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Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/fall08/ann/

Artificial Neural Networks ECE.09.454/ECE.09.560 Fall 2008. Lecture 11 November 17, 2008. Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/fall08/ann/. Plan. ANN Pre-processing Feature Extraction Approximation Theory Universal approximation

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Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/fall08/ann/

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  1. Artificial Neural NetworksECE.09.454/ECE.09.560Fall 2008 Lecture 11November 17, 2008 Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/fall08/ann/

  2. Plan • ANN Pre-processing • Feature Extraction • Approximation Theory • Universal approximation • Final Project Discussion

  3. Feature Extraction Objective: • Increase information content • Decrease vector length • Parametric invariance • Invariance by structure • Invariance by training • Invariance by transformation

  4. Approximation Theory:Distance Measures • Supremum Norm • Infimum Norm • Mean Squared Norm

  5. Recall: Metric Space • Reflexivity • Positivity • Symmetry • Triangle Inequality

  6. K Approximation Theory: Terminology • Compactness • Closure F

  7. E min M f u0 Approximation Theory: Terminology • Best Approximation • Existence Set E ALL f min M u0

  8. F e g f Approximation Theory: Terminology • Denseness

  9. Fundamental Problem E • Theorem 1: Every compact set is an existence set (Cheney) • Theorem 2: Every existence set is a closed set (Braess) min M g ?

  10. F e g f x x1 x2 1 f(x1) f(x2) Stone-Weierstrass Theorem • Identity • Separability • Algebraic Closure F af+bg

  11. Final Project Discussion

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