This paper introduces a non-parametric learning framework to combat outliers in online, multi-output, and nonlinear regression tasks. A hierarchical-optimization problem underpins the learning task: Search in a reproducing kernel Hilbert space (RKHS) for a function that minimizes a sample average $\ell_p$-norm ($1 \leq p \leq 2$) error loss on data contaminated by noise and outliers, subject to side information that takes the form of affine constraints defined as the set of minimizers of a quadratic loss on a finite number of faithful data devoid of noise and outliers. To surmount the computational obstacles inflicted by the choice of loss and the potentially infinite dimensional RKHS, approximations of the $\ell_p$-norm loss, as well as a novel twist of the criterion of approximate linear dependency are devised to keep the computational-complexity footprint of the proposed algorithm bounded over time. Numerical tests on datasets showcase the robust behavior of the advocated framework against different types of outliers, under a low computational load, while satisfying at the same time the affine constraints, in contrast to the state-of-the-art methods which are constraint agnostic.

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