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2010. 12. 9. 19:00 Footmarks
서강대 수학과 세미나: Neural Network Approximation 
강연: 함남우 교수님 (인천대학교) 

2010-12-09 나무 16:30-17:30 @R1418


함남우 교수님 주전공: 수치해석, 응용수학, 해석학...


> Approximation Theory

- To construct a model for an input/output process
x -> actual process -> f(x)
x -> computed model -> P_f(x)   다항함수

- Error 
Intrinsic Error <= model design
Noise Error <= observation


> Density Problem (접근가능성을 결정)
En(f) ~ inf { || f - p || : p, a subset of Pu }
To decide if En(f) -> 0 as n -> inf
eg. 중간값 정리
n이 얼마나 커야 하는지의 문제


> Complexity Problem
the rate at En(f) -> 0 ( when target fn. f(x) is generally unknown )


* Theory of Best Approximation




ref. G. G. Lorentz, Approximation of Functions


> Brief History of NN
- McCulloch & Pitts, 1943
- Rosenblatt, 1958 "perceptron"
- Minsky & Papert, 1969 "limitation of perceptron"
- Rumelhart, 1986 "hidden layer" (<- 연사님  전공)


> Neural Network (with One Hidden Layer)

x -> Sigma(i=1,...,n) c_i * sigma * (a_i*x + b_i)
a: weight, b: threshold or bias


algebraic polynomial -- approx. --> any continuous fn.


> Complexity Result


* Simultaneous Approximation  (cf. Whitney extension Theorem)

ref. Constructive Function Theory



  


posted by maetel