Week4 Machine Learning - ประวัติรุ่นแก้ไข

Parinya: /* Properties of r.k. Hilbert spaces */

2007-04-20T03:26:47Z

Properties of r.k. Hilbert spaces

2007-04-20T03:25:02Z

Reproducing kernel Hilbert spaces

2007-04-20T03:19:14Z

Reproducing kernel Hilbert spaces

2007-04-20T03:17:22Z

2007-04-20T00:04:39Z

หน้าใหม่

Let's first review some linear algebra concepts.

@@ แถว 30: / แถว 30: @@
 * 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 ==

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 03:25, 20 เมษายน 2550
แถว 25:		แถว 25:
	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>
		+
		+	== 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
		+
		+	*
		+
		+	== Eigenvector systems ==
		+
		+
		+	== Example 1: finite domain ==
		+	We show a motivating example when <math>\mathcal{X}</math> is finite.

@@ แถว 19: / แถว 19: @@
 == 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
+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>
 *'' (Reproducing property):'' For all y, for all <math>f \in \mathcal{H}</math>,

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 03:17, 20 เมษายน 2550
แถว 1:		แถว 1:
	Let's first review some linear algebra concepts.		Let's first review some linear algebra concepts.
		+
		+	== Hilbert spaces ==
		+	Recall some basic definitions:
		+
		+	'''Def:''' An inner product is<math> <\cdot, \cdot></math> is a bilinear form on a pair of vectors satisfying
		+	* <math><v, v> \ge 0</math> and <v,v> <math>= 0 \iff v=0</math>
		+	* <math><u,v> = \bar{<v,u>}</math>
		+	*<math> <ru + sv, w> = r<u,w> + s<v,w></math>
		+
		+	Note that every inner product space is a normed linear space with the norm
		+	:<math> \|\|v\|\| = \sqrt{<v, v>}</math>
		+	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 <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
		+	* 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>,
		+	: <math>f(y) = <f, K_y></math>
		+	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>