Generality of Lp-type RKBS and existence of embedding

Determine whether Lp-type reproducing kernel Banach spaces provide a sufficiently general modeling framework for machine learning hypothesis spaces; specifically, ascertain whether for any reproducing kernel Banach space E of real-valued functions on a domain Ω there exists an Lp-type reproducing kernel Banach space Bp on the same domain such that the embeddings E → Bp → L∞(Ω) hold.

Background

The paper investigates whether Lp-type reproducing kernel Banach spaces (RKBSs), where the feature space is an Lp(μ), are broad enough to encompass function classes relevant to machine learning. While much of the literature focuses on Lp-type constructions, the authors note an explicit uncertainty about the sufficiency of this modeling paradigm.

To make this uncertainty concrete, the authors pose a domain-specific embedding question: given any RKBS E of functions Ω → ℝ, does there exist an Lp-type RKBS Bp on the same domain such that E embeds into Bp and Bp embeds into L∞(Ω)? Their main results establish such embeddings under metric entropy growth conditions, but the general question—without auxiliary assumptions—remains explicitly raised in the text as an uncertainty.