Abstract
In Euclidean vector spaces, dimensionality reduction can be centered at the data mean. In contrast, distances do not split into orthogonal components and centered analysis distorts inter-point distances in the presence of curvature. In this paper, we define a dimensionality reduction procedure for data in Riemannian manifolds that moves the analysis from a center point to local distance measurements. Horizontal component analysis measures distances relative to lower-order horizontal components providing a natural view of data generated by multimodal distributions and stochastic processes. We parametrize the non-local, low-dimensional subspaces by iterated horizontal development, a constructive procedure that generalizes both geodesic subspaces and polynomial subspaces to Riemannian manifolds. The paper gives examples of how low-dimensional horizontal components successfully approximate multimodal distributions.
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Sommer, S. (2013). Horizontal Dimensionality Reduction and Iterated Frame Bundle Development. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2013. Lecture Notes in Computer Science, vol 8085. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40020-9_7
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DOI: https://doi.org/10.1007/978-3-642-40020-9_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-40019-3
Online ISBN: 978-3-642-40020-9
eBook Packages: Computer ScienceComputer Science (R0)