Wang Kunbo, Xu Yanxun
3400 N. Charles Street, Baltimore, MD 21218.
Stat Interface. 2024;17(2):199-217. doi: 10.4310/23-sii786. Epub 2024 Feb 1.
We propose a Bayesian tensor-on-tensor regression approach to predict a multidimensional array (tensor) of arbitrary dimensions from another tensor of arbitrary dimensions, building upon the Tucker decomposition of the regression coefficient tensor. Traditional tensor regression methods making use of the Tucker decomposition either assume the dimension of the core tensor to be known or estimate it via cross-validation or some model selection criteria. However, no existing method can simultaneously estimate the model dimension (the dimension of the core tensor) and other model parameters. To fill this gap, we develop an efficient Markov Chain Monte Carlo (MCMC) algorithm to estimate both the model dimension and parameters for posterior inference. Besides the MCMC sampler, we also develop an ultra-fast optimization-based computing algorithm wherein the maximum a posteriori estimators for parameters are computed, and the model dimension is optimized via a simulated annealing algorithm. The proposed Bayesian framework provides a natural way for uncertainty quantification. Through extensive simulation studies, we evaluate the proposed Bayesian tensor-on-tensor regression model and show its superior performance compared to alternative methods. We also demonstrate its practical effectiveness by applying it to two real-world datasets, including facial imaging data and 3D motion data.
我们提出了一种贝叶斯张量对张量回归方法，用于从任意维度的另一个张量预测任意维度的多维数组（张量），该方法基于回归系数张量的塔克分解。利用塔克分解的传统张量回归方法要么假设核心张量的维度已知，要么通过交叉验证或一些模型选择标准来估计它。然而，现有的方法都无法同时估计模型维度（核心张量的维度）和其他模型参数。为了填补这一空白，我们开发了一种高效的马尔可夫链蒙特卡罗（MCMC）算法，用于估计模型维度和参数以进行后验推断。除了MCMC采样器，我们还开发了一种基于超快速优化的计算算法，其中计算参数的最大后验估计器，并通过模拟退火算法优化模型维度。所提出的贝叶斯框架为不确定性量化提供了一种自然的方法。通过广泛的模拟研究，我们评估了所提出的贝叶斯张量对张量回归模型，并展示了其相对于其他方法的优越性能。我们还通过将其应用于两个真实世界的数据集，包括面部成像数据和3D运动数据，证明了其实际有效性。
