We present the first optimal rates for infinite-dimensional vector-valued ridge regression on a continuous scale of norms that interpolate between $L_2$ and the hypothesis space, which we consider as a vector-valued reproducing kernel Hilbert space. These rates allow to treat the misspecified case in which the true regression function is not contained in the hypothesis space. We combine standard assumptions on the capacity of the hypothesis space with a novel tensor product construction of vector-valued interpolation spaces in order to characterize the smoothness of the regression function. Our upper bound not only attains the same rate as real-valued kernel ridge regression, but also removes the assumption that the target regression function is bounded. For the lower bound, we reduce the problem to the scalar setting using a projection argument. We show that these rates are optimal in most cases and independent of the dimension of the output space. We illustrate our results for the special case of vector-valued Sobolev spaces.

翻译：我们首次提出了无穷维向量值岭回归在一系列连续范数（插值于$L_2$与假设空间之间，我们将该假设空间视为向量值再生核希尔伯特空间）上的最优收敛率。这些收敛率适用于真实回归函数不在假设空间中的错误设定情形。我们通过结合假设空间容量的标准假设与向量值插值空间的新型张量积构造方法，来刻画回归函数的平滑性。我们的上界不仅达到了与实值核岭回归相同的收敛率，还去除了目标回归函数有界的假设条件。对于下界，我们利用投影论证将问题简化为标量情形。我们证明这些收敛率在大多数情况下是最优的，且与输出空间的维数无关。我们以向量值Sobolev空间的特例展示了结果。