Yi Zhang
Machine Intelligence Laboratory, College of Computer Science, Sichuan University, Chengdu 610065, China.
IEEE Trans Neural Netw. 2010 Mar;21(3):494-507. doi: 10.1109/TNN.2009.2039758. Epub 2010 Feb 5.
The competitive layer model (CLM) can be described by an optimization problem. The problem can be further formulated by an energy function, called the CLM energy function, in the subspace of nonnegative orthant. The set of minimum points of the CLM energy function forms the set of solutions of the CLM problem. Solving the CLM problem means to find out such solutions. Recurrent neural networks (RNNs) can be used to implement the CLM to solve the CLM problem. The key point is to make the set of minimum points of the CLM energy function just correspond to the set of stable attractors of the recurrent neural networks. This paper proposes to use Lotka-Volterra RNNs (LV RNNs) to implement the CLM. The contribution of this paper is to establish foundations of implementing the CLM by LV RNNs. The contribution mainly contains three parts. The first part is on the CLM energy function. Necessary and sufficient conditions for minimum points of the CLM energy function are established by detailed study. The second part is on the convergence of the proposed model of the LV RNNs. It is proven that interesting trajectories are convergent. The third part is the most important. It proves that the set of stable attractors of the proposed LV RNN just equals the set of minimum points of the CLM energy function in the nonnegative orthant. Thus, the LV RNNs can be used to solve the problem of the CLM. It is believed that by establishing such basic rigorous theories, more and interesting applications of the CLM can be found.
竞争层模型（CLM）可以用一个优化问题来描述。该问题可以在非负象限子空间中通过一个称为CLM能量函数的能量函数进一步公式化。CLM能量函数的最小点集构成了CLM问题的解集。求解CLM问题意味着找出这样的解。递归神经网络（RNN）可用于实现CLM以解决CLM问题。关键在于使CLM能量函数的最小点集恰好对应于递归神经网络的稳定吸引子集。本文提出使用Lotka-Volterra递归神经网络（LV RNN）来实现CLM。本文的贡献在于建立了用LV RNN实现CLM的基础。该贡献主要包含三个部分。第一部分是关于CLM能量函数。通过详细研究建立了CLM能量函数最小点的充要条件。第二部分是关于所提出的LV RNN模型的收敛性。证明了有趣的轨迹是收敛的。第三部分也是最重要的。证明了所提出的LV RNN的稳定吸引子集恰好等于非负象限中CLM能量函数的最小点集。因此，LV RNN可用于解决CLM问题。相信通过建立这样严谨的基础理论，可以找到更多有趣的CLM应用。
