regularization perspectives were developed independently, they are in fact closely related. Notation In this context, the supervised learning problem is to learn the function which best predicts vector-valued outputs given inputs (data) . for , an input space (e.g. ) In general, each component of (), could have different input data () with different cardinality () and even different input spaces (). Geostatistics literature calls this case heterotopic, and uses isotopic to indicate that the each component of the output vector has the same set of inputs. Here, for simplicity in the notation, we assume the number and sample space of the data for each output are the same. Regularization perspective From the regularization perspective, the problem is to learn belonging to a reproducing kernel Hilbert space of vector-valued functions (). This is similar to the scalar case of Tikhonov regularization, with some extra care in the notation. It is possible, though non-trivial, to show that a representer theorem also holds for Tikhonov regularization in the vector-valued setting. Note, the matrix-valued kernel can also be defined by a scalar kernel on the space . An isometry exists between the Hilbert spaces associated with these two kernels: Gaussian process perspective The estimator of the vector-valued regularization framework can also be derived from a Bayesian viewpoint using Gaussian process methods in the case of a finite dimensional Reproducing kernel Hilbert space. The derivation is similar to the scalar-valued case Bayesian interpretation of regularization. The vector-valued function , consisting of outputs , is assumed to follow a Gaussian process: where is now a vector of the mean functions for the outputs and is a positive definite matrix-valued function with entry corresponding to the covariance between the outputs and . For a set of inputs , the prior distribution over the vector is given by , where is a vector that concatenates the mean vectors associated to the outputs and is a block-partitioned matrix. The distribution of the outputs is taken to be Gaussian: where is a diagonal matrix with elements specifying the noise for each output. Using this form for the likelihood, the predictive distribution for a new vector is: where is the training data, and is a set of hyperparameters for and . Equations for and can then be obtained: where has entries for and . Note that the predictor is identical to the predictor derived in the regularization framework. For non-Gaussian