Approximations from Anywhere and General Rough Sets
This work addresses a theoretical problem in rough set theory, with potential applications in machine learning, but appears incremental as it builds on the author's prior frameworks.
The paper tackles the inverse problem of fitting approximations to information systems in rough set theory, proving necessary conditions from number-theoretic and combinatorial perspectives in a higher-order variant of granular operator spaces. The results are intended to be useful in unsupervised and semi-supervised learning contexts.
Not all approximations arise from information systems. The problem of fitting approximations, subjected to some rules (and related data), to information systems in a rough scheme of things is known as the \emph{inverse problem}. The inverse problem is more general than the duality (or abstract representation) problems and was introduced by the present author in her earlier papers. From the practical perspective, a few (as opposed to one) theoretical frameworks may be suitable for formulating the problem itself. \emph{Granular operator spaces} have been recently introduced and investigated by the present author in her recent work in the context of antichain based and dialectical semantics for general rough sets. The nature of the inverse problem is examined from number-theoretic and combinatorial perspectives in a higher order variant of granular operator spaces and some necessary conditions are proved. The results and the novel approach would be useful in a number of unsupervised and semi supervised learning contexts and algorithms.