Selecting the independent coordinates of manifolds with large aspect ratios
Selecting the independent coordinates of manifolds with large aspect ratios
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发表时间：
2019-07
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通讯作者：
Yu-Chia Chen;M. Meilă
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作者：
Yu-Chia Chen;M. Meilă
当数据歧管具有较大的纵横比（例如长条带）时，许多歧管嵌入算法显然会失败。在这里，我们在寻找平滑的嵌入方面取得了成功和失败，还表明该问题比以前认识到的更为普遍，更复杂。从数学上讲，在非常广泛的条件下，如果嵌入是通过精心选择的Laplace-Beltrami操作员$ \ delta $完成的。因此，我们提出了一种双晶型独立特征分级选择（IES）算法，该算法选择具有少量特征向量的光滑嵌入。该算法基于理论上的基础，其计算开销较低，并且在合成和大型实际数据上取得了成功。
Many manifold embedding algorithms fail apparently when the data manifold has a large aspect ratio (such as a long, thin strip). Here, we formulate success and failure in terms of finding a smooth embedding, showing also that the problem is pervasive and more complex than previously recognized. Mathematically, success is possible under very broad conditions, provided that embedding is done by carefully selected eigenfunctions of the Laplace-Beltrami operator $\Delta$. Hence, we propose a bicriterial Independent Eigencoordinate Selection (IES) algorithm that selects smooth embeddings with few eigenvectors. The algorithm is grounded in theory, has low computational overhead, and is successful on synthetic and large real data.