Some Remarks on Diffusion Distances
Some Remarks on Diffusion Distances
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DOI：
10.1155/2010/464815
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发表时间：
2010-01-01
影响因子：
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通讯作者：
Kim, Seonja
中科院分区：
文献类型：
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作者：
Goldberg, Maxim J.;Kim, Seonja
作为扩散距离，我们建议使用一个度量（密切相关的余弦相似性），这是定义为两个L-2归一化向量之间的L-2距离。我们提供了一个数学解释，为什么归一化使扩散距离更有意义。我们的建议与几年前R. Coifman，它找到了某些L-1单位向量之间的L-2距离。在本文的第二部分中，我们给出了两个证明，平均首次通过时间的平均首次通过成本的扩展满足三角不等式，我们不假设潜在的马尔可夫矩阵是对角化的。最后，我们展示了一个有趣的连接（归一化）的平均首次通过时间和离散化的解决方案，一定的狄利克雷-泊松问题，并验证我们的结果数值单位圆的简单情况下。
As a diffusion distance, we propose to use a metric (closely related to cosine similarity) which is defined as the L-2 distance between two L-2-normalized vectors. We provide a mathematical explanation as to why the normalization makes diffusion distances more meaningful. Our proposal is in contrast to that made some years ago by R. Coifman which finds the L-2 distance between certain L-1 unit vectors. In the second part of the paper, we give two proofs that an extension of mean first passage time to mean first passage cost satisfies the triangle inequality; we do not assume that the underlying Markov matrix is diagonalizable. We conclude by exhibiting an interesting connection between the (normalized) mean first passage time and the discretized solution of a certain Dirichlet-Poisson problem and verify our result numerically for the simple case of the unit circle.