Improved Privacy-Preserving PCA Using Space-optimized Homomorphic Matrix Multiplication

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.