IEEE Trans Neural Netw Learn Syst. 2018 Jan;29(1):238-244. doi: 10.1109/TNNLS.2016.2574963.
Echo state network is a novel kind of recurrent neural networks, with a trainable linear readout layer and a large fixed recurrent connected hidden layer, which can be used to map the rich dynamics of complex real-world data sets. It has been extensively studied in time series prediction. However, there may be an ill-posed problem caused by the number of real-world training samples less than the size of the hidden layer. In this brief, a Laplacian echo state network (LAESN), is proposed to overcome the ill-posed problem and obtain low-dimensional output weights. First, an echo state network is used to map the multivariate time series into a large reservoir. Then, assuming that an unknown underlying manifold is inside the reservoir, we employ the Laplacian eigenmaps to estimate the manifold by constructing an adjacency graph associated with the reservoir states. Finally, the output weights are calculated by the low-dimensional manifold. In addition, some criteria of transient stability, local controllability, and local observability are given. Experimental results based on two real-world data sets substantiate the effectiveness and characteristics of the proposed LAESN model.
回声状态网络是一种新型的递归神经网络，具有可训练的线性读取层和固定的大型递归连接隐藏层，可以用于映射复杂的真实世界数据集的丰富动态。它在时间序列预测中得到了广泛的研究。然而，由于实际训练样本的数量少于隐藏层的大小，可能会出现不适定问题。在这份简短的文件中，提出了拉普拉斯回声状态网络（LAESN）来克服不适定问题并获得低维输出权重。首先，使用回声状态网络将多元时间序列映射到一个大型储层中。然后，假设未知的潜在流形在储层内部，我们通过构建与储层状态相关联的邻接图来利用拉普拉斯特征映射来估计流形。最后，通过低维流形计算输出权重。此外，还给出了暂态稳定性、局部可控性和局部可观性的一些准则。基于两个真实世界数据集的实验结果证实了所提出的 LAESN 模型的有效性和特点。
