Targeting at sparse multi-task learning, we consider regularization models with an ${\ell }^1$ penalty on the coefficients of kernel functions. In order to provide a kernel method for this model, we construct a class of vector-valued reproducing kernel Banach spaces with the ${\ell }^1$ norm. The notion of multi-task admissible kernels is proposed so that the constructed spaces could have desirable properties including the crucial linear representer theorem. Such kernels are related to bounded Lebesgue constants of a kernel interpolation question. We study the Lebesgue constant of multi-task kernels and provide examples of admissible kernels. Furthermore, we present numerical experiments for both synthetic data and real-world benchmark data to demonstrate the advantages of the proposed construction and regularization models.

针对稀疏的多任务学习，我们考虑将正则化模型与 ${\ell }^{\mathrm{1个}}$核函数系数的损失。为了提供此模型的内核方法，我们构造了一类矢量值的再现内核Banach空间，其中${\ell }^{\mathrm{1个}}$规范。提出了多任务可允许核的概念，以便构造的空间可以具有理想的属性，包括关键的线性表示定理。这样的内核与内核插值问题的有界Lebesgue常数有关。我们研究了多任务内核的Lebesgue常数，并提供了可接受的内核的示例。此外，我们目前对合成数据和实际基准数据进行数值实验，以证明所提出的构造和正则化模型的优势。