The remainder of this paper is structured as follows. Section 2 gives the problem setting and fixes some basic notation for the rest of the paper. Section 3 formally introduces the precomposition operator, investigates its mathematical properties and its establishes its connection to the infinite-dimensional regression problem. In Section 4, we lay the groundwork for a detailed statistical analysis of regression with Hilbert-Schmidt operators by combining spectral regularisation techniques with concentration bounds for sub-exponential Hilbertian random variables. In particular, we derive basic rates under the assumption of Hölder-type source conditions. Section 5 discusses the consequences of our results for specific statistical fields and connects them to the relevant literature. Section 6 recapitulates the key points of our work and their implications in the abstract context of learning infinite-dimensional information from **data** by comparing the infinite-dimensional response setting to standard regression with real-valued responses.