Data-driven modeling with fuzzy sets and manifolds

Abstract

The manifold hypothesis states that the shape of the observed data is relatively simple and that it lies on a low-dimensional manifold embedded in a high-dimensional space. We contribute to the problem of data-driven modeling by treating it as an inverse problem where the model defines a Euclidean space with a Riemannian manifold structure. In particular, our contribution shows that a fuzzy set on a bounded support defines a Riemannian manifold that can be embedded in a multidimensional space where dimension is a model parameter. A noticeable advantage of the proposed approach is its connection with the values of the membership function and independence from the dimension of the data being modeled.Last but not least, we have found various formal representations of the Laplace-Beltrami operator and use its values as an estimate of the quality of the approximate solution to the inverse problem.