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Targeting at sparse multi-task learning, we consider regularization modelswith an $\ell^1$ penalty on the coefficients of kernel functions. In order toprovide a kernel method for this model, we construct a class of vector-valuedreproducing kernel Banach spaces with the $\ell^1$ norm. The notion ofmulti-task admissible kernels is proposed so that the constructed spaces couldhave desirable properties including the crucial linear representer theorem.Such kernels are related to bounded Lebesgue constants of a kernelinterpolation question. We study the Lebesgue constant of multi-task kernelsand provide examples of admissible kernels. Furthermore, we present numericalexperiments for both synthetic data and real-world benchmark data todemonstrate the advantages of the proposed construction and regularizationmodels.

Multi-task Learning in Vector-valued Reproducing Kernel Banach Spaces with the $\ell^1$ Norm