The local minima-free condition of feedforward neural networks for outer-supervised learning

In this paper, the local minima-free conditions of the outer-supervised feedforward neural networks (FNN) based on batch-style learning are studied by means of the embedded subspace method. It is proven that only if the rendition that the number of the hidden neurons is not less than that of the training samples, which is sufficient but not necessary, is satisfied, the network will necessarily converge to the global minima with null cost, and that the condition that the range space of the outer-supervised signal matrix is included in the range space of the hidden output matrix Is sufficient and necessary condition for the local minima-free in the error surface. In addition, under the condition of the number of the hidden neurons being less than that of the training samples and greater than the number of the output neurons, it is demonstrated that there will also only exist the global minima with null cost in the error surface if the first layer weights are adequately selected.