Horizontal dimensionality reduction and iterated frame bundle development

TY - GEN

T1 - Horizontal dimensionality reduction and iterated frame bundle development

AU - Sommer, Stefan Horst

N1 - Conference code: 1

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

U2 - 10.1007/978-3-642-40020-9_7

DO - 10.1007/978-3-642-40020-9_7

M3 - Article in proceedings

SN - 978-3-642-40019-3

T3 - Lecture notes in computer science

SP - 76

EP - 83

BT - Geometric Science of Information

A2 - Nielsen, Frank

A2 - Barbaresco, Frédéric

PB - Springer

T2 - First International Conference on Geometric Science of Information

Y2 - 28 August 2013 through 30 August 2013

ER -