Manifold learning with bi-stochastic kernels
Manifold learning with bi-stochastic kernels
复制标题
使用双随机核的流形学习
DOI：
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复制
发表时间：
2017
影响因子：
1.2
通讯作者：
R. Coifman
中科院分区：
文献类型：
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作者：
Nicholas F. Marshall;R. Coifman
在本文中，我们回答了以下问题：什么是由一个内核，规范化，使它是双随机相对于一个指定的措施所定义的扩散过程的无穷小发电机？更确切地说，在假设数据是从黎曼流形采样的情况下，我们确定所得到的无穷小生成元如何依赖于样本点的潜在非均匀分布，以及双随机归一化的指定度量。在一个特殊的情况下，我们证明了一个连接到热核。我们考虑的情况下，只有一个单一的数据集，并给出一个数据集和一个参考集的情况。研究了所构造算子的谱理论，计算了本征函数梯度的Nystr“om延拓公式。离散点集和流形学习的应用进行了讨论。
In this paper we answer the following question: what is the infinitesimal generator of the diffusion process defined by a kernel that is normalized such that it is bi-stochastic with respect to a specified measure? More precisely, under the assumption that data is sampled from a Riemannian manifold we determine how the resulting infinitesimal generator depends on the potentially nonuniform distribution of the sample points, and the specified measure for the bi-stochastic normalization. In a special case, we demonstrate a connection to the heat kernel. We consider both the case where only a single data set is given, and the case where a data set and a reference set are given. The spectral theory of the constructed operators is studied, and Nystr"om extension formulas for the gradients of the eigenfunctions are computed. Applications to discrete point sets and manifold learning are discussed.