A Riemannian geometric framework for manifold learning of non-Euclidean data
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A Riemannian geometric framework for manifold learning of non-Euclidean data Cheongjae Jang1
· Yung-Kyun Noh2
· Frank Chongwoo Park1
Received: 5 May 2020 / Revised: 17 September 2020 / Accepted: 4 November 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract A growing number of problems in data analysis and classification involve data that are non-Euclidean. For such problems, a naive application of vector space analysis algorithms will produce results that depend on the choice of local coordinates used to parametrize the data. At the same time, many data analysis and classification problems eventually reduce to an optimization, in which the criteria being minimized can be interpreted as the distortion associated with a mapping between two curved spaces. Exploiting this distortion minimizing perspective, we first show that manifold learning problems involving non-Euclidean data can be naturally framed as seeking a mapping between two Riemannian manifolds that is closest to being an isometry. A family of coordinate-invariant first-order distortion measures is then proposed that measure the proximity of the mapping to an isometry, and applied to manifold learning for nonEuclidean data sets. Case studies ranging from synthetic data to human mass-shape data demonstrate the many performance advantages of our Riemannian distortion minimization framework. Keywords Manifold learning · Non-Euclidean data · Riemannian geometry · Distortion · Harmonic map Mathematics Subject Classification 53A35 · 53B21 · 58C35 · 58E20
Cheongjae Jang and Frank Chongwoo Park were supported in part by the NAVER LABS’ AMBIDEX Project, MSIT-IITP (2019-0-01367, BabyMind), SNU-IAMD, SNU BK21+ Program in Mechanical Engineering, SNU Institute for Engineering Research, the National Research Foundation of Korea (NRF-2016R1A5A1938472), the Technology Innovation Program (ATC+, 20008547) funded by the Ministry of Trade, Industry, and Energy (MOTIE, Korea), and SNU BMRR Grant DAPAUD190018ID. Yung-Kyun Noh was supported by Samsung Research Funding & Incubation Center of Samsung Electronics under Project Number SRFC-IT1901-13 and by Hanyang University (HY-2019). (Corresponding author: Frank Chongwoo Park.). Extended author information available on the last page of the article
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1 Introduction 1.1 Motivation and contribution A growing number of problems in data analysis and classification involve data that are non-Euclidean in nature. Examples include human shape and motion data (Vinué et al. 2016; Barahona et al. 2018), data collected from sensor networks, social networks in computational social sciences (Bronstein et al. 2017), and MRI imaging data (Fletcher and Joshi 2007). In all these cases it is a priori known that the collected data are drawn from a space that is not a vector space, but a curved space possessing additional geometric structure. Given that most data analysis algorithms are formulated in a vector space setting, common practice when encountering non-Euclidean data is to ignore the fact that t
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