离散LTI系统的非负输入$l_2$诱导范数分析及其在循环神经网络稳定性分析中的应用
$l_2$ Induced Norm Analysis of Discrete-Time LTI Systems for Nonnegative Input Signals and Its Application to Stability Analysis of Recurrent Neural Networks
摘要：
本文主要研究离散线性定常系统的“正”$l_2$诱导范数，其中输入信号被限制为非负。为了处理输入信号的非负性，我们采用共正规划作为分析的数学工具。然后，通过应用内逼近的共正锥，我们得到数值上易处理的半定规划问题的“积极的”$l_2$诱导范数的上限和下界计算。这个范数通常用于由LTI系统和非线性元件仅提供非负信号的非线性构造的反馈系统的稳定性分析。作为一个具体的例子，我们说明了有用的“积极的”$l_2$诱导范数的稳定性分析的激活函数的循环神经网络的整流线性单元。
英文摘要：
In this paper, we focus on the "positive" $l_2$ induced norm of discrete-time linear time-invariant systems where the input signals are restricted to be nonnegative. To cope with the nonnegativity of the input signals, we employ copositive programming as the mathematical tool for the analysis. Then, by applying an inner approximation to the copositive cone, we derive numerically tractable semidefinite programming problems for the upper and lower bound computation of the "positive" $l_2$ induced norm. This norm is typically useful for the stability analysis of feedback systems constructed from an LTI system and nonlinearities where the nonlinear elements provide only nonnegative signals. As a concrete example, we illustrate the usefulness of the "positive" $l_2$ induced norm for the stability analysis of recurrent neural networks with activation functions being rectified linear units.
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