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In this paper, we provide a Banach-space formulation of supervised learning with generalized total-variation (gTV) regularization. We identify the class of kernel functions that are admissible in this framework. Then, we propose a variation of supervised learning in a continuous-domain hybrid search space with gTV regularization. We show that the solution admits a multikernel expansion with adaptive positions. In this representation, the number of active kernels is upper-bounded by the number of data points while the gTV regularization imposes an l(1) penalty on the kernel coefficients. Finally, we illustrate numerically the outcome of our theory.
Giuseppe Carleo, Riccardo Rossi, Clemens Giuliani, Filippo Vicentini