基于向量场热流的测地线距离函数学习
Geodesic Distance Function Learning via Heat Flow on Vector Fields
摘要：
学习给定数据流形上的距离函数或度量在机器学习和模式识别中非常重要。以前的许多工作首先将流形嵌入到欧氏空间，然后学习距离函数。然而，如果原始流形不是欧几里德流形，这样的方案可能无法忠实地保持距离函数。注意，流形上的距离函数总是可以定义良好的。在本文中，我们提出直接在流形上学习距离函数，而无需嵌入。首先，我们提供了一个理论上的距离函数的梯度场的特征。基于我们的理论分析，我们建议先学习距离函数的梯度场，然后再学习距离函数本身。具体来说，我们将局部距离函数的梯度场设置为初始向量场。然后我们通过向量场上的热流将其转移到整个流形上。最后，测地距离函数可以通过要求其梯度场接近归一化向量场来获得。在模拟数据和真实数据上的实验结果表明了该算法的有效性。
英文摘要：
Learning a distance function or metric on a given data manifold is of great importance in machine learning and pattern recognition. Many of the previous works first embed the manifold to Euclidean space and then learn the distance function. However, such a scheme might not faithfully preserve the distance function if the original manifold is not Euclidean. Note that the distance function on a manifold can always be well-defined. In this paper, we propose to learn the distance function directly on the manifold without embedding. We first provide a theoretical characterization of the distance function by its gradient field. Based on our theoretical analysis, we propose to first learn the gradient field of the distance function and then learn the distance function itself. Specifically, we set the gradient field of a local distance function as an initial vector field. Then we transport it to the whole manifold via heat flow on vector fields. Finally, the geodesic distance function can be obtained by requiring its gradient field to be close to the normalized vector field. Experimental results on both synthetic and real data demonstrate the effectiveness of our proposed algorithm.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
