IEEE Trans Cybern. 2021 Oct;51(10):5008-5020. doi: 10.1109/TCYB.2020.2982901. Epub 2021 Oct 12.
Common spatial pattern (CSP) is one of the most successful feature extraction algorithms for brain-computer interfaces (BCIs). It aims to find spatial filters that maximize the projected variance ratio between the covariance matrices of the multichannel electroencephalography (EEG) signals corresponding to two mental tasks, which can be formulated as a generalized eigenvalue problem (GEP). However, it is challenging in principle to impose additional regularization onto the CSP to obtain structural solutions (e.g., sparse CSP) due to the intrinsic nonconvexity and invariance property of GEPs. This article reformulates the CSP as a constrained minimization problem and establishes the equivalence of the reformulated and the original CSPs. An efficient algorithm is proposed to solve this optimization problem by alternately performing singular value decomposition (SVD) and least squares. Under this new formulation, various regularization techniques for linear regression can then be easily implemented to regularize the CSPs for different learning paradigms, such as the sparse CSP, the transfer CSP, and the multisubject CSP. Evaluations on three BCI competition datasets show that the regularized CSP algorithms outperform other baselines, especially for the high-dimensional small training set. The extensive results validate the efficiency and effectiveness of the proposed CSP formulation in different learning contexts.
共空间模式（CSP）是脑机接口（BCI）中最成功的特征提取算法之一。它旨在找到空间滤波器，这些滤波器可以最大化对应于两个心理任务的多通道脑电图（EEG）信号协方差矩阵之间的投影方差比，这可以表示为广义特征值问题（GEP）。然而，由于 GEPs 的内在非凸性和不变性，在 CSP 上施加额外的正则化以获得结构解（例如稀疏 CSP）在原则上具有挑战性。本文将 CSP 重新表述为约束最小化问题，并建立了重新表述的和原始的 CSP 之间的等价性。提出了一种有效的算法，通过交替执行奇异值分解（SVD）和最小二乘法来解决这个优化问题。在这个新的公式下，可以很容易地实现各种用于线性回归的正则化技术，以正则化用于不同学习范式的 CSP，例如稀疏 CSP、迁移 CSP 和多主体 CSP。在三个 BCI 竞赛数据集上的评估表明，正则化的 CSP 算法优于其他基线，特别是对于高维小训练集。广泛的结果验证了所提出的 CSP 公式在不同学习环境下的效率和有效性。
