Abstract
The normal distribution in Euclidean space is used widely for statistical models. However, for pattern recognition, since pattern vectors are often normalized by their norm, they are on a hyper-spherical surface. Therefore, we have to study a normal distribution in a non-Euclidean space. Here, we provide the new concept of geometrically local isotropic independence and define the Maxwell normal distribution in a manifold. We also define the Mahalanobis metric, which is an extension of the Mahalanobis distance in Euclidean space. We provide the Mahalanobis metric equation, which is covariant with coordinate transformation. Furthermore, we show its experimental results.
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Keywords
- Riemannian Manifold
- Euclidean Space
- Central Limit Theorem
- Coordinate Transformation
- Mahalanobis Distance
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© 2006 Springer-Verlag Berlin Heidelberg
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Yamashita, Y., Numakami, M., Inoue, N. (2006). Maxwell Normal Distribution in a Manifold and Mahalanobis Metric. In: Yeung, DY., Kwok, J.T., Fred, A., Roli, F., de Ridder, D. (eds) Structural, Syntactic, and Statistical Pattern Recognition. SSPR /SPR 2006. Lecture Notes in Computer Science, vol 4109. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11815921_66
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DOI: https://doi.org/10.1007/11815921_66
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-37236-3
Online ISBN: 978-3-540-37241-7
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