Sparse Bayesian learning has promoted many effective frameworks for brain
activity decoding, especially for the reconstruction of muscle activity.
However, existing sparse Bayesian learning mainly employs Gaussian distribution
as error assumption in the reconstruction task, which is not necessarily the
truth in the real-world application. On the other hand, brain recording is
known to be highly noisy and contains many non-Gaussian noises, which could
lead to significant performance degradation for sparse Bayesian learning
method. The goal of this paper is to propose a new robust implementation for
sparse Bayesian learning, so that robustness and sparseness can be realized
simultaneously. Motivated by the great robustness of maximum correntropy
criterion (MCC), we proposed an integration of MCC into the sparse Bayesian
learning regime. To be specific, we derived the explicit error assumption
inherent in the MCC and then leveraged it for the likelihood function.
Meanwhile, we used the automatic relevance determination (ARD) technique for
the sparse prior distribution. To fully evaluate the proposed method, a
synthetic dataset and a real-world muscle activity reconstruction task with two
different brain modalities were employed. Experimental results showed that our
proposed sparse Bayesian correntropy learning framework improves significantly
the robustness in a noisy regression task. The proposed method can realize
higher correlation coefficient and lower root mean squared error in the
real-world muscle activity reconstruction tasks. Sparse Bayesian correntropy
learning provides a powerful tool for neural decoding which can promote the
development of brain-computer interfaces.

通过集成最大正相关性准则（MCC）于稀疏贝叶斯学习方法中，本文提出了一种新的稀疏贝叶斯相关性学习框架，以实现同时的鲁棒性和稀疏性，通过实验结果验证了该方法在噪声回归任务和真实肌肉活动重建任务中的鲁棒性提升，为神经解码提供了强大的工具，可推动脑 - 机接口的发展。

稀疏贝叶斯 Correntropy 学习：从嘈杂的脑信号重建强壮肌肉活动

Sparse Bayesian Correntropy Learning for Robust Muscle Activity  Reconstruction from Noisy Brain Recordings

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.

我们开发了一个强大的四元数循环神经网络（QRNN），用于实时处理具有异常值的三维和四维数据。这通过将实时递归学习（RTRL）算法与最大相关性准则（MCC）作为损失函数相结合来实现。通过基于广义 HR（GHR）微积分推导出这两种算法，GHR 微积分允许对四元数变量的实函数求导，并提供乘积和链式法则，从而实现了优雅而紧凑的推导。通过在肺癌放疗中对胸部内部标记物的运动预测进行模拟实验，包括常规和不规则的呼吸序列，支持了分析的结论。