We consider the problem of learning a linear operator $\theta$ between two Hilbert spaces from empirical observations, which we interpret as least squares regression in infinite dimensions. We show that this goal can be reformulated as an inverse problem for $\theta$ with the undesirable feature that its forward operator is generally non-compact (even if $\theta$ is assumed to be compact or of $p$-Schatten class). However, we prove that, in terms of spectral properties and regularisation theory, this inverse problem is equivalent to the known compact inverse problem associated with scalar response regression. Our framework allows for the elegant derivation of dimension-free rates for generic learning algorithms under H\"older-type source conditions. The proofs rely on the combination of techniques from kernel regression with recent results on concentration of measure for sub-exponential Hilbertian random variables. The obtained rates hold for a variety of practically-relevant scenarios in functional regression as well as nonlinear regression with operator-valued kernels and match those of classical kernel regression with scalar response.

翻译：我们考虑从经验观测学习两个希尔伯特空间之间的线性算子$\theta$的问题，并将其解释为无限维中的最小二乘回归。我们表明，这一目标可以重新表述为关于$\theta$的反问题，但其前向算子通常具有非紧性的不良特征（即使假设$\theta$是紧算子或属于$p$-Schatten类）。然而，我们证明，在谱性质和正则化理论方面，该反问题等价于与标量响应回归相关的已知紧反问题。我们的框架允许在Hölder型源条件下，为通用学习算法优雅地推导出与维数无关的收敛速率。证明过程结合了核回归技术以及次指数希尔伯特型随机变量测度集中的最新结果。所获得的速率适用于函数回归以及基于算子值核的非线性回归中的多种实际相关场景，并与经典标量响应核回归的速率相匹配。