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 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中学习两个希尔伯特空间之间的线性算子$\theta$的问题，这可以解释为无限维空间中的最小二乘回归。我们证明该目标可以重新表述为关于$\theta$的逆问题，其特征是其前向算子通常是非紧的（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的意义上，该逆问题等价于标量响应回归对应的已知紧逆问题。我们的框架允许在H\"older型源条件下优雅地推导通用学习算法的无维度收敛率。证明依赖于核回归技术与次指数希尔伯特随机变量测度集中最新结果的结合。所获得的收敛率适用于函数回归及算子值核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收敛率相匹配。