Visualizing data sets on the Grassmannian using self-organizing mappings
Visualizing data sets on the Grassmannian using self-organizing mappings
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DOI：
10.1109/wsom.2017.8020003
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发表时间：
2017-06
期刊：
影响因子：
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通讯作者：
M. Kirby;C. Peterson
中科院分区：
文献类型：
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作者：
M. Kirby;C. Peterson
我们将自组织映射算法扩展到格拉斯曼流形上的数据可视化问题。在此设置中，n 维中的 k 个点的集合由 k 维子空间表示，例如通过奇异值或 QR 分解。以这种方式组装的数据很难可视化，因为格拉斯曼上的抽象点并不位于欧几里得空间中。将 SOM 算法扩展到此几何设置仅需要可以测量两点之间的距离，并且可以将任何给定点移向呈现的图案。格拉斯曼上两点之间的相似性是根据子空间之间的主角（例如弦距离）来测量的。此外，我们采用一个公式沿着最短路径（即格拉斯曼曲线上两点之间的测地线）将一个子空间移动到另一个子空间。这使得能够忠实地实现 SOM 方法，用于可视化由 n 维欧几里德空间的 k 维子空间组成的数据。我们在高光谱成像应用中说明了所得算法。
We extend the self-organizing mapping algorithm to the problem of visualizing data on Grassmann manifolds. In this setting, a collection of k points in n-dimensions is represented by a k-dimensional subspace, e.g., via the singular value or QR-decompositions. Data assembled in this way is challenging to visualize given abstract points on the Grassmannian do not reside in Euclidean space. The extension of the SOM algorithm to this geometric setting only requires that distances between two points can be measured and that any given point can be moved towards a presented pattern. The similarity between two points on the Grassmannian is measured in terms of the principal angles between subspaces, e.g., the chordal distance. Further, we employ a formula for moving one subspace towards another along the shortest path, i.e., the geodesic between two points on the Grassmannian. This enables a faithful implementation of the SOM approach for visualizing data consisting of k-dimensional subspaces of n-dimensional Euclidean space. We illustrate the resulting algorithm on a hyperspectral imaging application.