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On Distance Mapping from non-Euclidean Spaces to Euclidean Spaces

Abstract : Most Machine Learning techniques traditionally rely on some forms of Euclidean Distances, computed in a Euclidean space (typically $$\mathbb {R}^{d}$$). In more general cases, data might not live in a classical Euclidean space, and it can be difficult (or impossible) to find a direct representation for it in $$\mathbb {R}^{d}$$. Therefore, distance mapping from a non-Euclidean space to a canonical Euclidean space is essentially needed. We present in this paper a possible distance-mapping algorithm, such that the behavior of the pairwise distances in the mapped Euclidean space is preserved, compared to those in the original non-Euclidean space. Experimental results of the mapping algorithm are discussed on a specific type of datasets made of timestamped GPS coordinates. The comparison of the original and mapped distances, as well as the standard errors of the mapped distributions, are discussed.
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Wei Ren, Yoan Miche, Ian Oliver, Silke Holtmanns, Kaj-Mikael Björk, et al.. On Distance Mapping from non-Euclidean Spaces to Euclidean Spaces. 1st International Cross-Domain Conference for Machine Learning and Knowledge Extraction (CD-MAKE), Aug 2017, Reggio, Italy. pp.3-13, ⟨10.1007/978-3-319-66808-6_1⟩. ⟨hal-01677143⟩

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