Joint Approximation Diagonalization of Eigen-matrices

<h2 id="algorithm">Algorithm</h2>
<p>Let 
  
    
      
        
          X
        
        =
        (
        
          x
          
            i
            j
          
        
        )
        ∈
        
          
            R
          
          
            m
            ×
            n
          
        
      
    
    {\displaystyle \mathbf {X} =(x_{ij})\in \mathbb {R} ^{m\times n}}
  
 denote an observed data matrix whose 
  
    
      
        n
      
    
    {\displaystyle n}
  
 columns correspond to observations of 
  
    
      
        m
      
    
    {\displaystyle m}
  
-variate mixed vectors. It is assumed that 
  
    
      
        
          X
        
      
    
    {\displaystyle \mathbf {X} }
  
 is <i>prewhitened</i>, that is, its rows have a sample mean equaling zero and a sample covariance is the 
  
    
      
        m
        ×
        m
      
    
    {\displaystyle m\times m}
  
 dimensional identity matrix, that is,
</p>

1
            n
          
        
        
          ∑
          
            j
            =
            1
          
          
            n
          
        
        
          x
          
            i
            j
          
        
        =
        0
        
        
          and
        
        
        
          
            1
            n
          
        
        
          X
        
        
          
            
              X
            
          
          
            ′
          
        
        =
        
          
            I
          
          
            m
          
        
      
    
    {\displaystyle {\frac {1}{n}}\sum _{j=1}^{n}x_{ij}=0\quad {\text{and}}\quad {\frac {1}{n}}\mathbf {X} {\mathbf {X} }^{\prime }=\mathbf {I} _{m}}
  
. 
<p>Applying JADE to 
  
    
      
        
          X
        
      
    
    {\displaystyle \mathbf {X} }
  
 entails
</p>
<ol><li>computing <i>fourth-order cumulants</i> of 
  
    
      
        
          X
        
      
    
    {\displaystyle \mathbf {X} }
  
 and then</li>
<li>optimizing a <i>contrast function</i> to obtain a 
  
    
      
        m
        ×
        m
      
    
    {\displaystyle m\times m}
  
 rotation matrix 
  
    
      
        O
      
    
    {\displaystyle O}
  
</li></ol>
<p>to estimate the source components given by the rows of the 
  
    
      
        m
        ×
        n
      
    
    {\displaystyle m\times n}
  
 dimensional matrix 
  
    
      
        
          Z
        
        :=
        
          
            O
          
          
            −
            1
          
        
        
          X
        
      
    
    {\displaystyle \mathbf {Z} :=\mathbf {O} ^{-1}\mathbf {X} }
  
.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Cardoso, Jean-François; Souloumiac, Antoine (1993). "Blind beamforming for non-Gaussian signals". IEE Proceedings F - Radar and Signal Processing. 140 (6): 362–370. CiteSeerX 10.1.1.8.5684. doi:10.1049/ip-f-2.1993.0054. <a href="/wiki/CiteSeerX_(identifier)" target="_blank">/wiki/CiteSeerX_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Cardoso, Jean-François (Jan 1999). "High-order contrasts for independent component analysis". Neural Computation. 11 (1): 157–192. CiteSeerX 10.1.1.308.8611. doi:10.1162/089976699300016863. <a href="/wiki/CiteSeerX_(identifier)" target="_blank">/wiki/CiteSeerX_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Joint Approximation Diagonalization of Eigen-matrices open-in-new

Joint Approximation Diagonalization of Eigen-matrices