Homework 4 - 1 2 3 4 g i ( x) = min x - m l =1, 2 , ,li l i...

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Unformatted text preview: 1 2 3 4 g i ( x) = min x - m l =1, 2 , ,li l i g j ( x) = min g j ( x) j =1, 2 , ,c 5 6 g i ( x) = min x - x , k = 1,2, k k i , Ni ,c g j ( x) = min g i ( x), i = 1,2, i x j 7 8 P ( m | x) = max[ P ( i | x)] i x P (e | x) = 1 - P ( m | x) P = P (e | x) p ( x)dx 9 Nxx'x PN (e | x ) = PN (e | x, x ) p ( x | x )dx ' ' ' N PN (e) = PN (e | x ) p ( x ) dx P = lim PN (e) N 10 p(x)>0Sx Ps = ' p( x )dx ' x s ' x1 , x2 , P ( x1 , x2 , , x N , N S , x N ) = (1 - Ps ) ' ' N N lim p( x | x) = ( x - x) 11 PN (e | x, x ) = 1 - P ( = i , = i | x, x ) ' ' ' i =1 c c = 1 - P ( i | x) P ( i | x ) ' i =1 x `x N lim PN (e | x, x ) = 1 - P ( i | x) ' 2 i =1 12 c N lim PN (e | x) = lim PN (e | x, x ) P ( x | x)dx ' ' N ' = [1 - P ( i | x)] ( x - x)dx 2 ' i =1 c ' = 1 - P ( i | x) 2 i =1 13 c P = lim PN (e) = lim PN (e | x) p( x)dx N N = lim PN (e | x) p ( x)dx N = [1 - P ( i | x)] p( x)dx 2 i =1 14 c P 1 P ( m P | x) = 1 P = [1 - 1] p ( x ) dx = 0 P = P (e | x) p ( x)dx = [1 - P ( m | x )] p ( x ) dx = 0 15 2 P(i | x) = 1 c c 1 2 c -1 P = [1 - ( ) ] p ( x)dx = c i =1 c 1 c -1 P = (1 - ) p ( x)dx = c c 16 P i =1 c 2 ( i | x) = P ( m | x) + P ( i | x) 2 2 im c P ( i | x ) = A, i = 1,2, ' c '2 2 , c; i m P i =1 ( i | x) = min P ( i | x) i =1 17 P ( ' im i | x) = (c - 1) P ( i | x) = P (e | x) ' P (e | x ) ,i m ' P ( i | x) = c - 1 1 - P (e | x), i = m 18 P i =1 c 2 ( i | x) = P ( m | x) + P ( i | x) 2 2 im P (e | x ) [1 - P (e | x)] + 2 i m (c - 1) 2 2 c 2 = 1 - 2 P (e | x ) + P (e | x ) c -1 c 2 1 - P ( i | x) 2 P (e | x) - P (e | x ) c -1 i =1 19 2 c E[ P (e | x)] = P Var [ P (e | x ) = [ P (e | x ) - P ] p ( x ) dx 2 = [ P (e | x ) p ( x ) - 2 P (e | x ) P p ( x ) + P p ( x )]dx 2 2 = P 2 (e | x ) p ( x ) dx - P 2 0 P 2 P (e | x) p ( x) dx 2 20 P = [1 - P ( i | x)] p ( x)dx 2 i =1 c c 2 [ 2 P (e | x ) - P (e | x)] p ( x)dx c -1 c 2 = 2 P (e | x ) p ( x )dx - P (e | x) p( x)dx c -1 c c 2 2P - P = P (2 - P ) c -1 c -1 21 c P P P (2 - P ) c -1 22 23 c P P P (2 - P ) c -1 P P 2P 24 25 K g i ( x) = ki , i = 1,2, ,c ki : ki g j ( x ) = max k i x j i k 26 27 c P P P (2 - P ) c -1 28 29 xN 1B + r < D ( x, M ) p p 2 + D ( x i , M p ) < D ( x , M p ) B 30 step1 step2 ' ' x x D ( x, x ) KxK 31 Kernel Nearest Neighbor ... 32 Spacefilling Curves (1) 33 Spacefilling Curves (2) Mappings between S and the Real Line 34 Problems of the algorithm Intolerable discontinuities in the mapping 35 Properties of mapping based on Spacefilling Curve Pairs of points which are close in the Real Line are guaranteed to come from nearby points in the original space. However, the converse is not true 36 Extended General Spacefilling Heuristic (1) Applying the spacefilling mapping r times with distinct submodels, r can be set equal to d (dimension) 37 Single Spacefilling Curves and multiple representations of data Several synthetic points around the original ones are the representatives.[5] By drawn from the distribution By perturbating 38 References (1) [1] Bartholdi, J.J.; Platzman, L.K. (1988). "Heuristics Based on Spacefilling Curves for Combinatorial Problems in Euclidean Space", Management Science, 34.pp.291-305 [2] Skubalska, E.; Krzyzak, A. (1996). "Fast k-NN Classification Rule Using Metric on Space-Filling Curves", Proceedings of ICPR-96, pp.121-125. [3] Perez, J.C.; Vidal, E. (1998). "An approximate Nearest Neighbours Search Algorithm based on the Extended General Spacefilling Curves Heuristic", SPR 98. Sydney. 39 References (2) [4] Perez, J.C.; Vidal, E. (1998). "The Extended General Spacefilling Curves Heuristic", Proceedings of ICPR-98 [5] "Approximate Nearest Neighbor Search using a Single Space-filling Curve and Multiple Representations of the Data Points", to be appeared in ICPR-2006. 40 1s Minkowski M ( xk , xl ) = [ xkj - xlj ] j =1 d s 1/ s s=1 c ( xk , xl ) = xkj - xlj j =1 41 d 2 s=2 M ( xk , xl ) = [ xkj - xlj ] j =1 d s 1/ s E ( xk , xl ) = [ ( xkj - xlj ) ] j =1 d 2 1/ 2 42 3Chebychev T ( xk , xl ) = max xkj - xlj j 4Q Q ( xk , xl ) = ( xk - xl ) Q( xk - xl ) T 43 2 ? 44 2 ? 45 46 Tangent distance in visual patterns Patrice Y. Simard, Yann A. Le Cun, John S. Denker, Bernard Victorri, "Transformation Invariance in Pattern Recognition -- Tangent Distance and Tangent Propagation". In Neural Networks: Tricks of the Trade, G. B. Orr and K-R Muller (Eds), Chapter 12, Springer, 1998. 47 Tangent distance in visual patterns 48 Tangent distance in visual patterns 49 Overview of TD 50 ...
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This note was uploaded on 06/02/2010 for the course ELECTRONIC PC2010S taught by Professor Zhangchangshui during the Spring '10 term at Tsinghua University.

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