Foundations of implementing the competitive layer model by Lotka-Volterra recurrent neural networks

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.