论文标题
由Tikhonov最小二乘训练的回声状态网络是Ergodic动力学系统的L2（μ）近似值
Echo State Networks trained by Tikhonov least squares are L2(μ) approximators of ergodic dynamical systems
论文作者
论文摘要
回声状态网络（ESN）是一类具有随机生成内部权重的单层复发性神经网络，以及一层可调的外部重量，通常通过正规化的线性最小二乘回归进行训练。值得注意的是，尽管训练程序完全是线性的，但ESN仍然享受通用近似属性。在本文中，我们证明了使用Tikhonov最小二乘回归对一组目标进行回归的ESN对一系列观测的ESN（具有不变的量$ $ $），将在$ L^2（μ）$ norm中近似目标函数。在目标是将来的观察结果的特殊情况下，ESN正在学习下一步图，该映射允许时间序列预测。我们通过使用Tikhonov最小二乘在Lorenz系统的标量观察序列训练ESN来以数值来证明该理论。
Echo State Networks (ESNs) are a class of single-layer recurrent neural networks with randomly generated internal weights, and a single layer of tuneable outer weights, which are usually trained by regularised linear least squares regression. Remarkably, ESNs still enjoy the universal approximation property despite the training procedure being entirely linear. In this paper, we prove that an ESN trained on a sequence of observations from an ergodic dynamical system (with invariant measure $μ$) using Tikhonov least squares regression against a set of targets, will approximate the target function in the $L^2(μ)$ norm. In the special case that the targets are future observations, the ESN is learning the next step map, which allows time series forecasting. We demonstrate the theory numerically by training an ESN using Tikhonov least squares on a sequence of scalar observations of the Lorenz system.
