Reproducing kernel Hilbert space | Wikipedia audio article
Channel: wikipedia tts
Date: 29/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Reproducing_kernel_Hilbert_space


00:04:48 1 Definition
00:12:26 2 Example
00:19:08 3 Moore–Aronszajn theorem
00:27:39 4 Integral operators and Mercer's theorem
00:37:48 5 Feature maps
00:43:54 6 Properties
00:50:37 7 Common Examples
00:50:48 7.1 Bilinear kernels
00:51:59 7.2 Polynomial kernels
00:52:49 7.3 Radial basis function kernels
00:55:38 7.4 Bergman kernels
00:58:32 8 Extension to vector-valued functions
01:09:20 9 See also



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SUMMARY
=======
In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Roughly speaking, this means that if two functions 
  
    
      
        f
      
    
    {\displaystyle f}
   and 
  
    
      
        g
      
    
    {\displaystyle g}
   in the RKHS are close in norm, i.e., 
  
    
      
        ‖
        f
        −
        g
        ‖
      
    
    {\displaystyle \|f-g\|}
   is small, then 
  
    
      
        f
      
    
    {\displaystyle f}
   and 
  
    
      
        g
      
    
    {\displaystyle g}
   are also pointwise close, i.e., 
  
    
      
        
          |
        
        f
        (
        x
        )
        −
        g
        (
        x
        )
        
          |
        
      
    
    {\displaystyle |f(x)-g(x)|}
   is small for all 
  
    
      
        x
      
    
    {\displaystyle x}
  . The reverse need not be true.
It is not entirely straightforward to construct a Hilbert space of functions which is not an RKHS. Note that L2 spaces are not  Hilbert spaces of functions (and hence not RKHSs), but rather Hilbert spaces of equivalence classes of functions (for example, the functions 
  
    
      
        f
      
    
    {\displaystyle f}
   and 
  
    
      
        g
      
    
    {\displaystyle g}
   defined by 
  
    
      
        f
        (
        x
        )
        =
        0
      
    
    {\displaystyle f(x)=0}
   and 
  
    
      
        g
        (
        x
        )
        =
        
          1
          
            
              Q
            
          
        
      
    
    {\displaystyle g(x)=1_{\mathbb {Q} }}
   are equivalent in L2). However, there are RKHSs in which the norm is an L2-norm, such as the space of band-limited functions (see the example below).
An RKHS is associated with a kernel that reproduces every function in the space in the sense that for any 
  
    
      
        x
      
    
    {\displaystyle x}
   in the set on which the functions are defined, "evaluation at 
  
    
      
        x
      
    
    {\displaystyle x}
  " can be performed by taking an inner product with a function determined by the kernel. Such a reproducing kernel exists if and only if every evaluation functional is continuous.
The reproducing kernel was first introduced in the 1907 work of Stanisław Zaremba concerning boundary value problems for harmonic and biharmonic functions.  James Mercer simultaneously examined functions which satisfy the reproducing property in the theory of integral equations. The idea of the reproducing kernel remained untouched for nearly twenty years until it appeared in the dissertations of Gábor Szegő, Stefan Bergman, and Salomon Bochner.  The subject was eventually systematically developed in the early 1950s by Nachman Aronszajn and Stefan Bergman.These spaces have wide applications, including complex analysis, harmonic analysis, and quantum mechanics.  Reproducing kernel Hilbert spaces are particularly important in the field of statistical learning theory because of the celebrated representer theorem which states that every function in an RKHS that minimises an empirical risk function can be written as  ...