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Operator-Valued Kernels (OVKs) and associated vector-valued ReproducingKernel Hilbert Spaces provide an elegant way to extend scalar kernel methodswhen the output space is a Hilbert space. Although primarily used in finitedimension for problems like multi-task regression, the ability of thisframework to deal with infinite dimensional output spaces unlocks many moreapplications, such as functional regression, structured output prediction, andstructured data representation. However, these sophisticated schemes cruciallyrely on the kernel trick in the output space, so that most of previous workshave focused on the square norm loss function, completely neglecting robustnessissues that may arise in such surrogate problems. To overcome this limitation,this paper develops a duality approach that allows to solve OVK machines for awide range of loss functions. The infinite dimensional Lagrange multipliers arehandled through a Double Representer Theorem, and algorithms for$\epsilon$-insensitive losses and the Huber loss are thoroughly detailed.Robustness benefits are emphasized by a theoretical stability analysis, as wellas empirical improvements on structured data applications.

Duality in RKHSs with Infinite Dimensional Outputs: Application to Robust Losses