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Quaternion recurrent neural network with real-time recurrent learning and maximum correntropy criterion

Pauline Bourigault, Dongpo Xu, Danilo P. Mandic

TL;DR

This work introduces a robust quaternion recurrent neural network (QRNN) trained online with real-time recurrent learning (RTRL) under the maximum correntropy criterion (MCC), enabled by Generalised HR (GHR) calculus. The approach targets real-time processing of multidimensional data and robustness to outliers, demonstrated on motion prediction of chest markers for lung cancer radiotherapy. The key contribution is deriving MCC-based gradient expressions for QRNNs via GHR calculus and showing improved resilience to irregular breathing compared with MSE-based methods and baselines, while remaining suitable for online learning with limited data. The results suggest practical impact for adaptive radiotherapy, where accurate, low-latency motion prediction can reduce healthy tissue exposure and support real-time control, backed by open data and detailed methodological appendices.

Abstract

We develop a robust quaternion recurrent neural network (QRNN) for real-time processing of 3D and 4D data with outliers. This is achieved by combining the real-time recurrent learning (RTRL) algorithm and the maximum correntropy criterion (MCC) as a loss function. While both the mean square error and maximum correntropy criterion are viable cost functions, it is shown that the non-quadratic maximum correntropy loss function is less sensitive to outliers, making it suitable for applications with multidimensional noisy or uncertain data. Both algorithms are derived based on the novel generalised HR (GHR) calculus, which allows for the differentiation of real functions of quaternion variables and offers the product and chain rules, thus enabling elegant and compact derivations. Simulation results in the context of motion prediction of chest internal markers for lung cancer radiotherapy, which includes regular and irregular breathing sequences, support the analysis.

Quaternion recurrent neural network with real-time recurrent learning and maximum correntropy criterion

TL;DR

This work introduces a robust quaternion recurrent neural network (QRNN) trained online with real-time recurrent learning (RTRL) under the maximum correntropy criterion (MCC), enabled by Generalised HR (GHR) calculus. The approach targets real-time processing of multidimensional data and robustness to outliers, demonstrated on motion prediction of chest markers for lung cancer radiotherapy. The key contribution is deriving MCC-based gradient expressions for QRNNs via GHR calculus and showing improved resilience to irregular breathing compared with MSE-based methods and baselines, while remaining suitable for online learning with limited data. The results suggest practical impact for adaptive radiotherapy, where accurate, low-latency motion prediction can reduce healthy tissue exposure and support real-time control, backed by open data and detailed methodological appendices.

Abstract

We develop a robust quaternion recurrent neural network (QRNN) for real-time processing of 3D and 4D data with outliers. This is achieved by combining the real-time recurrent learning (RTRL) algorithm and the maximum correntropy criterion (MCC) as a loss function. While both the mean square error and maximum correntropy criterion are viable cost functions, it is shown that the non-quadratic maximum correntropy loss function is less sensitive to outliers, making it suitable for applications with multidimensional noisy or uncertain data. Both algorithms are derived based on the novel generalised HR (GHR) calculus, which allows for the differentiation of real functions of quaternion variables and offers the product and chain rules, thus enabling elegant and compact derivations. Simulation results in the context of motion prediction of chest internal markers for lung cancer radiotherapy, which includes regular and irregular breathing sequences, support the analysis.
Paper Structure (22 sections, 45 equations, 2 figures, 2 tables)

This paper contains 22 sections, 45 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quaternion algebra
  4. The GHR calculus
  5. Maximum Correntropy criterion
  6. Quaternion Recurrent Neural Network with RTRL and Maximum Correntropy Criterion
  7. Forward pass
  8. Correntropy cost function
  9. Backpropagation
  10. Simulation results
  11. Motion prediction of chest internal points for lung cancer radiotherapy
  12. 3D marker position data
  13. Algorithms & Training
  14. Prediction performance
  15. Discussion of simulation limits
  16. ...and 7 more sections

Figures (2)

  • Figure 1: Mean square error (MSE) (left) and correntropy (right) in the 3D space where each point represents a pure quaternion. We assume that the true quaternion is (0, 0i, 0j, 0k) for simplicity.
  • Figure 2: A general architecture for QRNN with RTRL.