A new generalized multilinear regression model, termed the Higher-Order Partial Least Squares (HOPLS), is introduced with the aim to predict a tensor (multiway array) $\tensor{Y}$ from a tensor $\tensor{X}$ through projecting the data onto the latent space and performing regression on the corresponding latent variables. HOPLS differs substantially from other regression models in that it explains the data by a sum of orthogonal Tucker tensors, while the number of orthogonal loadings serves as a parameter to control model complexity and prevent overfitting. The low dimensional latent space is optimized sequentially via a deflation operation, yielding the best joint subspace approximation for both $\tensor{X}$ and $\tensor{Y}$. Instead of decomposing $\tensor{X}$ and $\tensor{Y}$ individually, higher order singular value decomposition on a newly defined generalized cross-covariance tensor is employed to optimize the orthogonal loadings. A systematic comparison on both synthetic data and real-world decoding of 3D movement trajectories from electrocorticogram (ECoG) signals demonstrate the advantages of HOPLS over the existing methods in terms of better predictive ability, suitability to handle small sample sizes, and robustness to noise.

提出了一种新的广义多线性回归模型——Higher-Order Partial Least Squares (HOPLS)，通过将数据投影到潜空间并在相应的潜变量上执行回归，从而预测张量（多维数组）从而解释数据的一系估计模型，该模型通过正交Tucker张量的总和来解释数据，正交负载的数量作为控制模型复杂度和防止过度拟合的参数。

Higher-Order Partial Least Squares (HOPLS): 一种广义多线性回归方法