1_3 hopfield模型.htm

企业数字神经网络功能开发文章

          <TD width="76%"><IMG height=95 
            src="1_3 Hopfield模型.files/4.3.ht29.gif" width=397 border=0></TD>
          <TD width="24%">(1-46)</TD></TR>
        <TR>
          <TD width="76%">对于神经元j，其能量函数可表示为</TD>
          <TD width="24%"></TD></TR>
        <TR>
          <TD width="76%"><IMG height=41 
            src="1_3 Hopfield模型.files/4.3.ht30.gif" width=307 border=0></TD>
          <TD width="24%">(1-47)</TD></TR></TBODY></TABLE></TD></TR>
  <TR>
    <TD width="100%" height=1>
      <P>也即是有 
      <P><IMG height=40 src="1_3 Hopfield模型.files/4.3.ht31.gif" width=76 
      border=0></P>
      <P>神经元j的能量变化量表示为<SPAN 
      style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">△</SPAN>E<SUB>j</SUB>：</P>
      <TABLE cellSpacing=0 cellPadding=0 width="95%" align=center border=0>
        <TBODY>
        <TR>
          <TD width="50%"><IMG height=164 
            src="1_3 Hopfield模型.files/4.3.ht32.gif" width=576 border=0></TD>
          <TD width="50%">(1-48)</TD></TR></TBODY></TABLE></TD></TR>
  <TR>
    <TD width="100%" height=18>
      <P>如果存在条件&nbsp;&nbsp; W<SUB>ii</SUB>=0,i=1,2,...,n 
      <P>W<SUB>ij</SUB>=W<SUB>ji</SUB>&nbsp; i=1,2,...,n&nbsp;&nbsp; 
      j=1,2,...,n</P>
      <P>则有：</P>
      <TABLE cellSpacing=0 cellPadding=0 width="95%" align=center border=0>
        <TBODY>
        <TR>
          <TD width="80%"><IMG height=117 
            src="1_3 Hopfield模型.files/4.3.ht33.gif" width=320 border=0></TD>
          <TD width="20%">(1-49)</TD></TR></TBODY></TABLE></TD></TR>
  <TR>
    <TD width="100%" height=18>
      <P>其中：E<SUB>j</SUB>为神经元j的能量； 
      <P><SPAN 
      style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">△</SPAN>E<SUB>j</SUB>为神经元j的能量变化；</P>
      <P>W<SUB>ij</SUB>为神经元i到神经元j的权系数：</P>
      <P>Y<SUB>i</SUB>为神经元j的输出；</P>
      <P>X<SUB>j</SUB>为神经元j的外部输入；</P>
      <P><SPAN 
      style="FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA"><FONT 
      size=3>θ<SUB></SUB></FONT></SPAN><SUB>j</SUB>为神经元j的阀值；</P>
      <P><SPAN 
      style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">△</SPAN>Y<SUB>j</SUB>为神经元j的输出变化。</P>
      <P>如果，令</P>
      <P><SPAN 
      style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">U<SUB>j</SUB>=ΣW<SUB>ij</SUB>Y<SUB>i</SUB>+X<SUB>j</SUB></SPAN></P>
      <P>则<SPAN 
      style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">△</SPAN>E<SUB>j</SUB>可表示为：</P>
      <P>考虑如下两种情况：</P>
      <P>1．如果U<SUB>j</SUB><SPAN 
      style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">≥</SPAN><FONT 
      size=3><SPAN 
      style="FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">θ</SPAN></FONT><SUB>j</SUB>，即神经元j的输入结果的值大于阀值，则U<SUB>j</SUB>-<FONT 
      size=3><SPAN 
      style="FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">θ</SPAN></FONT><SUB>j</SUB><SPAN 
      style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">≥0</SPAN>，则从二值神经元的计算公式知道：Y<SUB>j</SUB>的值保持为1，或者从0变到1。这说明Y<SUB>j</SUB>的变化<SPAN 
      style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">△</SPAN>Y<SUB>j</SUB>只能是0或正值。这时很明显有<SPAN 
      style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">△</SPAN>E<SUB>j</SUB>：</P>
      <P><SPAN 
      style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">△</SPAN>E<SUB>j</SUB><SPAN 
      style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">≤0</SPAN></P>
      <P>这说明Hopfield网络神经元的能量减少或不变。</P>
      <P>2．如果U<SUB>j</SUB><SPAN 
      style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">≤</SPAN><FONT 
      size=3><SPAN 
      style="FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">θ</SPAN></FONT><SUB>j</SUB>，即神经元j的输入结果的值小于阀值，则U<SUB>j</SUB>-<FONT 
      size=3><SPAN 
      style="FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">θ</SPAN></FONT><SUB>j</SUB><SPAN 
      style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">≥0</SPAN>，则从二值神经元的计算公式可知：Y<SUB>j</SUB>的值保持为0，或者从1变到0。这说明Y<SUB>j</SUB>的变化<SPAN 
      style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">△</SPAN>Y<SUB>j</SUB>只能是零或负位。这时则有<SPAN 
      style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">△</SPAN>E<SUB>j</SUB>：</P>
      <P><SPAN 
      style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">△</SPAN>E<SUB>j</SUB><SPAN 
      style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">≤0</SPAN></P>
      <P>这也说明Hopfield网络神经元的能量减少。</P>
      <P>上面两点说明了Hopfield网络在权系数矩阵W的对角线元素为0，而且W矩阵元素对称时，Hopfield网络是稳定的。</P>
      <P>Coben和Grossberg在1983年给出了关于Hopfield网络稳定的充分条件，他们指出：</P>
      <P>如果Hopfield网络的权系数矩阵w是一个对称矩阵，并且，对角线元素为0．则这个网络是稳定的。即是说在权系数矩阵W中，如果</P>
      <P>i=j时，W<SUB>ij</SUB>=0</P>
      <P>i<SPAN 
      style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">≠</SPAN>j时，W<SUB>ij</SUB>=W<SUB>ji</SUB></P>
      <P>则Hopfield网络是稳定的。</P>
      <P>应该指出：这只是Hopfield网络稳定的充分条件．而不是必要条件。在实际中有很多稳定的Hopfield网络，但是它们并不满足权系数矩阵w是对称矩阵这一条件。</P>
      <P>上面的分析可知：</P>
      <P>无自反馈的权系数对称Hopfield网络是稳定的网络。它如图1—16，图1—17所示。</P>
      <P align=center><IMG height=376 src="1_3 Hopfield模型.files/4.3.ht1.gif" 
      width=420 border=0></P></TD></TR>
  <TR>
    <TD width="100%" height=10>
      <P align=center>图1-16&nbsp; 对角线权系数为0的对称Hopfield网络</P>
      <P align=center><IMG height=338 src="1_3 Hopfield模型.files/4.3.ht2.gif" 
      width=409 border=0></P></TD></TR>
  <TR>
    <TD width="100%" height=6>
      <P align=center>图1-17&nbsp; 对角线权系数为0的对称网另一图示</P></TD></TR>
  <TR>
    <TD width="100%" height=12>
      <P>Hopfield网络的一个功能是可用于联想记忆，也即是联想存储器。这是人类的智能特点之一。人类的所谓“触景生情”就是见到一些类同过去接触的景物，容易产生对过去情景的回昧和思忆。对于Hopfield网络，用它作联想记忆时，首先通过一个学习训练过程确定网络中的权系数，使所记忆的信息在网络的n维超立方体的某一个顶角的能量最小。当网络的权系数确定之后，只要向网络给出输入向量，这个向量可能是局部数据．即不完全或部分不正确的数据，但是网络仍然产生所记忆的信息的完整输出。1984年Hopfield开发了一种用n维Hopfield网络作联想存储器的结构。在这个网络中，权系数的赋值规则为存储向量的外积存储规则(out 
      product storage prescription)，其原理如下： 
      <P>设有m个样本存储向量x<SUB>1</SUB>，x<SUB>2</SUB>，…，x<SUB>m</SUB> </P>
      <P>X<SUB>1={</SUB>X<SUB>11,</SUB>X<SUB>21,</SUB>...<SUB>,</SUB>X<SUB>m1}</SUB></P>
      <P>X<SUB>2={</SUB>X<SUB>12,</SUB>X<SUB>22,...,</SUB>X<SUB>m2}</SUB></P>
      <P><SUB>......</SUB></P>
      <P>X<SUB>m={</SUB>X<SUB>m1,</SUB>X<SUB>m2,...,</SUB>X<SUB>mm}</SUB></P>
      <P>把这m个样本向量存储人Hopfield网络中，则在网络中第i，j两个节点之间权系数的值为：</P>
      <P><IMG height=62 src="1_3 Hopfield模型.files/4.3.ht3.gif" width=295 
      border=0></P>
      <P>其中：k为样本向量X<SUB>k</SUB>的下标，k＝1，2，…m；</P>
      <P>i，j分别是样本向量X<SUB>k</SUB>的第i，j分量X<SUB>i</SUB>，X<SUB>j</SUB>的下标；i，j＝1，2，…n。</P>
      <P>对联想存储器的联想检索过程如下：</P>
      <P>给定一个向量X。进行联想检索求取在网络中的存储内容。这时，把向量</P>
      <P>X={X<SUB>1</SUB>,X<SUB>2</SUB>,...X<SUB>n</SUB>}</P>
      <P>的各个分量x<SUB>1</SUB>，x<SUB>2</SUB>，…，x<SUB>n</SUB>赋于相对应的节点j，(j＝1，2，…，n)，则节点有相应的初始状态Y<SUB>j</SUB>(0)，则有</P>
      <P>Y<SUB>j</SUB>(0)=X<SUB>j</SUB>，j＝1，2，…，n</P>
      <P>接着，在Hopfield网络中按动力学系统原则进行计算，得</P>
      <P>Y<SUB>j</SUB>(t+1)=f[<SPAN 
      style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">ΣW<SUB>ij</SUB>Y<SUB>j</SUB>(0)-</SPAN><FONT 
      size=3><SPAN 
      style="FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">θ</SPAN></FONT><SUB>j</SUB>]&nbsp; 
      , i,j＝1，2，…，n</P>
      <P>其中，f[·]是非线性函数，可取阶跃函数。</P>
      <P>通过状态不断变化，最后状态会稳定下来．最终的状态是和给定向量x最接近的样本向量。所以，Hopfield网络的最终输出也就是给定向量联想检索结果。这个过程说明，即使给定向量并不完全或部分不正确，也能找到正确的结果。在本质上，它也有滤波功能。</P>
      <P>1．3．2连续Hopfield网络</P>