The ability to express a learning task in terms of a primal and a dual optimization problem lies at the core of a plethora of machine learning methods. For example, Support Vector Machine (SVM), Least-Squares Support Vector Machine (LS-SVM), Ridge Regression (RR), Lasso Regression (LR), Principal Component Analysis (PCA), and more recently Singular Value Decomposition (SVD) have all been defined either in terms of primal weights or in terms of dual Lagrange multipliers. The primal formulation is computationally advantageous in the case of large sample size while the dual is preferred for high-dimensional data. Crucially, said learning problems can be made nonlinear through the introduction of a feature map in the primal problem, which corresponds to applying the kernel trick in the dual. In this paper we derive a primal-dual formulation of the Multilinear Singular Value Decomposition (MLSVD), which recovers as special cases both PCA and SVD. Besides enabling computational gains through the derived primal formulation, we propose a nonlinear extension of the MLSVD using feature maps, which results in a dual problem where a kernel tensor arises. We discuss potential applications in the context of signal analysis and deep learning.

翻译：将学习任务表达为原始和对偶优化问题的能力是众多机器学习方法的核心。例如，支持向量机（SVM）、最小二乘支持向量机（LS-SVM）、岭回归（RR）、Lasso回归（LR）、主成分分析（PCA），以及近年来的奇异值分解（SVD）都已被定义为原始权重或对偶拉格朗日乘数的形式。在大样本情况下，原始形式在计算上具有优势，而对偶形式则更适用于高维数据。关键的是，通过在原始问题中引入特征映射，可以使这些学习问题非线性化，这对应于在对偶问题中应用核技巧。本文推导了多线性奇异值分解（MLSVD）的一种原始-对偶形式，该形式将PCA和SVD都作为特例包含在内。除了通过推导的原始形式实现计算增益外，我们还利用特征映射提出了MLSVD的非线性扩展，这产生了一个出现核张量的对偶问题。我们讨论了该方法在信号分析和深度学习背景下的潜在应用。