用解耦可分离的矩阵值核进行插值
Interpolation with uncoupled separable matrix-valued kernels
AI总结：
本文研究了如何利用矩阵值再生核对域Ω上的向量值函数进行近似，提出了一种解耦可分离的矩阵值核插值方法，并推导了插值误差的误差界，同时通过人工数据验证了该方法在处理向量值函数时的优势。
摘要：
本文研究了在域Ω上近似向量值函数的问题。为此，我们使用矩阵值再生核，这些核可以与向量函数的再生核希尔伯特空间相关联，并可视为标量值情况的扩展。这些空间在建模目标函数组件之间的相关性时显得很有前景，因为组件并非独立学习。我们专注于使用此类矩阵值核进行插值。我们推导了插值误差的误差界，以广义幂函数的形式，并引入了一种子类的矩阵值核，其幂函数可以追溯到标量值再生核的幂函数。最后，我们将这些核应用于一些人工数据，以说明与逐组件方法相比，使用矩阵值核进行插值的优势。
英文摘要：
In this paper we consider the problem of approximating vector-valued functions over a domain $Ω$. For this purpose, we use matrix-valued reproducing kernels, which can be related to Reproducing kernel Hilbert spaces of vectorial functions and which can be viewed as an extension to the scalar-valued case. These spaces seem promising, when modelling correlations between the target function components, as the components are not learned independently of one another. We focus on the interpolation with such matrix-valued kernels. We derive error bounds for the interpolation error in terms of a generalized power-function and we introduce a subclass of matrix-valued kernels whose power-functions can be traced back to the power-function of scalar-valued reproducing kernels. Finally, we apply these kind of kernels to some artificial data to illustrate the benefit of interpolation with matrix-valued kernels in comparison to a componentwise approach.
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