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Decentralized Singular Value Decomposition for Large-scale Distributed Sensor Networks

Yufan Fan, Marius Pesavento

TL;DR

This work tackles decentralized Singular Value Decomposition for large-scale distributed sensor networks under two data-partition scenarios: direct row-wise partitioning and implicit outer-product formation. It introduces rational function approximation (RA) based algorithms (d-raSVD1 and d-raSVD2) with a parallel consensus framework, plus a truncation strategy (d-TraEVD/d-TraSVD2) that focuses on the principal subspace to cut communication and storage. The methods are validated through decentralized sensor localization via low-rank matrix completion and decentralized passive radar detection, showing convergence to centralized solutions with significantly reduced communication compared to decentralized power methods. The proposed approach enables scalable PCA/CCA-type processing in distributed networks, delivering practical gains in both accuracy and efficiency for large-scale sensor deployments.

Abstract

This article studies the problem of decentralized Singular Value Decomposition (d-SVD), which is fundamental in various signal processing applications. Two scenarios are considered depending on the availability of the data matrix under consideration. In the first scenario, the matrix of interest is row-wisely available in each local node in the network. In the second scenario, the matrix of interest implicitly forms an outer product from two different series of measurements. By combining the lightweight local rational function approximation approach with parallel averaging consensus algorithms, two d-SVD algorithms are proposed to cope with the two aforementioned scenarios. We evaluate the proposed algorithms using two application examples: decentralized sensor localization via low-rank matrix completion and decentralized passive radar detection. Moreover, a novel and non-trivial truncation technique, which employs a representative vector that is orthonormal to the principal signal subspace, is proposed to further reduce the communication cost associated with the d-SVD algorithms. Simulation results show that the proposed d-SVD algorithms converge to the centralized solution with reduced communication cost compared to those facilitated with the state-of-the-art decentralized power method.

Decentralized Singular Value Decomposition for Large-scale Distributed Sensor Networks

TL;DR

This work tackles decentralized Singular Value Decomposition for large-scale distributed sensor networks under two data-partition scenarios: direct row-wise partitioning and implicit outer-product formation. It introduces rational function approximation (RA) based algorithms (d-raSVD1 and d-raSVD2) with a parallel consensus framework, plus a truncation strategy (d-TraEVD/d-TraSVD2) that focuses on the principal subspace to cut communication and storage. The methods are validated through decentralized sensor localization via low-rank matrix completion and decentralized passive radar detection, showing convergence to centralized solutions with significantly reduced communication compared to decentralized power methods. The proposed approach enables scalable PCA/CCA-type processing in distributed networks, delivering practical gains in both accuracy and efficiency for large-scale sensor deployments.

Abstract

This article studies the problem of decentralized Singular Value Decomposition (d-SVD), which is fundamental in various signal processing applications. Two scenarios are considered depending on the availability of the data matrix under consideration. In the first scenario, the matrix of interest is row-wisely available in each local node in the network. In the second scenario, the matrix of interest implicitly forms an outer product from two different series of measurements. By combining the lightweight local rational function approximation approach with parallel averaging consensus algorithms, two d-SVD algorithms are proposed to cope with the two aforementioned scenarios. We evaluate the proposed algorithms using two application examples: decentralized sensor localization via low-rank matrix completion and decentralized passive radar detection. Moreover, a novel and non-trivial truncation technique, which employs a representative vector that is orthonormal to the principal signal subspace, is proposed to further reduce the communication cost associated with the d-SVD algorithms. Simulation results show that the proposed d-SVD algorithms converge to the centralized solution with reduced communication cost compared to those facilitated with the state-of-the-art decentralized power method.
Paper Structure (29 sections, 1 theorem, 48 equations, 5 figures, 1 table)

This paper contains 29 sections, 1 theorem, 48 equations, 5 figures, 1 table.

Table of Contents

  1. Motivation and Introduction
  2. Preliminaries
  3. EVD of the rank-one modification of a Hermitian matrix
  4. Decentralized rational function approximation based eigenvalue decomposition (d-raEVD)
  5. Remark on the implementation of d-raEVD
  6. D-SVD with Distributedly Partitioned Measurements
  7. Alternative: the decentralized Power Method (d-PM)
  8. Communication cost and storage requirement of d-raSVD1 and d-pmSVD1
  9. Communication cost
  10. Storage requirement
  11. Truncate d-raEVD
  12. D-SVD of an Implicit Outer Product
  13. Update of the left singular vectors
  14. Update of the right singular vectors
  15. Communication cost and storage requirement of d-raSVD2 and d-pmSVD2
  16. ...and 14 more sections

Key Result

Theorem 1

Suppose $\boldsymbol{\Lambda} = \operatorname{diag}\left( \lambda_1,\ldots,\lambda_N \right)\in\mathbb{R}^{N\times N}$ where the diagonal entries are distinct and are sorted in descending order, i.e., $\lambda_1>\cdots>\lambda_N$. Further assume that $\rho\neq 0$ and $\boldsymbol{z} = \left[z_1, \ld with $\bar{\lambda}_1>\cdots>\bar{\lambda}_N$, then

Figures (5)

  • Figure 1: Error performance of the d-TraEVD with different $\delta$, where $N=100, T = 500$. Results are averaged over $200$ random realizations of $\boldsymbol{R}$, and the relative error is examined only on the principal singular values.
  • Figure 2: (a): A sensor network with $N=30$ nodes (black dots) and $6$ obstacles (blue circles). The percentage of missing entries in the EDM is $22.5\%$. (b): Estimation of the true coordinates (black dots) using the d-raSVD1 (red $+$) and the d-pmSVD1 (blue $\star$). Recovery errors are $\epsilon_{\boldsymbol{R},\text{d-EVD}} = 0.0858$, $\epsilon_{\boldsymbol{R},\text{d-PM}} = 0.1060$, $\epsilon_{\boldsymbol{X},\text{d-EVD}} = 0.0447$ and $\epsilon_{\boldsymbol{X},\text{d-PM}} = 0.0539$.
  • Figure 3: An example of distributed passive radar system with $L=1$ non-cooperative illuminator.
  • Figure 4: ROC curves for $L = 1, N = 10, T = 5,$ SNR $=-10$ dB, $P = 10, \alpha=0.1$.
  • Figure 5: ROC curves for $L = 1, N = 100, T = 50,$ SNR $=-10$ dB, $P = 10, \alpha=0.1$.

Theorems & Definitions (3)

  • Remark 1
  • Remark 2
  • Theorem 1