Multitask learning (MTL) leverages task-relatedness to enhance performance. With the emergence of multimodal data, tasks can now be referenced by multiple indices. In this paper, we employ high-order tensors, with each mode corresponding to a task index, to naturally represent tasks referenced by multiple indices and preserve their structural relations. Based on this representation, we propose a general framework of low-rank MTL methods with tensorized support vector machines (SVMs) and least square support vector machines (LSSVMs), where the CP factorization is deployed over the coefficient tensor. Our approach allows to model the task relation through a linear combination of shared factors weighted by task-specific factors and is generalized to both classification and regression problems. Through the alternating optimization scheme and the Lagrangian function, each subproblem is transformed into a convex problem, formulated as a quadratic programming or linear system in the dual form. In contrast to previous MTL frameworks, our decision function in the dual induces a weighted kernel function with a task-coupling term characterized by the similarities of the task-specific factors, better revealing the explicit relations across tasks in MTL. Experimental results validate the effectiveness and superiority of our proposed methods compared to existing state-of-the-art approaches in MTL. The code of implementation will be available at https://github.com/liujiani0216/TSVM-MTL.

翻译：多任务学习利用任务相关性提升性能。随着多模态数据的出现，任务现在可以通过多个索引进行引用。本文采用高阶张量（其每个模式对应一个任务索引）自然地表示被多个索引引用的任务，并保持其结构关系。基于此表示，我们提出了一种通用的低秩多任务学习方法框架，融合张量化支持向量机（SVM）和最小二乘支持向量机（LSSVM），其中系数张量上部署了CP分解。我们的方法允许通过由任务特定因子加权的共享因子的线性组合来建模任务关系，并推广到分类和回归问题。通过交替优化方案和拉格朗日函数，每个子问题被转化为凸问题，在对偶形式中表述为二次规划或线性系统。与以往的多任务学习框架相比，我们对偶形式中的决策函数引入了一个加权核函数，其任务耦合项由任务特定因子的相似性刻画，更好地揭示了多任务学习中任务间的显式关系。实验结果表明，与现有最先进的多任务学习方法相比，我们提出的方法具有有效性和优越性。实现代码将在 https://github.com/liujiani0216/TSVM-MTL 提供。