{R}^{d}$. Following Dwork-Talwar-Thakurta-Zhang (STOC 2014), we consider the privacy model where neighboring inputs differ by one single row. We give a novel algorithm that achieves beyond-worst-case guarantees for input matrices with low coherence, which is a structural property of matrices in many applications, including but not limited to i.i.d. data. Our algorithm contributes to the extensive literature on private power iteration methods, where we introduce a new filtering technique which adapts to this coherence parameter. Our work departs from and complements the work by Hardt-Roth (STOC 2013) which achieves beyond-worst-case guarantees for the more restrictive privacy model where neighboring inputs differ in one single entry by at most 1.

翻译：我们研究$\left(ε,δ\right)$-差分隐私算法，用于近似计算矩阵$A\in\mathbb{R}^{n\times d}$的右奇异向量问题，其中$A$的每一行是$\mathbb{R}^{d}$中的一个数据点。遵循Dwork-Talwar-Thakurta-Zhang（STOC 2014）的工作，我们考虑相邻输入仅相差单行的隐私模型。针对具有低相干性的输入矩阵（该矩阵结构性质广泛存在于包括独立同分布数据在内的众多应用中），我们提出了一种超越最坏情况保证的新型算法。该算法丰富了私有幂迭代方法的相关文献，其中我们引入了一种能够自适应相干性参数的新型过滤技术。本研究与Hardt-Roth（STOC 2013）的工作有所不同并进行互补——后者针对相邻输入在单一元素上至多相差1这一更为严格的隐私模型实现了超越最坏情况保证。