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This paper presents a framework for computing random operator-valued featuremaps for operator-valued positive definite kernels. This is a generalization ofthe random Fourier features for scalar-valued kernels to the operator-valuedcase. Our general setting is that of operator-valued kernels corresponding toRKHS of functions with values in a Hilbert space. We show that in general, fora given kernel, there are potentially infinitely many random feature maps,which can be bounded or unbounded. Most importantly, given a kernel, we presenta general, closed form formula for computing a corresponding probabilitymeasure, which is required for the construction of the Fourier features, andwhich, unlike the scalar case, is not uniquely and automatically determined bythe kernel. We also show that, under appropriate conditions, random boundedfeature maps can always be computed. Furthermore, we show the uniformconvergence, under the Hilbert-Schmidt norm, of the resulting approximatekernel to the exact kernel on any compact subset of Euclidean space. Ourconvergence requires differentiable kernels, an improvement over thetwice-differentiability requirement in previous work in the scalar setting. Wethen show how operator-valued feature maps and their approximations can beemployed in a general vector-valued learning framework. The mathematicalformulation is illustrated by numerical examples on matrix-valued kernels.

Operator-Valued Bochner Theorem, Fourier Feature Maps for Operator-Valued Kernels, and Vector-Valued Learning