Non-manifold factors Euclidean spaces

Multi-Manifold Modeling in Non-Euclidean spaces

This paper advocates a novel framework for segmenting a dataset on a Riemannian manifold M into clusters lying around lowdimensional submanifolds of M . Important examples of M , for which the proposed algorithm is computationally efficient, include the sphere, the set of positive definite matrices, and the Grassmannian. The proposed algorithm constructs a data-affinity matrix by thoroughly exp...

Spatial Analysis in curved spaces with Non-Euclidean Geometry

The ultimate goal of spatial information, both as part of technology and as science, is to answer questions and issues related to space, place, and location. Therefore, geometry is widely used for description, storage, and analysis. Undoubtedly, one of the most essential features of spatial information is geometric features, and one of the most obvious types of analysis is the geometric type an...

Windows into non-Euclidean spaces

Metric Derivations on Euclidean and Non-euclidean Spaces

As introduced by Weaver, (metric) derivations extend the notion of differential operators on Euclidean spaces and provide a linear differentiation theory that is well-defined on all metric spaces with Borel measures. The main result of this paper is a rigidity theorem for measures on Euclidean spaces that support a maximal number of derivations. The proof relies substantially on ideas from geom...

Function approximation on non-Euclidean spaces

This paper presents a family of layered feed-forward networks that is able to uniformly approximate functions on any metric space, and also on a wide variety of non-metric spaces. Non-Euclidean input spaces are frequently encountered in practice, while usual approximation schemes are guaranteed to work only on Euclidean metric spaces. Theoretical foundations are provided, as well as practical a...