Principal Component Analysis (PCA) is a pivotal technique in the fields of machine learning and data analysis. It aims to reduce the dimensionality of a dataset while minimizing the loss of information. In recent years, there have been endeavors to utilize homomorphic encryption in privacy-preserving PCA algorithms. These approaches commonly employ a PCA routine known as PowerMethod, which takes the covariance matrix as input and generates an approximate eigenvector corresponding to the primary component of the dataset. However, their performance and accuracy are constrained by the incapability of homomorphic covariance matrix computation and the absence of a universal vector normalization strategy for the PowerMethod algorithm. In this study, we propose a novel approach to privacy-preserving PCA that addresses these limitations, resulting in superior efficiency, accuracy, and scalability compared to previous approaches. We attain such efficiency and precision through the following contributions: (i) We implement space optimization techniques for a homomorphic matrix multiplication method (Jiang et al., SIGSAC 2018), making it less prone to memory saturation in parallel computation scenarios. (ii) Leveraging the benefits of this optimized matrix multiplication, we devise an efficient homomorphic circuit for computing the covariance matrix homomorphically. (iii) Utilizing the covariance matrix, we develop a novel and efficient homomorphic circuit for the PowerMethod that incorporates a universal homomorphic vector normalization strategy to enhance both its accuracy and practicality.

翻译：主成分分析（PCA）是机器学习和数据分析领域中的关键技术，旨在降低数据集维度的同时最小化信息损失。近年来，已有研究尝试在同态加密的隐私保护PCA算法中应用该技术。这些方法通常采用称为PowerMethod的PCA流程，该流程以协方差矩阵为输入，生成与数据集主成分对应的近似特征向量。然而，由于同态协方差矩阵计算的不可行性以及PowerMethod算法缺乏通用向量归一化策略，其性能与精度受到限制。本研究提出一种新颖的隐私保护PCA方法，通过克服上述限制，在效率、精度和可扩展性方面均优于先前方法。我们通过以下贡献实现高效性与精确性：（i）针对同态矩阵乘法方法（Jiang等，SIGSAC 2018）实现空间优化技术，降低其在并行计算场景中的内存饱和风险；（ii）利用优化后的矩阵乘法优势，设计高效的协方差矩阵同态计算电路；（iii）基于该协方差矩阵，开发新颖高效的PowerMethod同态电路，并引入通用同态向量归一化策略以提升其精度与实用性。