ผลต่างระหว่างรุ่นของ "Week4 Machine Learning"

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== Reproducing kernel Hilbert spaces ==  
 
== Reproducing kernel Hilbert spaces ==  
  
Let <math>\mathcal{H}</math> be a Hilbert space consisting of functions on <math>\mathcal{X}</math>. A function <math>K: \mathcal{X} \times \mathcal{X} \rightarrow \mathbb{R}</math> is called a reproducing kernel if  
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Let <math>\mathcal{H}</math> be a Hilbert space consisting of functions on <math>\mathcal{X}</math>. A function <math>K: \mathcal{X} \times \mathcal{X} \rightarrow \mathbb{R}</math> is called a reproducing kernel for<math> \mathcal{H}</math> if  
 
* For all y, <math>K_y = K(\cdot, y)</math> belongs to <math>\mathcal{H}</math>  
 
* For all y, <math>K_y = K(\cdot, y)</math> belongs to <math>\mathcal{H}</math>  
 
*'' (Reproducing property):'' For all y, for all <math>f \in \mathcal{H}</math>,  
 
*'' (Reproducing property):'' For all y, for all <math>f \in \mathcal{H}</math>,  
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Note that, in this case, dot-product is defined in a natural way, i.e.  
 
Note that, in this case, dot-product is defined in a natural way, i.e.  
 
:<math> <f,g> = \int_{X} f(x) \bar{g}(x) </math>
 
:<math> <f,g> = \int_{X} f(x) \bar{g}(x) </math>
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== Properties of r.k. Hilbert spaces ==
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* It is known that if a r.k. exists for a given Hilbert space, then it is unique. The proof is two-line
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* (Existence of K):
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== Eigenvector systems ==
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== Example 1: finite domain ==
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We show a motivating example when <math>\mathcal{X}</math> is finite.

รุ่นแก้ไขปัจจุบันเมื่อ 03:26, 20 เมษายน 2550

Let's first review some linear algebra concepts.

Hilbert spaces

Recall some basic definitions:

Def: An inner product is is a bilinear form on a pair of vectors satisfying

  • and <v,v>

Note that every inner product space is a normed linear space with the norm

And with this norm, the inner product space forms a metric.

Def: A metric space is complete if every cauchy sequence converges to an element in the space

Def: A Hilbert space is a complete inner product space

Reproducing kernel Hilbert spaces

Let be a Hilbert space consisting of functions on . A function is called a reproducing kernel for if

  • For all y, belongs to
  • (Reproducing property): For all y, for all ,

Note that, in this case, dot-product is defined in a natural way, i.e.

Properties of r.k. Hilbert spaces

  • It is known that if a r.k. exists for a given Hilbert space, then it is unique. The proof is two-line
  • (Existence of K):

Eigenvector systems

Example 1: finite domain

We show a motivating example when is finite.