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Improved Privacy-Preserving PCA Using Optimized Homomorphic Matrix Multiplication

Xirong Ma

TL;DR

The paper tackles privacy-preserving PCA in a cloud setting by removing the bottleneck of homomorphic covariance computation and by delivering a robust normalization strategy for the PowerMethod. It introduces optimized homomorphic matrix multiplication with hoisting and diagonal convergence decomposition to speed up large-scale matrix operations and reduce rotation-key requirements, enabling scalable covariance computation. A Lazy Normalization-based InvSRT approach provides a universal, accurate vector normalization within the homomorphic PowerMethod, together with an EigenShift mechanism to extract successive principal components. Implemented on full-RNS CKKS via Lattigo, the approach demonstrates improved efficiency and accuracy over prior work (e.g., Panda 2021) across multiple datasets, highlighting practical viability for privacy-preserving cloud PCA. The work emphasizes modularity, parallelism, and modulus-refresh strategies to balance security, performance, and accuracy in real-world deployments.

Abstract

Principal Component Analysis (PCA) is a pivotal technique widely utilized in the realms 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 for the secure cloud computing scenario. 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 is constrained by the absence of an efficient homomorphic covariance matrix computation circuit and an accurate homomorphic vector normalization strategy in 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

Improved Privacy-Preserving PCA Using Optimized Homomorphic Matrix Multiplication

TL;DR

The paper tackles privacy-preserving PCA in a cloud setting by removing the bottleneck of homomorphic covariance computation and by delivering a robust normalization strategy for the PowerMethod. It introduces optimized homomorphic matrix multiplication with hoisting and diagonal convergence decomposition to speed up large-scale matrix operations and reduce rotation-key requirements, enabling scalable covariance computation. A Lazy Normalization-based InvSRT approach provides a universal, accurate vector normalization within the homomorphic PowerMethod, together with an EigenShift mechanism to extract successive principal components. Implemented on full-RNS CKKS via Lattigo, the approach demonstrates improved efficiency and accuracy over prior work (e.g., Panda 2021) across multiple datasets, highlighting practical viability for privacy-preserving cloud PCA. The work emphasizes modularity, parallelism, and modulus-refresh strategies to balance security, performance, and accuracy in real-world deployments.

Abstract

Principal Component Analysis (PCA) is a pivotal technique widely utilized in the realms 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 for the secure cloud computing scenario. 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 is constrained by the absence of an efficient homomorphic covariance matrix computation circuit and an accurate homomorphic vector normalization strategy in 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
Paper Structure (59 sections, 6 theorems, 16 equations, 2 figures, 4 tables, 14 algorithms)

This paper contains 59 sections, 6 theorems, 16 equations, 2 figures, 4 tables, 14 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Work
  4. Privacy-preserving PCA in cloud computing scenario
  5. Homomorphic Encryption Matrix Multiplication
  6. Preliminaries
  7. General Notation
  8. Homomorphic Encryption for Arithmetic of Approximate Numbers
  9. Homomorphic Matrix Operation
  10. Homomorphic Linear Transformation
  11. Hommorphic Matrix Multiplication by Jiang et al.
  12. Principal Component Analysis using PowerMethod
  13. Iterative algorithm of inverse square root function
  14. Optimized Homomorphic Matrix Multiplication
  15. Homomorphic Matrix Multiplication Equipped with Hoisting Techniques
  16. ...and 44 more sections

Key Result

theorem 1

Let $A$ be a boolean matrix that can be decomposed into $A = A_2A_1$, where $A_1$ and $A_2$ are both boolean matrices. For any 1 element $A_{N(i,j)}$ in $A$, there exists a unique index $0\leq a<n$ in $A_1$ and $A_2$ such that $A_{1N(a,j)}$ and $A_{2N(i,a)}$ are both 1. Furthermore, considering $D(k

Figures (2)

  • Figure 1: PowerMethod Simulations for Vector Normalization Parameterization. The figures presented on the left and right respectively illustrate the evolution of vector lengths and the number of iterative InvSRT iterations per PowerMethod iteration in 4 and 14 rounds of simulations. The simulations are conducted with an error bound of $e=1e-5$ and a precision bound of $p=1e-3$.
  • Figure 2: Flow chart of the modulus refresh strategy in homomorphic PowerMethod

Theorems & Definitions (16)

  • definition 1
  • definition 2
  • definition 3
  • theorem 1
  • proof
  • definition 4
  • theorem 2
  • proof
  • theorem 3
  • proof
  • ...and 6 more