College of Electronic Science and Engineering, National University of Defense Technology, Changsha 410073, China.
IEEE Trans Image Process. 2012 Feb;21(2):816-27. doi: 10.1109/TIP.2011.2165291. Epub 2011 Aug 18.
In this paper, we exploit the advantages of tensorial representations and propose several tensor learning models for regression. The model is based on the canonical/parallel-factor decomposition of tensors of multiple modes and allows the simultaneous projections of an input tensor to more than one direction along each mode. Two empirical risk functions are studied, namely, the square loss and ε -insensitive loss functions. The former leads to higher rank tensor ridge regression (TRR), and the latter leads to higher rank support tensor regression (STR), both formulated using the Frobenius norm for regularization. We also use the group-sparsity norm for regularization, favoring in that way the low rank decomposition of the tensorial weight. In that way, we achieve the automatic selection of the rank during the learning process and obtain the optimal-rank TRR and STR. Experiments conducted for the problems of head-pose, human-age, and 3-D body-pose estimations using real data from publicly available databases, verified not only the superiority of tensors over their vector counterparts but also the efficiency of the proposed algorithms.
在本文中，我们利用张量表示的优势，提出了几种用于回归的张量学习模型。该模型基于多元张量的典范/平行因子分解，允许在每个模式下同时将输入张量投影到多个方向。研究了两种经验风险函数，即平方损失函数和 ε-不敏感损失函数。前者导致更高阶张量岭回归（TRR），后者导致更高阶支持张量回归（STR），均使用 Frobenius 范数进行正则化。我们还使用组稀疏性范数进行正则化，从而有利于张量权重的低阶分解。通过这种方式，我们在学习过程中实现了秩的自动选择，并获得了最优秩的 TRR 和 STR。使用来自公开可用数据库的真实数据进行的头部姿态、人类年龄和 3-D 身体姿态估计问题的实验不仅验证了张量相对于其向量对应物的优越性，而且验证了所提出算法的效率。
