Applying inverse stereographic projection to manifold learning and clustering

Abstract

In machine learning, a data set is often viewed as a point set distributed on a manifold. Using Euclidean norms to measure the proximity of this data set reduces the efficiency of learning methods. Also, many algorithms like Laplacian Eigenmaps or spectral clustering that require to measure similarity assume the k-Nearest Neighbors of any point are quite equal to the local neighborhood of the point on the manifold using Euclidean norms. In this paper, we propose a new method that intelligently transforms data on an unknown manifold to an n-sphere by the conformal stereographic projection, which preserves the angles and similarities of data in the original manifold. Therefore similarities represent actual similarities of the data in the original space. Experimental results on various problems, including clustering and manifold learning, show the effectiveness of our method.

Feature Guided Path Redirection for VR Navigation

“…Then, with stereographic projection (STP) [38], we can project a point (x, y, z) of the Riemann sphere S 2 onto the complex plane and obtain the projected point (x ′ , y ′ ). Let point (0, 0, 1) be the pole, STP can be formulated as:…”

Quantum and Quantum-Inspired Stereographic K Nearest-Neighbour Clustering

“…On the other hand, angle embedding loses all amplitude information due to the normalisation of all points. A method to transform an unknown manifold into an n-sphere using ISP is proposed in [ 36 ]—here, however, the property of their concern was the conformality of the projection since subsequent learning is performed upon the surface. In [ 37 ], a parallelised version of [ 2 ] is developed using the FF-QRAM procedure [ 38 ] for amplitude encoding and the ISP to ensure a injective embedding.…”