Dr. Yuesheng Xu, COS Department of Math & Statistics, published a ground-breaking paper on Reproducing Kernel Banach Spaces. The paper has just appeared (February 2019) in the Memoirs of the American Mathematical Society, one of the leading journals in Mathematics. The publication is co-authored with Dr. Qi Ye of South China Normal University. The title of the paper is Generalized Mercer Kernels and Reproducing Kernel Banach Spaces and is with 122 pages a huge compendium of the subject. The notion of reproducing kernel Banach spaces was first introduced by Yuesheng and his collaborators in 2009 in order to provide a mathematical foundation for sparse machine learning in Banach spaces. Since 2009, the theory of reproducing kernel Banach spaces has received much attention from both the mathematics and machine learning community.
Classical machine learning methods such as support vector machines and other kernel based methods are used for pattern analysis or classification for collected data. Frequently, kernel based methods are applied in a setting called reproducing kernel Hilbert spaces. It turns out that an intrinsic nature of reproducing kernel Hilbert space prevents us from obtaining a classification function that is clean and simple. In order to obtain a more simplified classification function, more technically speaking, a sparse solution vectors of a machine learning problem, it is necessary to work in a setting that induces such simplified functions. It turns out that Banach spaces are appropriate spaces. Banach spaces offer much richer geometric properties than Hilbert spaces, which enable us to construct more simpler classification solutions. Thus, the study of reproducing kernel Banach spaces which was done in this article, leads to fast computing algorithms for machine learning.