用局部橢圓來分類資料, 基於布雷格曼矩陣分歧

Piecewise elliptical classification based on Bregman matrix divergence

陳宥任(You-Ren Chen)

指導教授 ： 陳素雲 ； 陳宏

國立臺灣大學/理學院/數學研究所/碩士(2013年)

https://doi.org/10.6342/NTU.2013.00394

摘要

無資料

關鍵字

內部資料 ； 外部資料 ； 布雷格曼 ； 橢圓 ； 對稱 ； 垂直 ； 非垂直射線分解

並列摘要

If a crowd of data (interior data) is encompassed by another set of data (exterior data) and we are
to find a closed surface centralized at the mean of interior data to wrap and separate it from exterior
data, the first candidate for the closed surface may be an ellipse. Kulis et al. (2006) applies Bregman
(1976), named Bregman's modification, trying to find a suitable ellipse to separate this kind of data.
To be more specific, the crowd of data (interior data) can be properly described by its mean and covariance
matrix, while the other set of data (exterior data) can be thought of as an unknown number
of crowds of data (e.g., a mixed multivariate skew normal). However, from the conventional PCA,
ellipses (i.e. semi positive definite matrices) exhibit rigid structures, that is, symmetry and orthogonality.
These two properties are too restrictive for the following situations, assuming that interior data
is like a unit ball (A) and exterior data is like two unit balls (B, C). First, B is 4 units far from A on the
left while C is 40 units far from A on the right. The best cuts should be 2 units far from A on the left
and 20 units far from A on the right. Unfortunately, an ellipse centralized at the center of A can only
have cuts equally far from the left and right of A. Second, B is 4 units far from A on the left while B
is 8 units far from A on the top right. Since an ellipse has orthogonal long and short axes while AB
and AC are not orthogonal, this makes ellipses unsuitable separators. Therefore, we offer a method-
Nonorthognal Ray Decomposition (NRD)- to improve this situation.

並列關鍵字

Interior data ； Exterior data ； Ellipse ； Symmetry ； Orthogonality ； Nonorthogonal Ray Decomposition

參考文獻

[1] A. Azzalini and A. Capitanio. Statistical applications of the multivariate skew-normal distribution.

J. Roy. Statist. Soc. Series B., 1999.

[2] A. Banerjee, S. Merugu, I. S. Dhillon, S. Inderjit, and J. Ghosh. Clustering with Bregman divergences.

Journal of Machine Learning Research, 2005.

[3] L. M. Bregman. The relaxation method of finding the common points of convex sets and its application