효율적인 고차원 데이터 처리를 위한 N차원 볼의 체적 집중에 대한 분석
Volume Concentration Analysis of an N-dimensional Ball for Efficient High Dimensional Data Processing
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This paper presents the volume concentration analysis of an n-dimensional ball (n-ball) defined in Euclidean space, near the surface and the equator, for efficient high-dimensional data processing. To quantify the volume concentration of an n-ball, two measures are defined: one measure as the volume ratio of a whole n-ball to the differential slice near the surface, and the other as the volume ratio of a whole n-ball to the differential slice near the equator. Without direct computation of the volumes of n-dimensional geometrical objects, both measures can be obtained as a function of the dimension of ball and the thickness of differential slice. According to computer simulation results for an n-ball, the surface volume concentration and the equator volume concentration show the changing patterns similar to each other, but the surface volume concentration is significantly stronger than the equator volume concentration. Finally, by interpreting the volume concentration of an n-ball as the distribution of high-dimensional data samples, a theoretical basis is provided for the planning of efficient high-dimensional data processing.
Abstract
This paper presents the volume concentration analysis of an n-dimensional ball (n-ball) defined in Euclidean space, near the surface and the equator, for efficient high-dimensional data processing. To quantify the volume concentration of an n-ball, two measures are defined: one measure as the volume ratio of a whole n-ball to the differential slice near the surface, and the other as the volume ratio of a whole n-ball to the differential slice near the equator. Without direct computation of the volumes of n-dimensional geometrical objects, both measures can be obtained as a function of the dimension of ball and the thickness of differential slice. According to computer simulation results for an n-ball, the surface volume concentration and the equator volume concentration show the changing patterns similar to each other, but the surface volume concentration is significantly stronger than the equator volume concentration. Finally, by interpreting the volume concentration of an n-ball as the distribution of high-dimensional data samples, a theoretical basis is provided for the planning of efficient high-dimensional data processing.
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