Lepetz Dominique, Némoz-Gaillard Max, Aupetit Michaël
EMA-DM, 6 Av. de Clavières, 30319 Alès cedex, France.
Neural Netw. 2007 Jul;20(5):621-30. doi: 10.1016/j.neunet.2006.11.006. Epub 2007 Apr 9.
The adaptation rule of Vector Quantization algorithms, and consequently the convergence of the generated sequence, depends on the existence and properties of a function called the energy function, defined on a topological manifold. Our aim is to investigate the conditions of existence of such a function for a class of algorithms including the well-known 'K-means' and 'Self-Organizing Map' algorithms. The results presented here extend several previous studies and show that the energy function is not always a potential but at least the uniform limit of a series of potential functions which we call a pseudo-potential. It also shows that a large number of existing vector quantization algorithms developed by the Artificial Neural Networks community fall into this class. The framework we define opens the way to studying the convergence of all the corresponding adaptation rules at once, and a theorem gives promising insights in that direction.
矢量量化算法的自适应规则，以及由此产生的序列的收敛性，取决于一个定义在拓扑流形上的称为能量函数的函数的存在性和性质。我们的目的是研究包括著名的“K均值”和“自组织映射”算法在内的一类算法中这种函数的存在条件。这里给出的结果扩展了先前的几项研究，并表明能量函数并不总是一个势函数，但至少是我们称为伪势的一系列势函数的一致极限。它还表明，人工神经网络社区开发的大量现有矢量量化算法都属于这一类。我们定义的框架为一次性研究所有相应自适应规则的收敛性开辟了道路，并且一个定理在该方向上给出了有前景的见解。
