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Federated Multilinear Principal Component Analysis with Applications in Prognostics

Chengyu Zhou, Yuqi Su, Tangbin Xia, Xiaolei Fang

TL;DR

The paper addresses privacy concerns when performing tensor dimension reduction with MPCA by introducing Federated MPCA (FMPCA). It develops three federated algorithms—Federated Centralization, Federated Initialization, and Federated Local Optimization—to replace the MPCA preprocessing, initialization, and optimization steps, respectively, while guaranteeing identical results to centralized MPCA. The approach is validated on simulated heat-transfer data and a rotating machinery case study in industrial prognostics, showing that FMPCA matches the performance of traditional MPCA and outperforms individual-user models in federated settings. This work enables privacy-preserving, collaborative tensor learning in data-sensitive industrial contexts, particularly where imaging-based degradation signals are involved. The methods are strengthened by theoretical results and proofs, and are demonstrated to be effective for forecasting time-to-failure under privacy constraints.

Abstract

Multilinear Principal Component Analysis (MPCA) is a widely utilized method for the dimension reduction of tensor data. However, the integration of MPCA into federated learning remains unexplored in existing research. To tackle this gap, this article proposes a Federated Multilinear Principal Component Analysis (FMPCA) method, which enables multiple users to collaboratively reduce the dimension of their tensor data while keeping each user's data local and confidential. The proposed FMPCA method is guaranteed to have the same performance as traditional MPCA. An application of the proposed FMPCA in industrial prognostics is also demonstrated. Simulated data and a real-world data set are used to validate the performance of the proposed method.

Federated Multilinear Principal Component Analysis with Applications in Prognostics

TL;DR

The paper addresses privacy concerns when performing tensor dimension reduction with MPCA by introducing Federated MPCA (FMPCA). It develops three federated algorithms—Federated Centralization, Federated Initialization, and Federated Local Optimization—to replace the MPCA preprocessing, initialization, and optimization steps, respectively, while guaranteeing identical results to centralized MPCA. The approach is validated on simulated heat-transfer data and a rotating machinery case study in industrial prognostics, showing that FMPCA matches the performance of traditional MPCA and outperforms individual-user models in federated settings. This work enables privacy-preserving, collaborative tensor learning in data-sensitive industrial contexts, particularly where imaging-based degradation signals are involved. The methods are strengthened by theoretical results and proofs, and are demonstrated to be effective for forecasting time-to-failure under privacy constraints.

Abstract

Multilinear Principal Component Analysis (MPCA) is a widely utilized method for the dimension reduction of tensor data. However, the integration of MPCA into federated learning remains unexplored in existing research. To tackle this gap, this article proposes a Federated Multilinear Principal Component Analysis (FMPCA) method, which enables multiple users to collaboratively reduce the dimension of their tensor data while keeping each user's data local and confidential. The proposed FMPCA method is guaranteed to have the same performance as traditional MPCA. An application of the proposed FMPCA in industrial prognostics is also demonstrated. Simulated data and a real-world data set are used to validate the performance of the proposed method.
Paper Structure (24 sections, 4 theorems, 19 equations, 4 figures, 5 algorithms)

This paper contains 24 sections, 4 theorems, 19 equations, 4 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. MPCA and its Challenges in Data Privacy Protection
  4. MPCA and Its Optimization Algorithm
  5. The Challenges in the Data Privacy Protection of MPCA
  6. Preprocessing
  7. Initialization
  8. Local Optimization
  9. Federated Multilinear Principal Component Analysis
  10. Preprocessing
  11. Initialization
  12. Local Optimization
  13. The Federated MPCA Algorithm
  14. The Application of FMPCA in Industrial Prognostics
  15. Numerical Study
  16. ...and 9 more sections

Key Result

Theorem 1

lu2008mpca Let $\lbrace \tilde{\bm{U}}^{(n)}, n = 1, \ldots, N \rbrace$ be the solution to (eq:MPCA 1). Then, given all the other projection matrices $\tilde{\bm{U}}^{(1)}, \ldots, \tilde{\bm{U}}^{(n-1)}, \tilde{\bm{U}}^{(n+1)}, \ldots, \tilde{\bm{U}}^{(N)}$, the matrix $\tilde{\bm{U}}^{(n)}$ consis where $\tilde{\bm{U}}_{\bm{\Phi}^{(n)}} = ( \tilde{\bm{U}}^{(n+1)} \otimes \tilde{\bm{U}}^{(n+2)} \

Figures (4)

  • Figure 1: An image stream.
  • Figure 2: Numerical Study.
  • Figure 3: An illustration of one infrared degradation image stream.
  • Figure 4: Case Study.

Theorems & Definitions (4)

  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Proposition 3