We study $(ε,δ)$-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 datapoint in $\mathbb{R}^{d}$. In our privacy model, neighboring inputs differ by one single row/datapoint. We study the private variant of the power iteration method, which is widely adopted in practice. Our algorithm is based on a filtering technique which adapts to the coherence parameter of the input matrix. This technique provides a utility that goes beyond the worst-case guarantees for matrices with low coherence parameter. Our work departs from and complements the work by Hardt-Roth (STOC 2013) which designed a private power iteration method for the privacy model where neighboring inputs differ in one single entry by at most 1.

翻译：我们研究用于近似计算矩阵 $A\in\mathbb{R}^{n\times d}$ 的顶部奇异向量的 $(ε,δ)$-差分隐私算法，其中 $A$ 的每一行是 $\mathbb{R}^{d}$ 中的一个数据点。在我们的隐私模型中，相邻输入仅相差一个行/数据点。我们研究实践中广泛采用的幂迭代方法的隐私变体。我们的算法基于一种自适应输入矩阵一致性参数的过滤技术。该技术为具有低一致性参数的矩阵提供了超越最坏情况保证的效用。我们的工作区别于并补充了 Hardt-Roth (STOC 2013) 的研究，后者针对相邻输入在单个条目上至多相差 1 的隐私模型设计了私有幂迭代方法。