状态方程:
st=(1−α)st−1+αtanh(Ast−1+Byt−1)
或者
st+1=(1−α)st+αtanh(Ast+Byt)≜f(st,yt)
∂st∂ft=(1−α)I+α∂st∂⎣⎢⎢⎢⎡tanh(∑iA1ist,i+∑iB1iyt,i)tanh(∑iA2ist,i+∑iB1iyt,i)⋮tanh(∑iAnist,i+∑iB1iyt,i)⎦⎥⎥⎥⎤=(1−α)I+α⎣⎢⎡[1−tanh2(∑iA1ist,i+∑iB1iyt,i)]A11⋮[1−tanh2(∑iAnist,i+∑iB1iyt,i)]An1⋯⋱⋯[1−tanh2(∑iA1ist,i+∑iB1iyt,i)]A1n⋮[1−tanh2(∑iAnist,i+∑iB1iyt,i)]Ann⎦⎥⎤=(1−α)I+α[I−diag(tanh2(Ast+Byt))]A
所以对于谱范数
∣∣∣∣∣∣∂st∂ft∣∣∣∣∣∣2≤(1−α)+α∣∣∣∣I−diag(tanh2(Ast+Byt))∣∣∣∣2⋅∣∣A∣∣2≤1−α+α∣∣A∣∣2≤1