Manifold Learning with Arbitrary Norms
Manifold Learning with Arbitrary Norms
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DOI：
10.1007/s00041-021-09879-2
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发表时间：
2021-10-01
影响因子：
1.2
通讯作者：
Singer, Amit
中科院分区：
文献类型：
--
作者：
Kileel, Joe;Moscovich, Amit;Singer, Amit
流形学习方法在非线性降维和其他涉及低内在维数的高维数据集的任务中发挥着重要作用。这些方法中的许多是基于图形的：它们将顶点与每个数据点关联，并将加权边与每对数据点关联。现有的理论表明，在成对仿射基于欧几里德范数的假设下，图的拉普拉斯矩阵收敛于数据流形的Laplace-Beltrami算子。本文确定了使用任意范数构造的图拉普拉斯算子的极限微分算子。我们的证明涉及到流形的第二基本形式和给定范数的单位球的凸几何之间的相互作用。为了证明非欧几里德范数在流形学习中的潜在好处，我们考虑了映射具有连续可变性的大分子运动的任务。在一个数值模拟中，我们表明，一个修改后的拉普拉斯特征映射算法，基于推土机的距离，优于经典的欧几里得拉普拉斯特征映射，无论是在计算成本和样本量需要恢复的内在几何。
Manifold learning methods play a prominent role in nonlinear dimensionality reduction and other tasks involving high-dimensional data sets with low intrinsic dimensionality. Many of these methods are graph-based: they associate a vertex with each data point and a weighted edge with each pair. Existing theory shows that the Laplacian matrix of the graph converges to the Laplace-Beltrami operator of the data manifold, under the assumption that the pairwise affinities are based on the Euclidean norm. In this paper, we determine the limiting differential operator for graph Laplacians constructed using any norm. Our proof involves an interplay between the second fundamental form of the manifold and the convex geometry of the given norm's unit ball. To demonstrate the potential benefits of non-Euclidean norms in manifold learning, we consider the task of mapping the motion of large molecules with continuous variability. In a numerical simulation we show that a modified Laplacian eigenmaps algorithm, based on the Earthmover's distance, outperforms the classic Euclidean Laplacian eigenmaps, both in terms of computational cost and the sample size needed to recover the intrinsic geometry.