Self-organizing mappings on the flag manifold with applications to hyper-spectral image data analysis
Self-organizing mappings on the flag manifold with applications to hyper-spectral image data analysis
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DOI：
10.1007/s00521-020-05579-y
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发表时间：
2021-03-27
影响因子：
6
通讯作者：
Peterson, Chris
中科院分区：
文献类型：
--
作者：
Ma, Xiaofeng;Kirby, Michael;Peterson, Chris
标志是向量空间的嵌套序列。标志的类型对构成标志的向量空间的维度序列进行编码。旗流形是一个流形，它的点参数化固定向量空间中所有固定类型的旗。本文提供了必要的数学框架，实现自组织映射的旗流形。标志隐含在许多数据分析上下文中，包括小波、傅立叶和奇异值分解。本文提出的几何框架，使旗帜之间的距离的计算，旗帜之间的测地线的计算，并能够移动一个标志的方向上的另一个标志规定的距离。使用这些操作作为构建块，我们实现了SOM算法上的旗流形。基本算法适用于参数化一组固定类型的标志的问题。
A flag is a nested sequence of vector spaces. The type of the flag encodes the sequence of dimensions of the vector spaces making up the flag. A flag manifold is a manifold whose points parameterize all flags of a fixed type in a fixed vector space. This paper provides the mathematical framework necessary for implementing self-organizing mappings on flag manifolds. Flags arise implicitly in many data analysis contexts including wavelet, Fourier, and singular value decompositions. The proposed geometric framework in this paper enables the computation of distances between flags, the computation of geodesics between flags, and the ability to move one flag a prescribed distance in the direction of another flag. Using these operations as building blocks, we implement the SOM algorithm on a flag manifold. The basic algorithm is applied to the problem of parameterizing a set of flags of a fixed type.